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We demonstrate a technique that enables lensless holographic imaging with extended reference structures, using ultra-broadband radiation sources for illumination. We show that this ’two-pulse imaging’ approach works with one- and two-dimensional HERALDO reference structures, and demonstrate that the obtained spectrally resolved data can be used to improve the signal-to-noise ratio in the final image. Intensity stitching of multiple exposures is applied to increase the detected dynamic range, leading to an improved image reconstruction. Furthermore, we show that a combination of holography and iterative phase retrieval can be used to obtain high-quality images quickly and reliably, by using the HERALDO reconstruction as the initial support constraint in the iterative phase retrieval algorithm. A signal-to-noise improvement of two orders of magnitude is achieved compared to the basic HERALDO result.
© 2014 Optical Society of America
1. Introduction
Lensless imaging is a powerful approach to high-resolution imaging in situations where imaging optics are not readily available. This is a situation typically encountered for short wavelength radiation such as extreme-ultraviolet or X-rays [1], or when optical components would be too large or costly [2, 3]. In lensless imaging, the imaging optics are essentially replaced by computer algorithms, and the ’microscope’ consists of only a light source and an image sensor. The main requirement for image reconstruction is that the phase of the recorded diffraction pattern can either be measured directly, or retrieved from the measured diffraction intensity through numerical means. Numerical phase retrieval algorithms have been developed and used successfully in coherent diffractive imaging (CDI) [4–6]. One important ingredient is that the imaging conditions are set up such that they fulfill some support requirement [4–8], or that multiple measurements are performed under different propagation conditions [9, 10]. Direct phase measurements can be performed in a lensless geometry by adding a reference scatterer close to the sample [11–13]. This reference scatterer provides a known reference wave which interferes with the light diffracted from the sample. The resulting far-field interference pattern is an off-axis hologram, and encodes the phase difference between the reference and object fields. A 2D Fourier transform of this hologram directly yields both amplitude and phase of the diffracted object field.
A major requirement of any lensless imaging method is that the light source is coherent (both spatially and temporally), as decoherence directly results in a decreased visibility of the diffraction pattern, although methods have been developed to handle some degree of partial coherence [14–16]. This has mostly limited the application of lensless X-ray imaging to large facilities such as synchrotrons and X-ray free electron lasers (XFELs), but recent breakthroughs in high-harmonic generation (HHG) have led to the development of fully coherent table-top soft-X-ray sources [17, 18]. While HHG sources exhibit high spatial coherence, their temporal coherence is very low. This property is intrinsic to the HHG process, which occurs on attosecond timescales and generates broadband radiation. In practice this means that lensless imaging with such a broadband source will result in a superposition of diffraction patterns at different wavelengths. While spectral filtering before imaging is possible, it results in a highly inefficient use of the available photon flux. We have recently developed a two-pulse lensless imaging method, which enables spectrally resolved lensless imaging with ultra-broadband light sources [19]. This approach combines lensless imaging with methods from Fourier-transform spectroscopy, and allows efficient use of the full spectrum of ultra-broadband sources for lensless imaging, which we demonstrated with both visible light and XUV radiation.
In the present paper we analyze the performance of two-pulse imaging in the far-field (Fraunhofer diffraction) using an octave-spanning supercontinuum light source, and demonstrate the possibility to improve the contrast of the final image by averaging multiple spectrally resolved diffraction patterns. Furthermore, we extend the method to Fourier-transform holography with extended reference structures (HERALDO), providing a flexible method to measure the phase of the diffraction pattern. We demonstrate the combination of two-pulse imaging and HERALDO using both one-dimensional (1D) and two-dimensional (2D) reference structures. In addition, we show that the combination of HERALDO and iterative phase retrieval algorithms provides a robust and relatively fast way to achieve image reconstruction from a diffraction pattern. This approach, which uses the HERALDO result as a robust support constraint for iterative phase retrieval, combines the flexibility and reliability of HERALDO with the high contrast and efficiency of iterative phase retrieval, thereby optimally exploiting the advantages of both methods. Finally, we highlight the importance of dynamic range for the quality of the images reconstructed from a HERALDO measurement.
