Abstract: Normal incidence Talbot-Lau interferometers in x-ray applications have the drawbacks of low fringe visibility with polychromatic sources when the wave propagation distance is increased to achieve higher phase sensitivity, and when fabrication limits the attainable grating density. In contrast, reflective gratings illuminated at grazing angles have dramatically higher effective densities than their physical values. However, new designs are needed for far field interferometers using grazing angle geometry with incoherent light sources. We show that, with the appropriate design and choice of reflective phase gratings, there exist pairs of interfering pathways of exactly equal lengths independent of the incoming beam’s incidence angle and wavelength. With a visible light grazing angle Mach-Zehnder interferometer, we show the conditions for achieving near ideal fringe visibility and demonstrate both absolute and differential phase-contrast imaging. We also describe the design parameters of an x-ray interferometer and key factors for its implementation.
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#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27481
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1. Introduction
Normal incidence, grating-based Talbot-Lau interferometers have a wide range of applications in matter-wave interference [1] and x-ray phase-contrast imaging [2]. However, a basic limitation of Talbot and Talbot-Lau interferometers is that they produce interference fringes at specific distances from the grating, and these distances are dependent on the wavelength of the light. To achieve higher phase contrast sensitivity, both the density of the gratings and the order of the Talbot fringes need to be increased together. This in turn demands increasingly narrower linewidths of the light source. In x-ray applications, laboratory sources usually have broad and continuous spectra, and most of the spectra will not satisfy the Talbot condition at a given distance, leading to low fringe visibility [3,4]. For example, Engelhardt and associates showed with simulations that the maximum fringe visibility of a polychromatic x-ray Talbot interferometer is approximately 30% if the wave propagations distance reaches 4 times the Talbot distance. Since the phase contrast sensitivity scales with (grating period)/(propagation distance), higher sensitivity prefers wave propagation out to the far-field where chromatic dispersion becomes severe, and fringe visibility is further reduced. Therefore, Talbot interferometers are less suitable for polychromatic light in the far-field range. Another significant challenge in x-ray applications is the fabrication of dense gratings with periods below several microns, which requires structures of extremely high aspect ratios [5].
Alternatively, it has been demonstrated with both visible light and matter waves that in the far field range, grating-based Mach-Zehnder interferometers can still produce strong interference fringes with polychromatic and spatially incoherent sources, if the layout meet certain symmetry conditions [6,7]. The symmetric layout produces pairs of mutually coherent diffracted beams at the image plane, which interfere to give high contrast fringes. A pre-requisite of this type of interferometer is that the grating density should be high enough such that on the image plane, the intersections of different orders of diffracted beams are separated. In this case the interference zones of coherent beam pairs do not overlap with other beams, and high fringe contrast is maintained. However, for x-ray applications in the normal incidence geometry, the required grating density can be prohibitively high.
Here we propose a solution to the above problem in the form of grazing-angle Mach-Zehnder interferometers using reflective phase gratings (Fig. 1). A distinct advantage of the grazing-angle geometry is that the effective grating period is much smaller than the physical period. We demonstrate with a visible light analog that the design works with polychromatic, spatially incoherent light sources. We learn from the visible light analog the necessary conditions for maximizing the visibility of the interference fringes, and apply these conditions to the design of an x-ray interferometer. X-ray reflective gratings rely on total internal reflection where the incidence angle is shallower than the critical angle of the surface material [8–10]. Filevich et al. realized a soft x-ray reflective grating Mach-Zehnder interferometer with a monochromatic parallel beam [9]. Their interferometer works over long path lengths
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27482
due to the coherence of the source. Stutman et al. demonstrated that reflective intensity gratings illuminated by a parallel polychromatic beam produce shadow lines in the near field, which can be observed with a second analyzer grating similar to transmission Talbot interferometers [10]. However, new designs are needed to produce interference effects beyond the near-field range using uncollimated, polychromatic sources.
2. Interferometer Design
A planar reflective grating can be represented as a planar transmission grating of zero thickness, where the virtual transmitted wave is the mirror image of the actual reflected wave (Fig. 2). In this paper a phase-shift reflective grating refers to a grating of interleaved strips of equal width and reflectivity, but the odd strips introduce a phase delay relative to the even strips.
