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Grating interferometry has been successfully adapted at standard X-ray tubes and is a promising candidate for a broad use of phase-contrast imaging in medical diagnostics or industrial testing. The achievable image quality using this technique is mainly dependent on the interferometer performance with the interferometric visibility as crucial parameter. The presented study deals with experimental investigations of the spectral dependence of the visibility in order to understand the interaction between the single contributing energies. Especially for the choice which type of setup has to be preferred using a polychromatic source, this knowledge is highly relevant. Our results affirm previous findings from theoretical investigations but also show that measurements of the spectral contributions to the visibility are necessary to fully characterize and optimize a grating interferometer and cannot be replaced by only relying on simulated data up to now.
© 2014 Optical Society of America
1. Introduction
To overcome the limited soft-tissue contrast in conventional absorption-based imaging, several new X-ray phase-contrast imaging methods have been developed [1]. While most of these techniques are restricted to highly brilliant X-ray sources like synchrotron radiation sources, grating-based phase-contrast imaging [2, 3] has been successfully adapted to work with conventional X-ray sources [4] and has become a promising candidate for medical diagnostics and industrial testing [5–15]. Maximum achievable visibility is the most important performance parameter of such a grating interferometer as it determines the quality of the exploited interference pattern. A high visibility is necessary for a reliable extraction of the phase signal and a good contrast to noise ratio [16].
Achieving a high visibility with a polychromatic spectrum requires a deep understanding of the complex interactions between several setup parameters and the contribution of each single wavelength of the spectrum to the overall visibility. The influence of polychromaticity on the visibility has been investigated in several theoretical studies [16–19], but only few experimental analyses exist. It has recently been shown that energy-resolved visibility measurements are a valuable tool for characterization of grating interferometers [20] and that the performance of a Talbot-Lau inteferometer can be extensively increased by adjusting the geometry of the setup, filter material and detector threshold regarding achieved visibility and photon flux [21–23]. Here we extend previous studies by directly comparing simulated data with experimental results and show the necessity of an experimental analysis for reliable setup characterization. We reveal the contribution of single energies of a polychromatic spectrum to the overall performance of a grating interferometer and demonstrate the importance of energy-resolved visibility measurements.
Our study particularly considers the differences between (so-called) π-shifting and π/2-shifting grating interferometer setups and their performance depending on spectral width and inter-grating distance. These findings confirm theoretical considerations [18] which have previously predicted considerable differences between these setup types for large inter-grating distances.
2. Materials and methods
2.1. Grating-based phase-contrast
Grating-based phase-contrast imaging (gbPCI) is a differential phase-contrast technique. This method relies on the detection of small refraction angles of the incoming X-ray beam caused by differences in the phase-shift induced by the object to the incoming wave. Grating interferometry allows detecting those small refraction angles even with a detector pixel size much larger than the spatial deviation of the X-ray beam.
Figure 1(a) shows a schematic setup of a grating interferometer. A phase grating (G1) is used to induce a periodical phase shift (typically π or π/2) to the incoming wavefront. Behind this grating a periodically self-repeating interference pattern occurs. This interference pattern and its relative lateral position can be analyzed using an absorption grating (G2). In case of a conventional X-ray tube as photon source, an additional absorption grating (the source grating (G0)) is installed to yield an array of virtual slit sources to enhance the spatial coherence. The analyzer grating (G2) is moved perpendicularly to the propagation direction of the beam (phase-stepping approach). As the period of the absorption grating is chosen to match the interference pattern period, a periodic intensity modulation I(x,y) is recorded at the detector plane (Fig. 1(b)).
Fig. 1 (a): Schematic principle of a grating interferometer. A source grating (G0, not shown in the scheme) has to be installed in case of photon sources with a low coherence. The phase grating (G1) induces a periodic phase shift to the incoming wavefront resulting in an interference pattern behind the grating (shown in Fig. 2). To analyze the relative position of the interference pattern an absorption grating is stepped perpendicular to the propagation direction to record a stepping curve. (b): Stepping curves for a reference scan (blue solid line) and a sample scan (green dashed line). From the stepping curves the mean intensity a0, the amplitude a1 and the displacement φ can be determined.