2. Principles
2.1. Holography with extended reference structures: HERALDO
In lensless imaging, rather than using a lens to create an image on a CCD, the intensity of the light diffracted on the sample is captured directly in the far field. An image can then be obtained through numerical back-propagation of the recorded field, provided that both its intensity and phase are known. Direct phase measurements can be performed by lensless Fourier transform holography, in which the phase is detected by interfering the diffraction pattern with a known reference wave. In many experiments this reference wave is generated by a known scatterer close to the sample, where the simplest realization is a pinhole in transmission, generating a spherical wave.
Interference between the object wave o(x, y) and the known reference wave r(x, y) provides a phase measurement. The superposition e(x, y) of both light fields at the sample plane propagates to the recording plane as E(u, v). When the light is recorded in the far field of the sample, the Fraunhofer approximation applies and so the propagation from the object plane to the recording plane can be described by a Fourier transformation. To propagate back the inverse Fourier transformation ℱ−1 can be applied. However, because we measure only the intensity I of the light field, the light field at the sample plane is not reconstructed directly:
where the cross-correlation function of o and r is defined as: where * denotes the complex conjugate. Notice that if r is chosen as a delta function, o ⊗ r and their complex conjugate reduces to o and o* and so the object can be retrieved directly. However, manufacturing a delta function or a pinhole next to the object is challenging, while creating an extended reference structure like a line segment or a trapezoid structure provides much more flexibility [20].A robust method to reconstruct images from holograms created with extended references is HERALDO, a differential operator based technique, which was developed by Podorov et al. [21] for square references and generalized to arbitrary references by Guizar-Sicairos et al. [22]. The key point of this method is the fact that a sharp feature in the reference object can be reduced to a delta function by applying differential operators. A 1D example is a line segment, which can be described as a rect function (rect(x) = 1 for |x| < 1/2 else 0). When applying a differential operator this reduces to two delta functions:
For higher-dimensional reference structures such as squares or triangles, multiple differential operators should be applied to reduce a corner to a delta function.The main advantage of HERALDO over iterative phase retrieval approaches is that it provides a direct phase measurement, and therefore does not suffer from complications related to iterative phase retrieval algorithms such as convergence and stagnation issues. Furthermore, HERALDO references have less manufacturing limitations than conventional pinhole holography, and require less extensive calculations and noise reduction compared to deconvolution-based methods [23–25]. HERALDO has been experimentally demonstrated with visible light [20], but also with X-rays from a synchrotron source [26] and with single pulses from a soft X-ray HHG setup [27]. It can also be combined with polarization dependent contrast techniques, such as X-ray magnetic circular dichroism [28].
2.2. Lensless imaging with broadband light: two-pulse imaging
As the diffraction angle of radiation from a sample is wavelength dependent, coherent diffractive imaging methods run into problems when the source spectrum is not monochromatic. Figure 1 presents a numerical simulation that shows the influence of the source spectrum on a diffraction pattern. A sample structure is depicted in Fig. 1(a), and a monochromatic diffraction pattern of this sample is displayed in Fig. 1(b). Since broadband diffraction patterns can be viewed as a superposition of monochromatic diffraction patterns, they will exhibit a structure that is smeared out in the radial direction. An example of such a broadband diffraction pattern is shown in Fig 1(c), in which this effect is clearly visible. Such smearing washes out the fine detail in the diffraction pattern, and limits the achievable resolution or even inhibits image reconstruction.
Fig. 1 (a) Simulated sample of a star shape placed beside a triangular HERALDO reference. (b) Simulation of a monochromatic far-field diffraction pattern of (a) at a wavelength of 800 nm. (c) Simulation of a broadband (400 nm wide spectrum, centered at a wavelength of 600 nm) far-field diffraction pattern of Fig. 1(a). While the global structure is similar to Fig. 1(b), any radial structure is hardly visible due to the strong spectral smearing that is present with such a broadband source.
