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We introduce depth-filtered digital holography (DFDH) as a method for quantitative tomographic phase imaging of buried layers in multilayer samples. The procedure is based on the acquisition of multiple holograms for different wavelengths. Analyzing the intensity over wavelength pixel wise and using an inverse Fourier transform leads to a depth-profile of the multilayered sample. Applying a windowed Fourier transform with a narrow window, we choose a depth-of interest (DOI) which is used to synthesize filtered interference patterns that just contain information of this limited depth. We use the angular spectrum method to introduce an additional spatial filtering and to reconstruct the corresponding holograms. After a short theoretical framework we show experimental proof-of-principle results for the method.
©2012 Optical Society of America
1. Introduction
Digital holography is a powerful technique for the acquisition of the morphology and shape of samples [1–3]. The numerical reconstruction allows for filtering procedures and numerical propagation of the wavefield to any plane of interest where both amplitude and phase information can be calculated. Especially the phase information is widely used in digital holography as it yields impressive depth resolutions of a few nanometer with a relatively simple setup [3].
On the other hand, the performance of digital holography is limited when the sample consists of multiple semi-transparent layers. The reflections at the different depths will interfere with each other and with the reference beam leading to a superposition of interference patterns on the digital camera. Phase analysis in such cases is a great challenge, especially when the depth of interest (DOI) is a buried layer and not at the surface of the sample. Nevertheless, tomographic measurements of multiple layer samples using digital holography have been performed with a similar approach as is used in low-coherence interferometry and time domain optical coherence tomography (OCT) [4–6]. In this approach the interferences created by the different layers can be separated exploiting the low coherence of broad-band light sources. Therefore multiple scattered photons outside the DOI can be suppressed by the coherence window [7–9]. The depth-resolution in such cases is limited by the coherence length of the light source which is typically a few micrometers. In combination with spatiotemporal and phase-shifting methods this led to first tomographic phase images with nm-resolution [10, 11] behind scattering layers.
Another approach is wavelength-scanning digital holography where holograms are recorded for multiple wavelengths and processed such that tomographic measurements are possible [12–15]. Most of these techniques just analyze the amplitude of the backscattered light. Tomographic phase reconstruction in general is rare and is mainly applied in transmission measurements e.g. through cells to obtain refractive index variations [16–19]. In OCT the phase analysis has mainly been used in topographic measurements [20–22] while tomographic phase imaging has just been reported for differential measurements [22].
In this paper we introduce an approach for wavelength scanning holography that combines depth sectioning by means of low-coherence interferometry in the spectral domain with digital holography. In this approach a Fourier analysis of multiple digital holograms is followed by a synthesis of interference patterns that are created at specific depths. This approach allows analyzing the depth-filtered phase and amplitude of selected depths in multilayer samples and thus yields tomographic nanometer resolution.
2. Depth-filtered digital holography
In digital holography a digital camera commonly acquires interference patterns utilizing monochromatic light. We extend this method and use a similar approach and data set as in wavelength-scanning digital holography. To do so we record a set of holograms with several distinct wavelengths. The data we obtain is thus the intensity over two lateral dimensions and wavelength. The interference pattern for each wavelength can be analyzed by means of digital holography, but as described above this will only allow for surface analysis. Samples that consist of semitransparent layers cannot be imaged with this approach because the reflections from the different depths overlap on the camera and lead to a superposition of interference patterns. Thus, standard digital holographic reconstruction techniques will fail.
In the following we introduce a technique that filters parasitic reflections and preserves the required information of the depth-of interest (DOI). This information is used to synthesize interference patterns which thus contain information for just the limited depth range. The procedure is depicted schematically in Fig. 1 . As a first step, multiple holograms are recorded for different wavelengths. Then the intensity is analyzed as a function of wavelength for each pixel of the digital holograms. Each path length mismatch between sample and reference wavefield leads to a spectral modulation which can be described by the law of spectral interference expressed as Eq. (1) [23]
Here, x and y describe the lateral dimensions, k is the wavenumber, S(k) is the power spectral density of the light source and RO is the reflectivity of the sample. As we use a Mach-Zehnder interferometer, TR is the transmission through the reference arm. In case of Michelson geometry this part has to be replaced by the reflectivity of the reference arm mirror RR. The frequency of the resulting modulation is proportional to the optical path-length mismatch between object and reference arm Δz. Each reflection in a certain sample-depth will consequently lead to a specific modulation frequency. Thus, in our technique the reference arm length is very important, since the resolvable modulation frequency is limited by the sampling theorem. This is in contrast to common digital holography, where the path lengths do not necessarily have to be adjusted as accurately.