Fig. 1. Setup of the dual phase grating grazing angle Mach-Zehnder interferometer (not to scale). White light from the flashlight is first reflected by a phase grating G1 of 1.0 mm period, then by another phase grating G2 of 0.5 mm period. The interference fringes at the image plane are recorded by a camera. The grating surfaces and the image plane are all parallel to the XY plane, while the image plane and grating G1 are coplanar. The grating lines are all parallel to the Y axis. Due to the grazing angle incidence, the effective grating period of G2 is approximately Psinθ. The two aperture slits are used to control the spread of the grazing angle and the width of the illuminated area on the grating G1.
For the grazing incidence geometry shown in Fig. 1, the Fresnel diffraction approximation is no longer valid, and thus the analysis of fractional Talbot-Lau effects which work for normal incidence interferometers no longer applies. However, when certain symmetry conditions are satisfied as illustrated in Fig. 2, there are always pairs of light paths of exactly equal lengths that coherently interfere with each other at the image plane, regardless of the incident beam angle or wavelengths. The grazing angle symmetric interferometer includes two π phase-shift gratings of periods 2P and P, respectively [11]. The transmission functions of the two gratings are
1
odd integers
2
odd integers
exp[ ],
2exp[ ],
m
m
n
n
T B i mxP
T C i nxP
(1)
where Bm and Cn are the coefficients of the Fourier series comprising the grating modulations. To obtain the intensity pattern on the image plane from a polychromatic, spatially incoherent source, we treat it as the sum of collimated monochromatic beams that are mutually uncorrelated. Then, the overall intensity is the sum of the intensities from individual beams. For a given collimated beam of amplitude A0, grazing angle θ and wave vector k, the various diffracted beams that reach the image plane are
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27483
0
,
2( ) exp{ [ cos )] }exp[ ( , )],m n Z
m n
m nA x A B C i k x i m n
P P
(2)
where φZ is the phase factor that is dependent on h (the Z separation of the gratings):
2 2 2 22( , ) { ( cos ) ( cos ) } .Z
m m nm n k k k k h
P P P
(3)
We denote each diffracted beam in Eq. (2) as Am,n. The critical factor of this design is the fact that
( 1,1) ( 1, 1).Z Zm m (4)
Equation (4) means that the phase accumulated along the (m-1, 1) and (m + 1, 1) pathways are exactly equal regardless of the incidence angle or wavelength. Thus the pair of beams (m-1,1) and (m + 1,-1) will coherently interfere at the image plane to give intensity fringes of period P:
1, 1 1,1 1, 1 1,1 1, 1 1,1 1,1 1, 1 1, 1
2
0 1 1 1 1 1 1 1 1
2 2
1 1 1 1
* * * *
2 2[ * * exp( ) * * exp( )
| | | | ].
m m m m m m m m m m
m m m m
m m
I A A A A A A A A
A B C B C i x B C B C i xP P
B C B C
(5)
This interference pattern is invariant with both the beam’s incidence angle and wavelength. A geometric interpretation of this symmetry is that these two pathways form a perfect parallelogram and have exactly equal path lengths (Fig. 2).
Fig. 2. Schematics of light diffraction in the dual-phase-grating grazing angle interferometer. The grazing angle θ shown here is much larger than the actual values for clear illustration. Inset: the reflective gratings are viewed as equivalent transmission gratings, such that the transmitted waves are exact mirror images of the actual reflected waves. The first grating G1 splits the incident beam into diffracted beams of orders m. Each is further split by the second
grating according to diffraction orders n. It can be seen that the (m = 1,n = 1) and (m = 1,n =
1) pair of beams each have the exact same path length together forms a parallelogram. Equal pathway length and parallelogram formation is independent of the incidence angle and wavelength. The two beams interfere at the image plane to give fringes of 100% visibility, which is defined as (Imax-Imin)/(Imax + Imin). The positions of the interference zones (Im,n) are dependent on the incidence angle. High fringe visibility is maintained if the I-1,1 zone does not substantially overlap with neighboring zones. For direct derivation of wave propagation see Eq. (1) – (5).
The most interesting beam pair is (1,1) and (1,-1) for two reasons. First, with the π phase-shift gratings the highest Fourier coefficients are B1, B-1 and C1, C-1, and thus these pathways
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27484
have the highest amplitudes. Second, these two beams have equal amplitudes because B1 = B-1 = C1 = C-1 = 2/π. Thus the interference pattern between them is simplified:
2
1,1 0 4
32 2[1 cos( )],I A x
P
(6)
This pattern should give the highest possible visibility. Fringe visibility is defined as (Imax-Imin)/(Imax + Imin) and the maximum value is 100%.