We recently introduced ’two-pulse imaging’ as a new approach to acquire spectrally resolved diffraction patterns using ultra-broadband sources [19]. This method employs two coherent pulses with a tunable time delay T, as shown in Fig. 2(a), and with them a series of diffraction patterns is recorded as a function of T. The resulting signal at each camera pixel then contains an interference pattern as a function of T, which encodes the spectrum of the light that has diffracted onto that pixel (Figs. 2(b)–2(d). By Fourier transforming this interference pattern, the spectrum can be retrieved in a manner analogous to Fourier-transform spectroscopy [19,29,30]. This approach results in a collection of quasi-monochromatic diffraction patterns that spans the full source spectrum, with a spectral resolution limited only by the scanned time delay. Each of these quasi-monochromatic diffraction patterns can then be used as a starting point for image reconstruction using any phase retrieval method, including HERALDO.
Fig. 2 (a) Schematic principle of two-pulse imaging. Two coherent pulses with a tunable time delay T are used to illuminate an object. A series of diffraction patterns is then recorded as a function of T, using an image sensor. (b) If the spectrum of the light source contains only a single frequency, each camera pixel will detect a sinusoidal interference pattern as a function of T. (c) If there are two frequency components present in the source spectrum, this will lead to an additional beating in the interference pattern. (d) A complex illumination spectrum results in a complex interference pattern. From this recorded interference pattern, the spectrum that illuminated this pixel can be retrieved.
An important advantage of this approach is that the full source spectrum is used throughout the entire measurement, which is a well-known property of Fourier-transform-based techniques [29, 30]. This results in an efficient use of the available photon flux, and makes the method suitable for implementation with low photon flux sources such as HHG [19]. Furthermore, the spectrum of the source does not have to be known in advance, and the method is remarkably robust against jitter and noise in the spectrum [19]. Moreover, two-pulse imaging has many other features in common with Fourier-transform-based methods [29, 30], including insensitivity for constant background stray light.
3. Experimental methods and results
3.1. Setup
We set up an experiment aimed at demonstrating lensless imaging with ultra-broadband light in combination with HERALDO. The setup used in the experiment is schematically depicted in Fig. 3(a). To investigate the influence of the shape of the reference object, we performed measurements using both a 1D and a 2D reference structure. The sample with the 1D reference structure is shown in Fig. 3(b), and a sample with the 2D reference structure is displayed in Fig. 3(c). In the experiment, a 15 fs pulsed (modelocked) titanium-sapphire laser running at 80 MHz repetition rate with a central wavelength of λ = 800 nm was used as light source. To generate a particularly broad spectrum, the light was coupled into a nonlinear photonic crystal fiber (PCF), in which supercontinuum generation broadened the spectrum to about an octave. The spectrum after the PCF is shown in Fig. 3(d). To produce a pair of coherent pulses with a controllable time delay, the output of the PCF is sent into a Michelson interferometer. By adjusting the length of one of the arms in the interferometer, the time delay between the two pulses can be scanned, which was achieved by mounting one of the interferometer end mirrors on a closed-loop piezo stage. After the interferometer, the pulse pair is used to illuminate the sample and the resulting diffraction pattern was captured with a 1936 × 1465 pixels, 14 bit, 4.54 μm pixel pitch CCD camera in the far field.
Fig. 3 (a) Schematic drawing of two-pulse Fourier transform imaging setup: ultrashort pulses are spectrally broadened in a photonic crystal fiber (PCF) to produce a spatially coherent white light continuum. A scanning Michelson interferometer produces a coherent pulse pair with tunable time delay, which is used for two-pulse imaging. BS: beamsplitter. See text for further details. (b) Scanning electron microscope picture of the 1D HERALDO holography sample, created by focused ion beam etching of a gold surface on a glass substrate. Besides the transmitting star pattern, horizontal and vertical line shaped reference structures were present. (c) Visible-light microscope picture of the 2D HERALDO holography sample, produced by laser micro machining. The corners of the transmitting square reference structure had different curvature. (d) Spectrum of the light after the PCF, reconstructed from a two pulse Fourier transform scan. The wavelengths of different reconstructions shown in Fig. 4 are marked with coloured squares. The grey area indicates the spectral information used in Fig. 5.