Fig. 1 Principle of depth-filtered digital holography (DFDH). The pixel-wise analysis of multiple holograms acquired at different wavelengths bears depth-information of the sample coded within a spectral modulation. Using a depth-filtering procedure, interference patterns can be synthesized that just contain information of the filtered depth, i.e. certain modulation frequencies. Digital holographic analysis of the filtered interference patterns allows for phase analysis in the depth.
For simplicity we now consider the sample to be just one reflecting surface at distance. Applying the inverse Fourier transform leads to Eq. (2):
wheredescribes the DC part that appears at z = 0. The second term in the parentheses creates peaks in the distance of around the center and thus gives access to the depth of the sample at this point in the xy-plane. Repeating this procedure for multiple pixels allows for reconstruction of both cross sectional images and full three dimensional information of the sample. This method is also used in full-field swept-source optical coherence tomography [24].In this paper we extend this method by using an approach which is usually performed in spectroscopic optical coherence tomography to calculate depth resolved spectroscopic data [25]. The depth of interest (DOI) in the depth profile is separated by means of a windowed Fourier transform, i.e. by multiplying the complex data with a window-function.
This window-function is only non-zero in the depth of interest and zero for other depths. The simplest form of such a window function is a rectangular window. More sophisticated approaches, with elaborate shapes of the window function can show a better performance because the Fourier transformed signal is convoluted with the Fourier Transform of the window function.
In the experiments described in this paper we use a Hann-window to filter a certain depth of the profile. The Hann-window can be expressed as Eq. (3):
with and being the window size. The procedure can in other words be understood as a band pass filter applied to the spectral modulation frequencies.It follows directly that the interference pattern at the filtered depth can be expressed as. After the Fourier transform we obtain which describes the filtered spectral modulations and can be used to form a 3D data-cube. The whole procedure leads to synthesized interference patterns, which just contain the information of the DOI. The filtered interference patterns are now reconstructed using the angular spectrum method [26, 27]. To do so, are analyzed at a single wave-number .
The complex object wave-field in the hologram plane which is the plane of the digital sensor placed at distance c = 0 is. The interference pattern we obtain by this procedure is complex. We drop the imaginary part and just use the real part for the reconstructions.
The angular spectrum of can be obtained by Fourier transform of the interference pattern [27] as described by Eq. (4):
where fx and fy are the spatial frequencies corresponding to x and y directions, respectively.We use an off-axis configuration as introduced by Leith and Upatnieks [28] meaning that reference and object beams enclose a small angle φ. The spatial frequencies that occur in the spectrum depend on this angle. Similar to the Fourier transform of the spectral modulation, a very strong DC term is present, positioned in the middle of the angular spectrum and surrounded by the real and the virtual image. A part of these spatial frequencies is filtered by a numerical band pass [29], in our case a Gaussian filter. By appropriate selection of the center frequency of this filter unwanted interferences of light reflected by the sample and the reference can be filtered out. The filtered field can be rewritten as the inverse Fourier transform of its filtered angular spectrum (Eq. (5)):
can now be propagated towards any plane of interest using Eq. (6):where [27].The reconstructed complex wavefield at any plane at the distance c = d perpendicular to the propagation axis is given by Eq. (7):
The amplitude and phase of can be reconstructed and analyzed.Both the depth-filtering procedure and the spatial filtering of the angular spectrum method have basically a comparable approach as just a small part of the spectrum is filtered and processed. A combination of both filtering-techniques leads to a very powerful tool that has the potential to filter object light travelling from a specific depth into a specific direction.
In the next section we present experimental proof-of-principle results for this technique and compare the results with those obtained without depth-filtering.
3. Experimental results
In order to analyze the performance of our technique we employ the sample configuration depicted in Fig. 2 . The metallic island we used in former experiments [26] is the object to be imaged. An exemplary reconstruction of the pure uncovered island is shown in Fig. 2(a). It has lateral dimensions of 1.0 mm x 1.3 mm and a height of 200 nm, as verified in [26] using a profilometer-scan.
