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Abstract
This paper describes on-chip digital holographic interferometry for measuring the wavefront deformation of transparent samples. The interferometer is based on a Mach-Zehnder arrangement with a waveguide in the reference arm, which allows for a compact on-chip arrangement. The method thus exploits the sensitivity of digital holographic interferometry and the advantages of the on-chip approach, which provides high spatial resolution over a large area, simplicity, and compactness of the system. The method’s performance is demonstrated by measuring a model glass sample fabricated by depositing SiO2 layers of different thicknesses on a planar glass substrate and visualizing the domain structure in periodically poled lithium niobate. Finally, the results of the measurement made with the on-chip digital holographic interferometer were compared with those made with a conventional Mach-Zehnder type digital holographic interferometer with lens and with a commercial white light interferometer. The comparison of the obtained results indicates that the on-chip digital holographic interferometer provides accuracy comparable to conventional methods while offering the benefits of a large field of view and simplicity.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Transparent materials, such as liquids, single crystals, plastics, biological cells and tissues, or gases, are frequently studied in scientific research. However, characterizing these materials using optical methods can be challenging, as optical sensors only measure the intensity of light, which remains constant as it passes through the sample. Therefore, small changes in the refractive index are often the only physical properties that can reveal internal inhomogeneities caused by variations in temperature, pressure, or density. Measuring the spatial distribution of the refractive index is thus a crucial task for characterizing transparent samples. This is particularly relevant in fields such as biology, biomedicine, and material engineering, where noninvasive imaging methods are needed for counting and differentiating cells, for example. However, achieving this requires optical imaging systems with a large field of view (FOV) and high lateral resolution, typically in the micrometer range, to capture detailed images with sufficient information about the intensity of light.
The requirement for large FOV measurements can be met by on-chip methods that use the entire area of the sensor to image the sample, which is positioned above the sensor as closely as possible. At current pixel densities of optical sensors, this approach allows high resolution images to be taken without the use of microscope lens with large numerical apertures, resulting in a significant simplification of the entire optical system. The contemporary on-chip techniques are based on in-line digital holographic recording. Several variations of lensless or on-chip techniques have been published [1–9], using different geometric arrangements of measurements (light source, illumination at different angles, multi-plane image capturing, special sample preparation) and reconstruction algorithms. The reported on-chip imaging system achieves lateral resolution up to the 350 nm within a FOV greater than 20 mm$^2$ with a pixel size of 1.1 $\mu$m and a wavelength of 530 nm using super-resolution techniques [9]. It should be also noted that the in-line holography (including on-chip arrangements) is based on the coherent superposition of the undiffracted component of the illuminating beam and the component diffracted by the measured object (particles).
Despite their advantages, current on-chip imaging methods have several drawbacks. Most reconstruction methods, which are required in on-chip imaging, are based on computationally intensive iterative algorithms. To workarround this issue, algorithms based on convolutional neural networks (CNNs) [10–12], have been used in the recent works, however, efficiency and accuracy of these methods are highly dependent on data size, quality and proper labeling. The biggest drawback of current on-chip imaging systems is their limitation to imaging samples that create contrast in the intensity of the propagating optical wave. This means that current on-chip imaging systems cannot be used to characterize transparent samples that require a purely interferometry-based imaging approach. Some methods use a diffraction grating [13,14] or reflective optical elements [15,16] to introduce the reference wave for holographic imaging of transparent samples, allowing a distance between the camera and the sample of tens of millimeters.
The above problems have led us to develop on-chip digital holographic interferometry, which extends the capabilities of on-chip techniques to characterize transparent samples. The main challenge to be solved is to create an optical interferometer by bringing a reference wave to the optical sensor while maintaining the on-chip arrangement. This task was solved by the design, construction and use of a thin waveguide placed between the optical sensor and the sample to be measured. Other key features of our method are the use of heterodyne detection using acousto-optic modulators (AOMs), which allows the use of phase-shifting (PS) algorithms [17,18] to calculate the phase of an optical wave passing through a transparent sample. In principle, any other PS technique or single image reconstruction with off-axis ordering can be used in our method. The calculated phase of the optical wave passing through the sample can then be used to calculate the spatial variations of the refractive index and possibly other physical quantities.