In a lensless imaging system, the resolution Λ is given by Λ > zλ/(2ξ), where z is the distance between the sample and the CCD, λ the wavelength of the light and ξ is the height of the CCD. In order to use HERALDO, the CCD must be placed in the far field, which implies that the Fraunhofer approximation πa2 ≪ λz must be fulfilled [13, Chapter 4], in which a is the radius of sample, λ the wavelength of the light and z the distance between sample and CCD. To achieve these far-field conditions with a sufficiently high NA, the CCD was placed in the Fourier plane of a lens with a focal length f = 5 cm, placed at 5 cm behind the sample. In this situation, z becomes the distance between the sample and the lens, which is equal to f. This geometry resulted in a resolution Λ = 4.5 μm for λ = 600 nm light. While this geometry is therefore not truly ’lensless’, the lens is not used for image formation but only to achieve farfield conditions at a practical distance from the sample. Therefore, our approach can be applied equally well to lensless HERALDO experiments.
In the two-pulse scan, the delay was scanned over a distance of 100 μm in 500 equidistant steps during the experiment, which resulted in a minimal resolvable wavelength of 400 nm (0.74 PHz) and a frequency resolution of 3 THz. After every step, an image of the diffraction pattern was taken with different exposure times between 0.01 and 3 ms.
3.2. Broadband HERALDO results
In Fig. 3(b) an electron microscope image is shown of the 1D HERALDO test sample, which consists of a star pattern with a diameter of 200 μm as object, and two line segments as HERALDO reference structures. This sample was etched in a 100 nm gold layer deposited on a fused silica substrate. We performed two-pulse lensless imaging with this sample, of which the results are shown in Fig. 4. The ultra-broadband diffraction pattern that is obtained when using single pulses from the white-light continuum source is shown in Fig. 4(a). A strong radial smearing is present and direct reconstruction from this diffraction pattern did not result in a proper image, as shown in Fig. 4(b). A quasi-monochromatic (Δλ = 5 nm) diffraction pattern retrieved by the two pulse Fourier transform imaging method is shown in Fig. 4(c). Note that the strong radial smearing has disappeared, and much more detailed features are visible in this diffraction pattern. In addition, a phase image of the object can be retrieved simultaneously (data not shown), which would enable quantitative phase contrast imaging if an object contains phase variations.
Fig. 4 Experimental results on spectrally resolved HERALDO combined with two-pulse imaging. (a) Ultra-broadband diffraction pattern from the sample shown in Fig. 3(b). (b) HERALDO reconstruction from this ultra-broadband diffraction pattern shown in Fig. 4(a). The vertical line (see Fig. 3(b)) is used as the reference object for reconstruction. (c) Quasi-monochromatic (λ = 559 nm, Δλ = 5 nm) diffraction pattern obtained from a two pulse Fourier transform scan. (d) HERALDO intensity reconstruction from Fig. 4(c) with a resolution of 6 μm. (e) Enlarged view of the reconstructed star pattern in Fig. 4(d). The sharp features in the center of the star are used to determine the resolution. (f) Three reconstructions at different wavelengths (λ =559, 695 and 885 nm, marked with coloured squares in Fig. 3(d) shown superimposed, using data obtained from a single two-pulse scan. The wavelength dependence of the diffraction pattern results in a different field-of-view for the different wavelengths. Figures 4(b), 4(d) and 4(e) are plotted with an unbiased linear gray scale.