Fig. 3. Illustration of the overlap of interference zones of different orders with a spread of incidence angles. The various interference zones have phase-locked fringes but different levels of visibility. The central I-1,1 zone has the highest fringe visibility of 100%, and the peripheral zones have lower fringe contrast. A finite source can be viewed as producing beams of a range of incidence angles. They are incoherent to each other such that their intensity patterns on the image plane simply add to each other. Here two sets of diffracted beams from incidence angles θ1 and θ2 are shown in red and maroon. The central I-1,1 zone of the red beams overlap the I-3,-1 zone of the maroon beams, and vice versa.
In practice, a finite source produces mutually incoherent parallel beams of a continuous range of incidence angles. An important and interesting fact revealed by Eq. (6) is that the positions of the peaks and valleys of the fringes are independent of the incidence angle, i.e. the fringes from different incidence angles are phased locked to each other. This is the reason why such symmetric Mach-Zehnder interferometers work with extended sources. However, the range of incidence angles needs to be controlled in order to achieve high fringe visibility. This is because the locations of the interference zones of various beam pairs are dependent on the incidence angle (Fig. 3). Thus the overall image is the superposition of arrays of interference zones whose positions are offset from each other according to their incidence angles, while their fringes are all phase locked to each other. Generally the central I-1,1 zone has the highest fringe visibility. Therefore, the overlap of the I-1,1 zone with the neighboring I-
3,-1 and I3,1 zones tends to reduce the fringe visibility. We can avoid the overlap by limiting the width of the incidence beam and the range of the incidence angle. In our setup this was realized by placing two aperture slits in front of the light source (Fig. 1). The range of incidence angle is given by
2 2 1cos( ) / ( );S SS L L (7)
the width of the illuminated area on the first grating is given by
1 1 1 2 1cos( )(1 / ) / / .S S SL S L L L (8)
Equations (7) and (8) were used to determine the slit sizes for a given set of incidence angle spread and light footprint on the first grating.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27485
3. Experimental Methods
3.1 Fabrication and testing of the phase gratings
The phase gratings were fully covered with a reflective aluminum layer. To cause a π phase difference between reflection from odd and even strips, their surface heights were offset by an abrupt step of
/ (4sin ),d (9)
where λ = 525 nm is the central wavelength of white light, and θ = 0.13 (radians) is the grazing angle, yielding d = 1.0 µm.
Fig. 4. The fabrication process of the phase gratings. An initial 1.0 µm of aluminum was deposited on a silica optical flat substrate. A grid pattern of photoresist was laid on the Al layer by photolithography. After etching away the unprotected Al, the photoresist was removed. An over layer of 0.1 µm Al was finally deposited to make the surface fully reflective. The height step d introduces the desired phase shift for a given incidence angle.
To realize this design, the first step was the deposition of an aluminum layer of 1.0 µm thickness onto a silica optical flat substrate by electron-beam physical vapor deposition. Then, the aluminum layer was patterned with grating lines using standard photolithographic techniques, and the aluminum was etched from the exposed areas using Transene Aluminum Etchant. After the photoresist was removed, a second 0.1 µm aluminum layer was deposited over the grid pattern making half of the area a 1 µm step below the other half. The second layer aluminum deposition over the grid pattern was the last step in producing the final grating (this fabrication process is outline in Fig. 4). Phase gratings of 0.5 mm and 1.0 mm periods were used. It is worth mentioning that due to the shallow incidence angle, the d steps cast shadows of 6.7 µm length. Since the shadow affects 2.6% and 1.3% of the lowered Al step for the 0.5 mm and 1.0 mm period gratings respectively, the shadow’s effects are negligible.
Each phase grating was tested by shining a collimated green beam on its surface at the targeted grazing angle, which produced the expected diffraction peaks (Fig. 5). Theoretically the central zeroth order peak should be null. The actual intensity of the zeroth order peak was approximately 10% of that of the m = + 1/-1 peaks.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27486
Fig. 5. Measured intensity distribution of the diffraction pattern of a single phase grating. The incidence light was a collimated green beam. Ideally the zeroth order peak should have zero
intensity. The measured area under the zeroth order peak is 10% of the + 1 and 1 peaks.