The lateral deviation of the stepping curve against a reference curve recorded without a sample allows calculation of the differential phase shift induced by the sample. Visibility is a main performance parameter for the accuracy of this measurement. With Imax as highest and Imin as lowest intensity value of the stepping curve the visibility is defined as:
2.2. Differences between π- and π/2-setup
For the following simulations and considerations the phase grating is assumed to be perfectly homogeneous with strictly rectangular grating bars. The duty cycle, defined as the ratio of grating-bar width to grating period, is set to 0.5. The Talbot-carpets in Fig. 2 illustrate the simulated intensity patterns for a monochromatic beam behind a π-shifting and a π/2-shifting grating along the propagation axis. Periodic intensity patterns can be observed at different fractions of the full Talbot distance dT for the two different grating-types [24, 25]:
Here, n is an odd integer. The patterns have a lateral period of p1/2 in case of a π-shifting phase grating and a period of p1 for a π/2-shifting grating. At the fractional Talbot distances where n is an even integer, no intensity variation is observable. The observable intensity patterns at odd fractional Talbot distances behind a π-shifting grating are all identical, whereas the intensity patterns at the 1st and 3rd fractional Talbot distance behind a π/2-shifting grating are shifted by half a period.Fig. 2 Monochromatic Talbot carpets for a π-shifting (a) and a π/2-shifting (b) phase grating. The fractional Talbot distances are n · dT/16 for a π-shifting grating and n · dT/4 for a π/2-shifting grating. At even fractional Talbot distances the incoming wavefront repeats itself. At odd fractional Talbot distances a periodic intensity pattern arises. The lateral period of this pattern is p/2 for the π-shifting grating. For the π/2-shifting grating the period is p and the patterns at the 1st and 3rd fractional Talbot distance are shifted half a period against each other.
For an interferometer operated at a monochromatic source (e.g. synchrotron radiation) the height of the phase grating bars is chosen to cause either a phase shift of π or π/2 to the incoming wave. The period of the analyzer grating p2 is adapted to the phase grating period p1 and the phase shift Δϕ induced to the incoming wave
where M is the magnifying factor, and depends on setup geometry [26].For an interferometer equipped with a source of a broad spectrum a certain energy, the design energy, has to be defined to choose the phase-grating height. In case of such a broad spectrum, a phase grating inducing a phase shift of π/2 at the design energy at the same time induces a phase shift of π to x-rays with half of this energy. This allows the analyzer grating to be adapted either to a π/2- or to a π-shifting grating. For simulations where the induced phase shift covers a range of 0 to 2π, the setup type cannot be defined by the phase grating and its induced phase shift, but by the combination of phase grating period and analyzer grating period. Further on we will differentiate between the setup-types in the following way:
2.3. Simulations
For theoretical considerations we simulated different so-called visibility carpets (Figs. 3 and 4). For each carpet we simulated a set of 500 Talbot carpets with different phase shifts in the range of 0 and 2π (examples shown in Fig. 2). For this we assumed an absolute coherent planar wavefront with its phase being periodically modified by the desired phase shift. The modified wavefront E0 was propagated to the full Talbot length by 500 separate steps (z) using the Fresnel transform [27]
To retrieve the visibility values, a Fourier analysis of the corresponding intensity values I = E2 was performed. The visibility V then is given by the Fourier coefficients: To distinguish between the visibility values for a π-setup and a π/2-setup the corresponding Fourier coefficient an has to be chosen. For a simulation with the number of grating periods g, the Fourier coefficient an is given by:Fig. 3 Simulated visibility carpet for a π/2-setup with a phase grating duty cycle of 0.5 (a) and the corresponding line plots for a constant phase shift of π/2 (b) and the 1st fractional Talbot distance as fixed distance (c) (both marked by white dashed lines in the visibility carpet). Negative values correspond to an opposite orientation of the intensity pattern.