As a first step towards the HERALDO reconstruction, a raised cosine filter is applied on the quasi-monochromatic data from Fig. 4(c), which reduces edge effects at the cost of reduced signal from the outermost 100 pixels. Subsequently, a HERALDO reconstruction is performed by multiplying the data with a differential phase mask and Fourier transforming back to the image plane [26]. In this reconstruction, shown in Fig. 4(d) and enlarged in Fig. 4(e), a sharp image of the star object is observed with a resolution of 6 μm, which approaches the diffraction limit of 4.5 μm that is theoretically expected for our system. From the two-pulse dataset, reconstructions can be performed throughout the whole source spectrum, and reconstructions at three different wavelengths are shown together in Fig. 4(f). Although the scattering intensity from this particular sample is not wavelength dependent, this experiment does demonstrate the ability of our two-pulse imaging method to obtain images at multiple wavelengths simultaneously. If a sample would display spectrally dependent scattering, the two-pulse method would allow spectroscopic imaging. In the case of Fresnel diffraction, this multi-wavelength data can be used for robust iterative phase retrieval [19,31]. It is worth mentioning that the spectral intensity of the white-light continuum behind the PCF was highly unstable, and fluctuated strongly during a two-pulse scan. These fluctuations did not influence the quality of the reconstructed images significantly, as most of this intensity noise was filtered by the Fourier-transform approach. This is a highly advantageous feature of two-pulse imaging, which we have investigated in detail in ref. [19].
3.3. Signal to noise ratio improvement through spectral averaging
In cases where the sample diffraction is not wavelength dependent, information about different wavelengths can be used to enhance the signal to noise ratio, since the respective diffraction patterns that are retrieved at different wavelengths are independent. To demonstrate this feature, reconstructions are performed for several different spectral components retrieved from a single two-pulse scan, scaled according their wavelength, and subsequently averaged. A reconstruction of a single wavelength component is shown in Fig. 5(a). Multiple frames in a small wavelength range (shown in grey in Fig. 3(d)) are combined, which results in the image shown in Fig. 5(b). Clearly visible are the smoother background and constant signal level in the combined reconstruction compared to the case of a single monochromatic image. This effect is highlighted in Fig. 5(c), where the image intensity across a line in both images is plotted: the spectrally averaged image shows much less deviations from the average value.
Fig. 5 Improving the signal to noise ratio by averaging images at multiple wavelengths. (a) Reconstructed intensity from a single monochromatic diffraction pattern obtained from a two-pulse scan. (b) Average of six different reconstructions at different wavelengths after proper scaling of the field-of-view. Intensities in both images are shown with the same unbiased linear gray scale. (c) Image intensity across the red line shown in Fig. 5(a) for both the monochromatic (black trace) and the spectrally averaged (red trace) image. A significant reduction in fluctuations around the average (i.e. higher signal-to-noise) is observed.
3.4. Contrast improvement by means of iterative phase retrieval algorithms
Compared to Fourier-transform holography, HERALDO offers improved flexibility in terms of reference structure fabrication, but also through its potential for providing a higher signal-to-noise, because of the much stronger reference wave. However, this signal-to-noise advantage is usually not fully exploited, which is due to the way in which the reconstruction is performed in HERALDO.
In HERALDO, the observed far-field holographic cross-correlation is the result of a convolution between the object and the reference structure. The strength of this cross-correlation is therefore proportional to the area of the reference structure. When taking the derivative in the HERALDO reconstruction procedure, however, the resulting field only contains a contribution from the edge of the reference structure (assuming that the reference structure has a constant intensity). In this sense, HERALDO itself does not make effective use of the available reference light in the way that a deconvolution would. The advantage of HERALDO over a deconvolution is in its simplicity: a deconvolution procedure is highly sensitive to numerical errors and artefacts, and requires accurate knowledge of the shape and intensity distribution of the reference structure. In contrast, HERALDO only requires the information that the reference structure contains a sharp edge in a specific direction.
The potential signal-to-noise advantage in HERALDO can be optimally exploited by combining the HERALDO reconstruction procedure with a Gerchberg-Saxton-type iterative phase retrieval algorithm. We use a two-step image reconstruction procedure to achieve this. First, we perform the known HERALDO reconstruction to obtain a first image of the sample. Second, we use the result of the HERALDO reconstruction to define a strong support constraint for an iterative phase retrieval algorithm. Such iterative algorithms retrieve the phase for the full diffraction pattern, which means that the full strength of the cross-correlation signal between reference and sample is optimally taken into account. Although this approach comes at the cost of increasing the complexity of the reconstruction procedure, it does provide both a large signal-to-noise improvement and the high resolution offered by HERALDO. A similar two-step procedure combining holographic methods with iterative phase retrieval has been shown to allow improved resolution and/or reconstruction of missing data caused by the presence of a beam stop in the central beam [26, 32].