3.2 Testing the interference fringe visibility of the grazing angle grating interferometer
We built a visible light dual phase grating grazing angle interferometer to test the design concept. The layout is shown in Fig. 1. The light source was a white LED flashlight of 265 lumens brightness and 16 mm exit window. A flat screen of fine-grain photo printing paper was placed at the image plane, and the image on the screen was photographed with a digital camera. A close-up lens was fitted to the camera to allow better light collection. The gratings were fixed in mirror mounts with micrometer adjustments of the Z position and tilt around Y and X axes. Experiments were performed in a dark room. To correct for residual ambient light, a dark field image was acquired by blocking the path between the grating G2 and the screen. This dark field was then subtracted from the data shots.
Fringe visibility was measured for two settings of the source apertures. In the narrow setting, S1 = 2.5 mm and S2 = 0.4 mm. An 8 mm wide section of the first grating was illuminated and the spread of the grazing angle was 5 milliradians. In the wide setting, S1 = 10.0 mm and S2 = 0.5 mm. A 20 mm wide section of the first grating was illuminated and the divergence of the grazing angle was 20 milliradians.
3.3 Absolute phase shift and differential phase-contrast imaging
The dual phase grating interferometer was tested for both absolute phase shift and differential phase-contrast imaging. Absolute phase measurement is possible since the two interfering beams of the I-1,1 pattern are separated in the X direction when emerging from the second grating (Fig. 2). By both placing the imaged object in one of the two beams and using the other beam as reference, the resulting shift of the interference fringes relative to those without the object yields an absolute measure of the phase shift introduced by the object (Fig. 6):
Light path in object
( ', ') 1[ ( , , ) 1] ,
X x yn x y z dx
P
(10)
where the coordinates (x’, y’) are on the image screen and correspond to coordinates (y, z) in the object by ray tracing along the object beam, P is the period of the fringes on the image screen, n(x, y, z) is the refractive index distribution in the object, and the integration is along light rays of the object beam and over the thickness of the object.
For differential phase contrast [2] the object was placed near the image plane where the two beams overlap (Fig. 6). In this case the shift of the fringes represents the gradient of phase shift by the object in the Z direction (Fig. 6):
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27487
Light path in object
( ', ') 2( , , ) ,
X x y Dn x y z dx
P P z
(11)
where D is the distance between the object and the center of the I-1,1 interference zone on the image screen.
Fig. 6. Placement of the imaged object for absolute phase delay and differential phase contrast
imaging. The two diffracted beams drawn in orange and green are the (1, 1) and (1, 1) diffractions from an incident beam of grazing angle θ. They interfere coherently at the image plane to produce fringes. In the absolute phase measurement setting, the maroon beam is used as the object beam and the green beam the reference beam. The sample is placed in object beam and as near as possible to the G2 grating so as not to intersect the reference beam. The resulting fringe shifts represent the absolute phase delay in the object relative to air. In the differential phase contrast setting, the object is placed in the intersecting zone of the two beams near the image plane, where the fringe shifts represent the local difference in phase delay between the two beams.
Experimentally, the sample for absolute phase measurement is a patch of oil sandwiched between two microscopy glass slides. The sample was positioned near the edge of the second grating, such that the oil patch intersected one of the two interfering beams. An absolute phase image was derived from the interferogram by harmonic analysis in the Fourier domain [12–14]. A calibration phase map without the oil patch was subtracted from the result to remove background phase caused by the imperfect flatness of the image screen. The oil patch needed to be thin enough such that the chromatic dispersion in Eq. (10) did not blur the fringes. On the other hand, differential phase contrast imaging can work with thicker samples. We demonstrated it with a clear acrylic layer painted on a glass slide. The sample was placed near the image screen and approximately 20 cm from the center of the second grating.