Fig. 2 a) Quantitative digital holographic phase reconstruction of the metallic island as a reference. b) The side-view and c) the front-view of the sample-configuration used in this paper.
In the following we place four 150 µm thick cover-glasses in front of the island that are arranged as stairs (Figs. 2(b) and 2(c)) to simulate a multiple layer sample as they create reflections at different depths. The cover-glasses are clamped together, so that they are separated by intermediate air layers. The distance from the last cover-glass to the metallic island is approximately 600 µm. This sample configuration is placed into a Mach-Zehnder like setup as sketched in Fig. 3 . The two lenses have a focal length of f = 150 mm and form a 4f-geometry. The CCD camera consists of 1376 x 1040 pixels with a pixel size of 6.45 µm x 6.45 µm. The laser source is a tuneable diode laser in Littman configuration as we already used in [30]. The whole setup is in off-axis geometry with an angle of approximately 1° between object and reference arm.
We record a digital hologram of the sample and use the angular spectrum method for reconstruction. The wavelength is set to be λ = 820 nm.
For quantitative measurements we acquire a reference phase of a plane mirror in order to remove the system errors introduced to the phase by the optical components [3,26]. The reconstructions of this hologram are depicted in Fig. 4 . The amplitude (Fig. 4(a)) and the wrapped phase (Fig. 4(b)) clearly show that the cover glasses strongly affect the reconstructions. Just the part of the island that is covered by only one glass is reconstructed correctly, as is shown by the line-scan of the unwrapped phase-profile (Fig. 4(c)) across the line marked in Fig. 4(a) with a red dotted line. The 200 nm height of the island is correctly reconstructed. The other parts of the island that are covered with two or three glasses are severely disturbed and cannot be reconstructed.
Fig. 4 Digital holographic reconstructions obtained without depth-filtering: Amplitude (a), wrapped phase (b) and line-scan over unwrapped phase-profile (c) across the line marked in (a).
Figure 5(a) shows the angular spectrum of the interference pattern used. The component in the middle is the DC term which is the centre of point-symmetry for the other parts that correspond to different spatial frequencies formed by different angles between the reference beam and the object beam. The frequency component used for reconstruction of the results in Fig. 4 is marked with the green arrow. As can be seen the cover glasses introduce many parasitic components to the interference pattern, which are caused by multiple reflections.
Fig. 5 Angular spectrum of the interference pattern of the hologram reconstructed in Fig. 4. The insets a-c show the reconstruction of the corresponding part of the spectrum.
These multiple reflections have different angles of incidence onto the CCD camera and thus lead to different positions in the angular spectrum. The angular spectrum method allows for selecting and reconstructing each component independently. This is shown in Fig. 5. The parasitic components marked with red arrows are each reconstructed with the angular spectrum method leading to the amplitude reconstructions depicted in the figure. The reconstruction in Fig. 5(a) shows a very smooth amplitude which is connected to the top-reflection on the surface of the cover glasses. The reconstruction in Fig. 5(b) clearly contains information of the sample and is created by multiple reflection of the sample light whereas 5(c) belongs to multiple reflections on the cover glasses. The reconstructions in Fig. 4 have been performed with the spectral components marked with the green arrow.
In the following we use the DFDH-approach to generate synthesized interference patterns that only contain information of the metallic island. We use our tunable laser to record 80 holograms at equidistantly spaced wavelengths between 820.0 nm and 827.9 nm with a wavelength spacing of Δλ = 0.1 nm. The holograms are recorded sequentially, i.e. we check the exact wavelength setting to have accurate values. We do not use a direct wavelength sweeping. The acquisition time for each hologram is set to 50 ms. The recording procedure is comparable to the approach described in [24]. The reconstruction of each individual hologram leads to results comparable to those shown in Figs. 4 and 5.
In order to increase the quality of the depth-profile calculated by the inverse Fourier transform of the modulated spectra we use zero-padding by a factor of ten. Thus we have 800 data points, from which only 400 can be used, due to the symmetry of the Fourier transform. An inverse Fourier transform of one spectral modulation leads to a peak in the depth of the corresponding layer. Repeating the procedure for all pixels, we obtain the depth profile of the whole sample as depicted in Fig. 6 . The obtained depth is proportional to the spectral modulation frequency and thus to the path-length mismatch between object and reference arms [23]. This is in contrast to other techniques, like e.g. standing wave microscopy, where the used modulation frequency is constant [31].