Fig. 4 Simulated visibility carpet for a π-setup with a phase grating duty cycle of 0.5 (a) and the corresponding line plots for a constant phase shift of π (b) and the 3rd fractional Talbot distance as fixed distance (c) (both marked by white dashed lines in the visibility carpet).
Please note that the visibility carpets were simulated ignoring influences other than the phase shift and propagation distance to become universally applicable. This makes the carpets insufficient for the calculation of absolute visibility values. Taking into account influences such as absorption by the grating bars, the visibility carpet becomes energy dependent and is only valid for one specific setup configuration. Nevertheless a lot of influences like grating bar errors are very hard to handle within the simulations and can only be approximated. Therefore all energy-dependent influences and grating imperfections have not been considered when simulating the carpets. Also, all shown visibility carpets will be scaled from 0 to 1 in case of a π-setup and from −1 to 1 in case of a π/2-setup.
2.4. Equipment and measurements
The experimental part of this study (chapter 4) consists of measurements with two different detector systems. An imaging detector was used to analyze the overall performance of the interferometer whereas a single-pixel spectrometer was used to analyze the energy-dependent contributions to the overall visibility.
As imaging detector we used a Dectris Pilatus II 100K equipped with a 1000μm thick silicon sensor. It is a photon counting detector with a pixel size of 172 x 172 mm2. For all measurements the threshold level was set to 16 keV to reduce effects resulting from charge sharing. The measurements with this detector consisted of 11 steps and were analyzed using a standard fast Fourier transform algorithm.
For our energy resolved measurements (chapter 4.2) we used an Amptek X-123 spectrometer. This detector consists of only one Si-PIN. The effective sensor area was reduced to 1mm2 by an additional Pb-shielding to avoid pile-up effects from a too high photon flux. For performance analysis with this detector we recorded the complete spectrum at each stepping position resulting in a stepping curve for each energy bin (detector channel C). As source a rotating anode (Enraf Nonius FR-591) equipped with a molybdenum target was used and operated at an acceleration Voltage of U = 40 kV.
The interferometer for this study consisted of three gratings arranged in a symmetric geometry. Due to the geometry and the magnifying factor of M = 2, all three gratings had the same period for the π-setup configuration. For the π/2-setup, the phase grating had half the period of the absorption gratings. All gratings were produced by the Institute for Microstructure Technology (IMT) of the Karlsruhe Institute of Technology (KIT) using LIGA-processing [28, 29]. The grating structures for source and analyzer grating were made of gold, the structures for the phase gratings were made of nickel.
For that part of this study dealing with the differences between a π- and a π/2-setup the phase grating period was always 2.4 μm whereas the absorption gratings’ period was 2.4 μm with a gold structure height of 50 μm (G0) and 80 μm (G2) in case of a π-setup and 4.8 μm with a gold structure height of 50 μm (G0) and 90 μm (G2) in case of a π/2-setup. The phase gratings for those setups all had a duty cycle of 0.5. Table 1 lists the phase grating heights used for this study.
Table 1. List of nickel made phase-gratings used for the comparison of the two different setup types for long distances.
The part of this study dealing with the comparison of imaging detector and spectrometer results was performed using a set of gratings with a period of 5.4 μm (π-setup). The nickel phase-grating had a structure height of 8 μm, giving a π-shift to a photon energy of 23 keV. The absorption gratings had a gold structure height of 65 μm (G0) and 70 μm (G2). This large grating periods were chosen due to the more reliable and homogeneous structures, which is essential for a direct comparison of the results from the spectrometer with its limited field of view to a large area detector. Visibility measurements have been performed with inter-grating distances of 40 cm, 60 cm, 67 cm, 70 cm, 80 cm, 90 cm and 100 cm.