4. Results
Figure 7 shows fringe visibility measurements from the dual phase grating interferometer. With 5 milliradians divergence of the grazing angle, there was no substantial overlap between the central interference zone (I-1,1) and the neighboring zones on the image plane. This setting attained the highest fringe visibility of 78%. In the wide setting of 20 milliradian spread of the grazing angle, the interference zones overlapped and merged. Interference fringes could be seen over the entire illuminated area on the image plane, and the fringe visibility was relatively uniform. At the center of the illuminated area (Fig. 7d) the fringe visibility was 13%.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27488
Fig. 7. Interferograms with narrow and wide divergence of the incidence angle. (a) At 5 milliradian spread of the incidence angles, the interference zones of different beam pairs are separated. (b) Magnified view of the central zone shows high contrast fringes. The “waviness” of the fringes comes from the imperfect flatness of the image screen. (c) Intensity plot of the central fringes show a fringe visibility of 78%. (d) At 20 milliradian spread of the incidence angles, the interference zones merge into an area of relatively uniform brightness and fringe visibility. (e) Magnified view of the central region and (f) intensity plot show decreased fringe visibility relative to the narrow angle setting.
Figure 8 summarizes results from both absolute phase and differential phase-contrast imaging experiments. In the absolute phase image, there was a global and relatively uniform shift of the interference fringes over the entire area of the oil patch (relative to the fringe pattern without the oil patch), which indicates the absolute phase delay of the light traversing the oil film. Fourier harmonic analysis showed that the phase delay was approximately 0.6π on average. Based on the the index of refraction of vegetable oil (n = 1.5), the average thickness of the oil film was estimated as wavelength*(phase shift)/2π/(n-1). This amounted to 0.35 microns.
In the differential phase contrast image of the clear acrylic layer, distortions of the fringe lines can be seen (Fig. 8c). According to Eq. (11) they indicate the presence of spatial gradients of the phase delay across the layer, and suggest that the thickness of the layer is not uniform. A corroborative result is shown in Fig. 8d. Here a grid pattern was photographed under ambient room lighting with the glass slide placed between the grid and the camera. The uneven refractive effects of the acrylic layer caused visible bending of the grid lines in that area.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27489
Fig. 8. (a) A patch of oil between two glass slides was placed in one of the two interfering beams to produce this interferogram. The fringes within the oil patch appear to be uniformly shifted to the right relative to the fringes outside the patch. (b) Absolute phase shift map obtained through Fourier fringe analysis quantifies the amount of fringe shift in the oil film, and shows the corresponding phase delay of the transmitted light. (c) A differential phase-contrast image of a clear acrylic layer on a glass slide shows fringe distortions due to gradients of phase shifts, which come from thickness variations of the layer. (d) Another way to show the uneven thickness of the acrylic layer is this photo of a grid pattern taken under normal room lighting. The glass slide with the acrylic layer was positioned at 2 cm in front of the grid. The acrylic layer is highlighted by the red line. Light refraction due to the uneven thickness causes the visible bending of the grid lines, which corroborates the inteferogram of (c).
5. Design of the x-ray version of the grazing angle interferometer
We performed design calculations for an x-ray interferometer of the dual phase grating configuration. The gratings were assumed to have gold reflective surfaces and have the same fabrication process as in Fig. 4. The spectrum of the x-ray source was assumed to be between 25 keV and 40 keV, which is the appropriate range for small animal imaging. In this energy range, the minimum critical grazing angle for total internal reflection by gold is 1.98 milliradians. We assumed a central grazing angle of 1.8 milliradians in our calculation. The period of the two phase gratings were set at 200 µm and 100 µm respectively. The separation between gratings in the X direction was kept at 25 cm. The calculated design parameters are listed in Table 1.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27490
Table 1. Design parameters of the x-ray grazing angle interferometer
Central grazing angle (milliradians)
Allowed grazing angle spread (milliradians)
Width of 1st grating
(mm)
Width of 2nd
grating (mm)
Grating Z separation h
(mm)
Grating height step δ (nm)
Effective grating
period (nm)
1.8 0.16 27 90 0.45 5.3 180
Given the spread of the grazing angle θ and x-ray energy, the actual phase shift from the height step in the gratings is given by the expression 4πdsinθ/λ. This value is not expected to be a constant but to vary over a range as Fig. 9 shows, of the calculated range of phase shifts (from 0.73π to 1.28π). Since the visibility of the interference fringes is dependent on the phase shift, the calculated visibility levels ranged between 60% and 100% as shown in Fig. 9. The visibility is mostly dependent on the x-ray energy.
Fig. 9. Calculated grating phase shift and visibility of the interference fringes for a range of grazing angles and x-ray energy. The left graph is a contour plot superimposed on a gray scale representation of the phase shift induced by the height step in the grating. The right graph shows the corresponding fringe visibility. The fringe visibility is expected to be between 60% and 100% for x-ray energy between 25 keV and 40 keV.