Fig. 6 Depth profile of the metallic island behind the cover glasses. The z-axis shows the mismatch between object- and reference arm-length.
The cover glasses appear in a depth between 0.8 and 1.3 mm. Due to multiple overlapping reflections and autocorrelations the signal is blurred. The island is placed between 0.1 and 0.4 mm. The performance of DFDH strongly depends on the quality of the depth-profile. The better the depth-separation in the profile the better the result should be. Since we use a narrow spectral bandwidth, which gives a low depth separation, the profile we obtain is of limited quality. In the experiment presented here the depth-separation is approximately 70 µm. As is shown by the angular spectrum in Fig. 5 the sample information is just a small fraction of the interference patterns that is superimposed by the DC part and many parasitic reflections at the different layers.
As we pointed out in [24] the procedure has inaccuracies as just 80 different wavelengths are used to scan the spectral modulation. Using a broader spectral bandwidth should lead to much better results. Typical depth-resolutions using this approach are around 10 µm and reach 2 µm in sophisticated setups [32, 33]. But the usage of broadband-sources will also increase the influence of dispersion, which will affect the calculation of the depth-profile. This will have an impact on the synthesized interference patterns and consequently on the digital holographic reconstruction.
Besides, numerical errors of the Fourier-transform and the signal processing could also degrade the results. But as we will show, for this proof of principle experiment the set up and signal processing is sufficient to analyse the phase and to obtain nanometer resolution in multilayer samples. As already mentioned we use a Hann-window to filter a specific depth of the profile. We combine the filtered modulations and synthesize interference patterns as described in section 2. Then we use the angular spectrum method for digital holographic reconstruction of the synthesized patterns.
In the following example we apply the depth-filtering for 8 different DOIs which we name window “a” to window “h” as shown in Fig. 7 . The depth-profile in Fig. 6 consists of 400 data-points into axial-direction. The windows we use have a width of 10 data-points except window “d” which has a width of 50 data-points. The different windows in Fig. 7 are marked with colored lines and are spread in different depths. Their center is given in the right column.
Fig. 7 The windows that are used in the following are marked in the depth-profile. We name them windows a-h.
The corresponding amplitude reconstruction from the synthesized interference pattern belonging to each named depth is plotted in Fig. 8 . Depending on the DOI, the reconstruction strongly changes. For windows “e” to “h” the amplitude reconstruction is strongly disturbed and the island is not reconstructed. This is different for windows “a”, “b” and “c”. Here the metallic island is clearly visible and the regions where the other reconstructions have their main distortions are clearly reconstructed. Moreover, due to the changing optical path length depending on the number of cover glasses in front of the corresponding part of the island, different parts of the island can be addressed using windows in different depths. The addressed parts are clearly highlighted in the reconstructions. Window “d” ranges over windows “a”, “b” and “c”. Here the whole sample is reconstructed. These results prove that the depth-filtering has advantageous impact on the digital holographic amplitude reconstruction. The whole island is clearly reconstructed using the DFDH-approach.
Fig. 8 The figure shows the reconstructed amplitude of the interference patterns synthesized for the windows a-h, as depicted in Fig. 7. Windows “a” to “d” are located in the depth of the metallic island. Consequently the reconstructions of the interference patterns synthesized for these depths clearly show a good reconstruction of the metallic island. In contrast, the amplitude-reconstructions for windows e to h are strongly affected by the cover glasses as the filtered depth used here is located away from the island.
In the following we analyze the reconstruction of the more important phase information. The results are depicted in Fig. 9 . The columns of Fig. 9 correspond to the amplitude, wrapped and unwrapped phase reconstructions, respectively. While the first row shows unfiltered data the second row corresponds to filtering with the broadest window “d” These results prove that the filtering procedure enables the phase reconstruction of the island.
Fig. 9 Digital holographic reconstruction without window (a-c) and with window “d” (d-f). From left to right: amplitude, wrapped and unwrapped phase.