3. Simulated visibility carpets
Working with a polychromatic spectrum, each wavelength of the beam creates a different intensity pattern at the analyzer grating position with an energy-specific visibility. Therefore, the observed intensity pattern at the detector plane becomes a superposition of the interference patterns from each single energy of the spectrum. Equally, the measured visibility becomes a combination of all the energy-specific visibilities.
To get an idea of the influence of the energy spectrum on the overall visibility, simulations were performed to create visibility carpets that provide a quick overview of the contribution of each energy on the interferometer performance. The visibility carpets shown in Figs. 3 and 4 were obtained by analyzing simulated monochromatic Talbot carpets along the full Talbot-distance for an induced phase shift between 0 to 2π to the incoming wave for both setup types. To get a uniform length scale we define the reduced propagation distance
as the ratio of the propagation distance d to the full Talbot-distance dT. This reduced propagation distance and the phase shift were used to scale the axes for the presented visibility carpets.Knowing the shape of the carpets, the change of visibility as a function of the energy and inter-grating distance can be estimated. As shown in Fig. 2, a shift of the lateral intensity pattern can be recognized in the Talbot carpet for a π/2-setup between the first and the third fractional Talbot distance. In the simulated visibility carpets, this inversion is taken into account with a negative sign of the visibility value, due to the fact that the stepping curves obtained at these positions will be shifted by half a period relative to each other.
3.1. Interpretation of simulated visibility carpets
For the correct interpretation of the simulated visibility carpets, we consider the monochromatic case first. The derived concepts can then be expanded to polychromatic spectra. The parameters of the simulation are the propagation distance behind the phase grating (η, Eq. (11)) and the phase shift for a given energy. For different propagation distances with a fixed phase shift, all visibility values lie on a horizontal line through the visibility carpet. This can be derived from Fig. 2 for the two presented phase shifts and an analyzer grating moving along the propagation axis. The strong maxima in the visibility carpets occur when the analyzer grating passes the position in the Talbot carpet where the highest contrast is observed. Various intermediate states between the two shown carpet types are observed dependent on the induced phase shift. Generally, these carpets show less contrast between intensity minima and maxima than the carpets for a π/2- or π-shifting phase grating. For a fixed distance the visibility values for different phase shifts lie on the corresponding vertical line on the visibility carpets. The line plots in Figs. 3 and 4 illustrate this for a constant phase shift of π/2 resp. π (Figs. 3(b) and 4(b)) and for a variable phase shift at the 1st resp. 3rd fractional Talbot distance (Figs. 3(c) and 3(c)).
The induced phase shift by the phase grating with height h is given by Δϕ(E) = 2πδh/λ (λ ∝ 1/E), with δ ∝ 1/E2 as the refractive index decrement. Thus the given phase shift is proportional to the grating height and inverse proportional to the photon energy. To change the induced phase shift in an experimental setup the phase grating height or the photon energy of the incoming beam can be altered. In the latter case, the Talbot distance changes as well and results in a different reduced propagation distance η. In the case of a polychromatic spectrum, the interferometer has to be defined for one specific combination of design energy ED and corresponding phase shift ΔϕD as well as corresponding reduced propagation distance ηD. For such a setup, the relation of phase shift and reduced propagation distance to any energy is determined by:
Both of these values scale linearly with energy. This implies that for a specific setup with design energy ED, all visibility values for any energy lie on a line crossing the origin of the visibility carpet. The induced phase shift as well as the reduced propagation distance is lower for higher energies. This means that the visibility values are closer to the origin for higher energies. Combining Eqs. (12) and (13) reveals the connection between Δϕ(E) and η(E) as: Therefore the slope of such a line is given by the ratio ΔϕD/ηD. For this examination the same design energy was assumed for the phase grating and the inter-grating distance. In general, the slope can be determined if exactly one pair of Δϕ(Ẽ) and η(Ẽ) is known: Here Ẽ is any energy and not restricted to the design energy ED of the interferomter.Figure 5 gives an example how the visibility distribution can be estimated from a visibility carpet. The example is valid for any π/2-setup that fulfills the condition Δϕ(E)/η(E) = 2π/5dT. For instance when the interferometer has a propagation distance of 5/4 dT for the design energy and induces a phase shift of π/2 to this energy.