The availability of a high-quality support constraint is important for robust image reconstruction with Gerchberg-Saxton type iterative phase retrieval algorithms [4]. The HERALDO reconstruction provides such a support, making this procedure reliable and reproducible. In our experiment, the HERALDO reconstruction shown in Fig. 4(e), combined with the knowledge that the reference structures are present near the sample, was sufficient as support constraint to ensure high quality reconstructions. The results of 150 runs of 200 iteration steps with random initial start phases were averaged. This two-step approach resulted in the reconstruction shown in Fig. 6, in which a contrast improvement of two orders of magnitude is achieved compared to the HERALDO reconstruction. It should be noted that in this image the resolution slightly deteriorated from 6 to 7 μm.
Fig. 6 Comparison of the HERALDO reconstruction (left half of the image) with the result of an iterative phase retrieval algorithm (right half) that uses the HERALDO reconstruction as a support constraint. The image is reconstructed after averaging 150 runs of 200 iterations of a phase retrieval algorithm. Both images are shown on a logarithmic intensity scale (scale bar indicates optical density). The use of the iterative phase retrieval algorithm results in a dramatic increase in signal-to-noise ratio, which is improved by two orders of magnitude.
3.5. Dynamic range considerations and 2-D HERALDO
As the two-pulse imaging shows encouraging results in combination with 1D HERALDO, a logical next step is to extend the approach to higher dimensional reference structures. For this purpose, we used the windmill target combined with a square reference structure, shown in Fig. 3(c). A two-pulse imaging procedure was used with nearly identical parameters as used in the 1D HERALDO experiment, again resulting in a collection of spectrally resolved diffraction patterns. By applying two derivatives to a spectrally resolved image from this dataset, the windmill was reconstructed on the corners of the square reference, as shown in Fig. 7. Not all corners of the reference structure had the same sharpness, which resulted in different qualities of reconstructions.
Fig. 7 Influence of the dynamic range of the recorded diffraction pattern on HERALDO reconstructions, demonstrated on a 2D HERALDO test sample (the ’windmill’ sample from Fig. 3(c)). (a) Diffraction pattern captured with exposure time of 0.021 ms. The center of the diffraction pattern is not saturated, but a low SNR is obtained for high spatial frequencies. (b,c) Diffraction pattern captured with exposure times of 0.300 and 3.000 ms, respectively. The center of the diffraction pattern is over-exposed, but the high spatial frequencies are detected with higher SNR compared to Fig. 7(a). (d) High dynamic range combination of Figs. 7(a) and 7(b). (e) High dynamic range combination of Figs. 7(a)–7(c). (f,g,h) Intensity of 2D HERALDO reconstruction of Figs. 7(a), 7(d) and 7(e) respectively, shown with an unbiased linear gray scale. Using data from longer exposure times enhances resolution and smoothness of the image. (i) Horizontal cut through the center of the diffraction patterns Figs. 7(a)–7(c). The thick part of the various lines indicate which data has been used in assembling Fig. 7(e).