The 3D view of the unwrapped phases is shown in Fig. 10 . Using DFDH the quality of the phase reconstruction is clearly enhanced. It is evident that the reconstructed phase at the covered parts of the island is not destroyed by the cover-glasses. But due to the change of the optical path-length introduced by the cover-glasses phase jumps occur that lead to ambiguous phase values in the borders of the glasses. This has to be taken into account for quantitative discussions. Additionally as can be seen, at the part of the island covered by one glass in Fig. 10, the phase has a small curvature which is introduced by the cover-glasses. This is especially a significant challenge for tomographic phase analysis of unknown samples. Phase distortions created by layers that are located above the DOI have a strong impact on the reconstruction. In such a case it is possible to use DFDH to analyse the phase distortions iteratively. Starting at the top surface of the sample and scanning the depth with small depth-steps, allows tracing the phase change into the depth dimension. This opens up the possibility to correct the phase in the DOI. But to do so, much better depth-profiles are needed that allow for error-free depth-selection.
In the example shown here the phase distortion can be reconstructed using a two-wavelength interferometry approach that applies reconstruction with a synthetic wavelength [34]. The wavelengths used in this example are λ1 = 821.0 nm and λ2 = 822.0 nm, which leads to a synthetic wavelength of Λ = 675 µm. The results for the unfiltered case are shown in Fig. 11(a) . The 3D view and the unwrapped 2D phase profile of the filtered island are shown in Figs. 11(b) and 11(c). As can be seen the 150 µm path-length change introduced by each cover-glass is correctly reconstructed (Fig. 11(d)). The contour of the island can be clearly seen in the unwrapped phase, while the 200 nm height is naturally not resolvable due to the much bigger synthetic wavelength. This shows that the usage of DFDH allows for correct reconstruction of the path-length changes and that using synthesized interference patterns and synthetic wavelength leads to successful reconstructions.
Fig. 11 Digital holographic reconstruction with a synthetic wavelength. 3D view for the unfiltered (a) and filtered case (b) (first row), unwrapped phase (c) (second row left) and line-scan over phase-profile (d) (second row right).
To reconstruct the height of the island quantitatively a different approach has to be used. We reconstruct the phase across the cover-glasses which means that we circumvent the optical path-length change. We use the line-scans marked in the amplitude reconstruction shown in Fig. 12(a) . The corresponding height-profile is plotted in Fig. 12(b). The green line across the region covered by one glass shows the 200 nm height of the island. The regions where two and three cover glasses prohibit the reconstruction with the common DH-approach show good quantitative results for the height profile using DFDH. The line-scans across the two and three glasses correctly reconstruct the 200 nm height, but are superimposed by a modulation which can be attributed to the limited quality of the depth-profile and thus not ideally windowed and synthesized interference patterns. This modulation degrades the phase resolution which in this example is in the region of around 50 nm. Nevertheless, the reconstruction is good enough to demonstrate that the usage of DFDH enables the tomographic phase analysis of our sample. Improved experimental conditions or advanced processing promise the resolution to be in the nanometer range.
Fig. 12 Digital holographic amplitude reconstruction using window “d” (a). Line-scan of the unwrapped phase profile for the marked lines (b).
4. Conclusions
Depth-filtered digital holography is a powerful technique that enables quantitative tomographic phase analysis of covered interfaces. This is possible due to band pass filtering of modulation frequencies of the recorded spectra and spatial frequencies of the interference patterns. The combination of the two filtering techniques enables the analysis of light coming from a certain depth and at a certain angle. The ultimate limits of this technique have still to be explored, yet first results are very promising. One main point will be the improvement of the depth-profiles which will allow for better filtering and will lead to higher quality of the synthesized interference patterns. However, the usage of broadband light-sources will lead to problems with dispersion, which will have to be addressed in detail. The application of our technique through weakly scattering layers should also be possible as the first processing steps are based on a similar approach as frequency domain OCT. However, the performance and limitations of our technique in scattering environment have to be analysed in detail. A priori information of the sample is not needed. The structure of the sample can be obtained iteratively by scanning the DOI through the sample and tracking the phase changes with depth. Depth-filtered digital holography has the potential to enable quantitative measurements with nm resolution in multilayer samples. We see enormous potential for applications in the semiconductor technology for analysis of buried samples and for biomedical applications.
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