Fig. 5 For a fixed setup all visibility values for any energy are located on a line through the origin. The example shown as white line in the carpet is valid for a setup built up at the 5th fractional Talbot distance and a phase shift of π/2 for the design energy ED. The plotted line shows the corresponding visibility values as a function of ED/E.
The most striking difference between the visibility-carpets of a π-setup and a π/2-setup takes effect for a broad spectrum and a long propagation distance. Here, the line through the carpet, representing the spectrum, can cross multiple maxima. In case of the π/2-setup those maxima can have a different sign, but not for the π-setup. The negative sign indicates that the interference patterns for the phase-shift and propagation distance pairings in these regions are shifted by half a period against those in the regions with a positive sign. Together with the analyzer grating period adapted to a π/2-setup this results in recorded phase-stepping curves shifted by half a period. In case of a π-setup (and the corresponding analyzer-grating period) the observed stepping curves are not shifted relative to each other. Therefore the visibility carpet of a π-setup does not show areas with different signs.
3.2. Spectrum weighted visibility
To predict the mean visibility of the complete spectrum, the simplest approach is to weight the visibility for each energy V(E) with its associated flux and to integrate over the whole spectrum:
The weighting factors w(E) are given by the intensity of the corresponding energy and the normalization condition ∫E w(E) = 1.From the different signs present in the visibility-carpet resulting from the lateral shift of the interference patterns in different parts of the spectrum for a π/2-setup, according to [20] we concluded that the contribution of these differently signed areas lead to annihilation and therefore an additional weighting factor of ±1 has to be considered.
4. Experimental verification
To verify the predictions based on simulated data, in particular the occurrence of annihilating visibilities, a series of measurements was performed.
4.1. Imaging detector results
The results shown in Fig. 6 demonstrate the rapid drop of the overall visibility for a π/2-shift setup due to the mutual annihilation of single visibilities, whereas the overall visibility of a π-shift setup does not show this effect. Additionally, the results show that a slight variation of the phase grating height has only little effect on the mean visibility when using a polychromatic beam.
Fig. 6 Comparison of π/2-setup (a) and a π-setup (b) for the same inter-grating distances and different phase gratings. The phase grating period was 2.4 μm for both setup types. The periods of the source and analyzer grating were 4.8 μm in case of the π/2-setup and 2.4 μm in case of the π-setup.. A monotonous decrease in the measured visibility is detectable for the π/2-setup whereas the π-setup visibility remains constant.
4.2. Energy-resolved results
Based on the recorded spectra we could analyze a stepping curve for each detector channel (energy bin). This gives us the visibility as a function of energy (Fig. 7). The mean visibility for the complete spectrum can then be calculated by either using the overall counts from each stepping position or by summing up the energy dependent visibility values weighted by the corresponding flux. Derived from Eq. (16) the mean visibility is then a sum over the detector channels C:
Fig. 7 Comparison of predicted visibilities from simulations (marked in the visibility carpet, black dashed lines in the visibility plots) and experimental energy-resolved measurements (blue solid lines in the visibility plots) of a π/2-setup configuration (compare section 2.2) using 4.8μm period absorption gratings and a 2.4μm period phase grating. The experimental setups were built up at the third fractional Talbot-distance with a phase grating inducing a π/2-shift to the design energy of 29 keV (a) and at the first fractional Talbot distance of 23 keV with a phase-grating inducing a phase-shift of π to this design energy and a π/2-shift to 46 keV. (b). The simulated visibility values in the comparison plots (a) and (b) are arbitrarily scaled for an easy comparison of the curves shape. The comparison shows that a rough approximation of the contribution from the single energies to the overall performance is possible. The insufficiency of the simulated data for an exact characterization of a setup becomes apparent as the inflection points and extreme values are not positioned at the same energies. This is attributed to the non-perfectly absorbing and inhomogeneous structures of the gold gratings which is not considered in the simulation, but also to possible deviations in the actual phase grating height and the duty cycles of all three gratings.