This sample with its relatively large open reference structure highlights another challenge that needs to be overcome in Fourier-transform holographic imaging, namely the limited dynamic range of the recorded diffraction pattern for a single camera exposure, which is usually limited by the CCD characteristics. In practice, capturing the Fourier-transform hologram with sufficient SNR is a challenging task due to the large dynamic range of the far-field diffraction pattern. The main cause is the strong zero-order signal, as the light that is directly transmitted by the object is concentrated in a few central pixels, while the information about the higher spatial frequencies is spread out over a much larger detector area. Figure 7 explains a procedure to improve the dynamic range. To achieve a sufficiently high dynamic range for image reconstruction with a conventional CCD camera, multiple exposures with different exposure times are taken, shown in Figs. 7(a)–7(c), and are stitched together numerically. This procedure starts with stitching the images with the shortest exposure time together. The overexposed areas in these images and their direct surroundings (about 10 pixels) are identified and masked out. In these areas, the data from the image with the shorter exposure time is used. Areas with moderately high signal (SNR larger than approximately 5) are selected in the diffraction patterns and are used to normalize the different exposure times. The different exposures are then combined, as shown in Figs. 7(d) and 7(e), and the effect of combining multiple exposures of 0.021, 0.3 and 3 ms is shown in Figs. 7(f)–7(h). By comparing these figures, it can be seen that there is a clear increase in resolution and contrast after combining images from all three exposure times compared to using only the shorter exposure times. To give an impression of the gain in dynamic range, horizontal cuts through the center of the diffraction patterns are shown in Fig. 7(i). This graph also indicates which part of each exposure image is used in the reconstruction of the final high-dynamic-range image.
In some cases, such as X-ray diffraction experiments, a beam block is used to protect the CCD from intense unscattered light and so the center of the diffraction pattern cannot be measured directly. The missing parts can then be reconstructed using iterative phase retrieval algorithms. Holographic and HERALDO reconstructions can be used as initial guess and as a solid basis for the support constraints [26, 32] required for these algorithms.
4. Outlook and conclusion
Several successful realizations of HERALDO have already been demonstrated [20, 26–28, 32–35], using a range of different support structures. Nevertheless, the ability to use broadband light can be a valuable extension, especially when the spectral characteristics of the light source are not well known or unstable, or in cases where spectrally narrow light sources are scarce. This is typically the case for shorter wavelength sources, such as extreme-ultraviolet and soft-X-ray radiation produced through HHG. At these short wavelengths, HERALDO has already been demonstrated as a powerful imaging method, and our two-pulse imaging approach can extend HERALDO to broadband soft-X-ray sources as well.
Our two-pulse imaging approach is photon-efficient, as the full source spectrum is used effectively for the full duration of the measurement. This makes the method ideally suited when dealing with low-flux broadband sources such as HHG systems. Two-pulse imaging has already been demonstrated with a table-top XUV source at a wavelength of 47 nm [19]. The present work enables extension of two-pulse XUV imaging to far-field holography and HERALDO, which has significant advantages for imaging weakly scattering objects such as biological samples. An important advantage of holography is that the measured signal strength is the product of the object field and the reference field, which gives the possibility of coherent amplification of the weak object wave by a strong reference. Therefore, weakly scattering samples can be imaged with sufficient signal-to-noise ratio by increasing the strength of the reference field rather than increasing the flux on the sample. Our approach of two-pulse imaging combined with HERALDO therefore provides a promising framework for ultrahigh-resolution imaging of biological samples with HHG-based soft-X-ray microscopes. In addition, two-pulse imaging can provide spectroscopic information throughout the entire source spectrum.
In conclusion, we have combined lensless imaging with coherent supercontinuum light sources and extended reference holography. Our two-pulse imaging method enables holographic imaging at all spectral components of the incident light. We have demonstrated ultra-broadband imaging using HERALDO with 1-D and 2-D reference structures. Moreover, the individual spectral components that are retrieved from a two-pulse scan can be combined to enhance the contrast in the final image reconstruction. We have investigated the effect of limited dynamic range of CCD detectors on the quality of the image reconstruction. Finally, we have demonstrated that iterative phase retrieval algorithms can increase the contrast of the reconstruction by orders of magnitude when the HERALDO reconstruction is used as support, thereby combining the flexibility and robustness of holographic methods with the higher photon efficiency of phase retrieval techniques.
Acknowledgments
The authors thank D.W.E. Noom for insightful discussions, and H. Zeijlemaker and M. Slaman for assistance in sample preparation. This work is (partly) financed by an NWO-groot investment grant of the Netherlands Organisation for Scientific Research (NWO) and Laserlab Europe (JRA Bioptichal). S.W. acknowledges support from NWO Veni grant 680-47-402.
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