The main challenge in fabricating such x-ray gratings is the extreme flatness that is required of the grating substrate surface. In order to limit the RMS phase errors due to surface height fluctuations to less than π/4, the RMS fluctuation of the substrate surface needs to be below λ/(22.6sinθ), or 1 nm for the above design. It is worth noting that the effective grating periods are more than an order of magnitude smaller than what is currently possible with normal incidence gratings.
6. Conclusion and discussion
We showed that grazing-angle Mach-Zehnder interferometers based on reflective phase gratings can produce interference fringes of near ideal visibility with white, un-collimated light sources. This method potentially overcomes the limitations in phase-contrast sensitivity of normal incidence Talbot-Lau interferometers, namely the need for monochromatic sources and difficulties in fabricating ultrahigh density gratings. The reflective gratings are straightforward to make provided that the substrates have sufficient surface flatness, and modest grating periods of tens of microns can produce effective grating periods of nanometers. It is clear that by choosing the appropriate incidence angle spread and beam width based on the system layout, the various interference zones on the image plane can be separated and the highest fringe visibility can be achieved.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27491
Grating-based Mach-Zehnder interferometers in the normal incidence geometry have been implemented with both visible light and atomic waves [6,7]. We extended the idea to the grazing-angle geometry and showed that with certain choices of gratings and symmetric layout, there exist interference pathways that are of exactly the same lengths without any approximation and are independent of incidence angle and wavelengths. Using this type of symmetric Mach-Zehnder interferometer with grazing angle geometry to obtain equal-distance interference pathways and separation of different diffraction orders is the two enabling factors of our method.
We showed that in addition to differential phase-contrast imaging, the dual phase grating interferometer can be used for absolute phase measurements if the object is small enough to fit into one of the two interfering beams. Previously this has been realized with single-crystal monochromatic x-ray interferometers, which produced high contrast images in soft tissue samples [15]. The current method potentially allows direct phase retrieval of imaging and diffraction experiments with laboratory x-ray sources.
The 78% fringe visibility of our visible light interferometer was less than the theoretical 100% and probably due to two instrumental effects: stray light from imperfections in the gratings and the residual zeroth-order diffraction peak (Fig. 5) contributed a background intensity to the interference fringes; the slight roughness of the surface of the image screen cast grainy textures on the image at grazing illumination, which also reduced the contrast of the fringes. At the wide aperture setting, the fringe visibility can be calculated from Eq. (2) and (5) assuming that all diffracted beams completely overlap. The result was approximately 10% and near the experimentally observed 13%.
The main limitation of this method is that due to the grazing-incidence geometry, the effective field-of-view (FOV) in the Z direction is small, limited by the grating size. In our experiments the Z field was just over 2 mm. In the x-ray design, 50 mm grating separated by 25 cm distance is expected to give Z FOV of just over 100 µm. While larger gratings yield proportionally larger Z FOV, the practical limit for x-ray is below 1.0mm. Such a fan beam geometry means that larger samples need to be scanned across the Z direction. The procedure of vertical scanning is similar to many diffraction-enhanced imaging (DEI) studies using single-crystal analyzers [16,17], where the crystal size and flatness also limit the fan beam height. Although DEI has achieved Z FOV of 10mm [18], an advantage of the current approach is that it is not limited to accept only monochromatic and highly collimated x-rays as is the case in DEI with single-crystal analyzers.
The primary challenge in implementing the x-ray version is the surface flatness of the grating substrates. Surface flatness includes long length scale variation (or warping of the surface) and short length scale fluctuations (or roughness). Commercially available high grade silica optical flats have roughness below 5Å, but the long range shape of the surface often fluctuates more than 5 nm. However, using deformable x-ray mirrors, mirror flatness with fluctuation of less than 1nm have been realized [19]. With such flat substrates it should be possible to implement the grazing angle Mach-Zehnder interferometer for both x-ray and matter wave applications.
Acknowledgements
The gratings were fabricated at the Nanofab Facility of the National Institute of Standards and Technology, Gaithersburg, Maryland, with the help of Chester Knurek and Dr. Gerard Henein.
#137617 - $15.00 USD Received 2 Nov 2010; revised 1 Dec 2010; accepted 9 Dec 2010; published 14 Dec 2010(C) 2010 OSA 20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27492