The energy-resolved measurements with different gratings were evaluated to prove the necessity of this sign (Table 2). The overall visibility calculated by analyzing the absolute photon counts of the detector was compared to the visibility calculated based on individual visibilities using Eq. (17) with and without taking the sign into account. As predicted, the overall visibility calculated from individual visibilities only match the value based on the absolute photon count when the single visibilities are sign-weighted. Without sign-weighting, the resulting values are significantly too high. This comparison lets conclude that single visibilities of high value are present but annihilate each other mutually, so that only a very low overall visibility is measurable and the interferometer will not be operable in this configuration. Having a look onto single stepping curves from different channels, the phenomenon of annihilating visibilities through shifted interference patterns becomes obvious. Figure 8 shows two stepping curves for different energies obtained with a π/2-setup at the 5th fractional Talbot distance for a design energy of 26 keV (phase grating A3). The combination of these two energies shows an immense decrease in visibility due to the shift of the intensity modulation.
Table 2. Calculated mean visibilities, with and without taking into account the intensity pattern orientation, compared to the analyzed stepping curve resulting from the overall counts. The necessity of the sign to determine the mean visibility becomes obvious. The huge difference of the calculated values imply that single visibilities of higher values are present, but annihilated by each other resulting in a not properly functioning interferometer.
Fig. 8 Comparison of two stepping curves from two different energies within the same measurement using the pi/2-setup configuration. The red dotted line gives the intensity modulation for 19 keV and the blue dashed line gives the intensity modulation for 27 keV. The black solid line gives the resulting stepping curve from both energies if not analyzed independently. The effect of annihilating visibilities through the shifted intensity patterns is clearly recognizable.
Figure 7 shows the energy distribution of the visibility from two different π/2-setups and the corresponding visibility carpet. The setups were positioned at the third fractional Talbot-distance with a phase grating inducing a π/2-shift to the design energy of 29 keV (Fig. 7(a)) and at the first fractional Talbot-distance with a phase grating inducing a π-shift to the design energy of 23 keV (Fig. 7(b)). As the simulated visibilities are not taking into account any effects from the absorption gratings, especially the non-perfect absorption, only the qualitative visibility distribution can be considered. The values of the simulations were rescaled to match approximately the experimental data for an easy comparison. By comparing simulated and experimental visibility distribution, it is shown that a rough approximation of the contribution from the single energies to the overall performance is possible. Nevertheless the differences in the inflection points and extreme values show that energy-resolved measurements are necessary for exact setup characterization due to possible deviations from the expected grating specifications, e.g. the phase grating’s height or duty cycle. The latter parameter is especially critical as demonstrated in the following and last example.
4.3. Comparison of imaging detector and spectrometer results
To verify the presumption that the overall visibility can be calculated by integrating the single energy-resolved recorded visibilities over the whole spectrum, we performed a series of measurements with a π-setup at different inter-grating distances between 40 cm and 100 cm for both detectors. For each distance we calculated the overall visibility from the spectrometer data using Eq. (17) and compared it to the visibility obtained with the imaging detector. The recorded spectra were recalculated to correct for the different absorption efficiency of the detectors due to the different sensor thicknesses. The results (Fig. 9) clearly show that the overall visibility can be derived from the energy-dependent visibilities. This reveals that the performance of an interferometer can be analyzed very precisely with a spectrometer and remains valid despite of a detector change, if it is taken care of the different detector efficiency.
Fig. 9 Comparison of the overall visibility determined from energy-resolved visibility analysis (red dashed line) and the mean visibility recorded with an imaging detector (blue solid line) using a π-setup consistent of three 5.4μm period gratings. The results affirm the theory to use spectrum weighted visibilities to determine the overall visibility.
Although we used a π-setup for this comparison the sign for the weighted visibilities was necessary. This seems to be in conflict with the previous results. The reason for this is that the used phase-grating did not have a duty cycle of 0.5 but between 0.6 and 0.7. The duty cycle of the phase grating has strong influence on the shape of the visibility carpets and also causes areas where the sign of the visibility changes. An example of a visibility carpet for a π-setup with a phase grating duty cycle of 0.7 is shown in Fig. 10. This carpet shows massive differences compared to the carpet shown in Fig. 4 with a phase grating duty cycle of 0.5. The two most remarkable characteristics are the presence of areas with a negative sign and the position and number of the absolute visibility maxima. Whereas the eight maxima were located at the odd fractional Talbot distances and a phase shift of π for a duty cycle of 0.5, now only 4 maxima are present and located at the 2nd, 6th, 10th, and 14th fractional Talbot distance with a phase shift of π. This phenomenon again emphasizes the importance of energy-resolved visibility analysis for an accurate interferometer characterization. Beyond this, the huge influence of the phase-grating duty cycle requires further investigations on this topic.
Fig. 10 Visibility carpet for a π-setup with a phase grating duty cycle of 0.7. A huge impact from the duty cycle is notable. Compared to a carpet for a phase grating duty cycle of 0.5, the number and location of the visibility maxima have changed. The white dashed line indicates that strong visibility maxima are present at even fractional Talbot-distances. The carpet also shows areas where the sign of the visibility changes. Due to the limited values of negative visibilities, this carped is scaled differently then the carpets before.
5. Conclusions and discussion
The presented results demonstrate how different energies of a polychromatic spectrum contribute to the overall performance of a grating-based interferometer. The introduced visibility carpets are a useful tool for a rough performance estimation for a polychromatic setup. The significant discrepancies that can occur between simulated and measured visibilities reveal the great value of energy-resolved visibility measurements to characterize an existing system. Especially for a setup with many uncertainties like exact grating heights and duty cycles, energy-resolved visibility-measurements are necessary for exact performance characterization. The results from energy-resolved measurements also can be easily adapted to be valid for use with a different detector. Our experiments also confirmed the previous prediction that, for a given phase grating and a broad spectrum, the inter-grating distance can be chosen independently from the design energy of the phase gratings [17].
We have shown that the performance of an inteferometer operated at a polychromatic source is highly dependent on its geometry and the effective spectrum. Our results show that a large propagation distance between the gratings can negatively affect the visibility for a π/2-setup when the spectrum extends over several peaks in the visibility carpet. In this case the visibility values have to be weighted with a sign according to the orientation of the observed intensity pattern. As annihilating single visibilities were not observed for the π-setup (when applying phase gratings with duty cylces of 0.5), a π-setup should be chosen for interferometers aiming at a high sensitivity through a large inter-grating distance. This is confirmed by recent theoretical investigations [20]. But we also showed that these findings are not valid for a phase-grating duty-cycle that differs too much from 0.5. Our energy-resolved measurements revealed that the phase-grating duty-cycle can have an immense effect on the performance of an interferometer and especially on the ideal geometry. The effect that the preferable inter-grating distances are strongly dependent on the phase-grating duty cycle requires further investigations of this phenomenon.
Acknowledgments
We acknowledge financial support through the DFG Cluster of Excellence Munich-Centre for Advanced Photonics (MAP), the DFG Gottfried Wilhelm Leibniz program and the European Research Council (ERC, FP7, StG 240142). This work was carried out with the support of the Karlsruhe Nano Micro Facility (KNMF, www.kit.edu/knmf), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT).
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