Thus, our method combines high axial resolution (on the order of nanometers) due to the use of digital holographic interferometry (DHI) principles with a large field of view (up to 400 mm$^2$ depending on the sensor used) and high lateral resolution (down to sub-wavelengths in extreme cases using super-resolution techniques) due to on-chip measurement techniques. In our presented system we achieve a lateral resolution of 3.5 $\mu$m over a FOV greater than 20 mm$^2$ using light source with a wavelength of 631.52 nm. In order to benchmark the imaging parameters of our developed system, we provide a comparison of our system with a conventional free-space digital holographic interferometer and with the commercial imaging system Zygo NewView 7200, white light interferometer (WLI). This WLI system achieves lateral resolution of 2.05 $\mu$m within an FOV of 1.05$\times$1.4 mm$^2$ using a 5x magnification objective with NA 0.13 and a wavelength of 532 nm.
The Article is divided into six sections. After the introduction, Section 2 describes the principles and methods. The development of the optical waveguide used in the on-chip arrangement is described in Section 3. Section 4 is devoted to describing on-chip DHI and Section 5 then describes the measurement details and experimental results. Finally, Section 6 concludes the results presented in this Article.
2. Principles and methods
In this work, we adopt the concept of conventional optical on-chip imaging as shown in Fig. 1(a). The conventional on-chip arrangement consists of a light source (LS) and a sample (S) placed directly on the camera sensor (Cam). We apply this concept in the design of the on-chip Mach-Zehnder digital holographic interferometer. The concept of the simplest and most compact arrangement is shown in Fig. 1(b). The on-chip interferometer is constructed by inserting the optical waveguide (WG) between the sample and the camera sensor. The waveguide is fabricated by means of transmission gratings tDG1 and tDG2 on opposite sides of a thin glass plate. By illuminating the system with a coherent light source, a micro-interference pattern is created on the camera sensor, which can be processed by well-known reconstruction algorithms of Digital Holographic Interferometry (DHI) [17,18].
Fig. 1. (a) Scheme of a conventional on-chip imaging arrangement and (b) conceptual scheme of the simplest and most compact arrangement of the digital holographic interferometer for on-chip imaging, consisting of a light source (LS), a sample (S), a camera (Cam) and an optical waveguide (WG) formed by transmission diffraction gratings (tDG1, tDG2). The coupling grating tDG1 can be replaced by a prism while maintaining functionality.
DHI is a sensitive and non-invasive method that uses coherent light to measure wavefront deformation. Samples examined by DHI can be reflective, where the aim of the interferometric measurement is usually to accurately evaluate the surface topography, or transparent, where changes in refractive index and derived quantities are measured. The wavefront deformation is calculated from the phase of the complex optical wave, which cannot be measured directly by an optical sensor. Current optical sensors can only measure the intensity of the optical wave. DHI can encode information about the phase of the optical wave in an intensity interference pattern, also known as a hologram. Mathematically, the hologram, i.e. the intensity of the microinterference pattern captured by the camera $I$, can be expressed as the sum of the following three terms:
where $I_r$ and $I_o$ are the intensities of the reference and object waves respectively. The term $\sqrt {I_r I_o}\Re \left [{e^{j(\phi + \delta )}}\right ]$ is the contribution to the measured intensity due to interference. The symbol $\phi$ is the phase of the object wave to be calculated, the symbol $\delta$ is the phase shift introduced by the phase shifting (PS) technique described in the next paragraph. Numerically, $I$, $I_o$, $I_r$ and $\phi$ are two-dimensional arrays where the row and column positions correspond to the coordinates $x$ and $y$ respectively. The physical quantity $\delta$ is a scalar. Symbol $j$ is $\sqrt {-1}$ and the symbol $\Re$ stands for the real part of the complex value.The PS method allows to calculate the phase $\phi$ by capturing several holograms with defined phase shifts $\delta$ in Eq. (1), we assemble a system of equations for three unknowns $I_r$, $I_o$ and $\phi$. To calculate the unknowns, it is necessary to record at least three phase-shifted holograms for three different values of $\delta$. Usually algorithms using a larger number of holograms are used, which are more robust and less sensitive to noise. In our approach, we used heterodyne detection for PS using acousto-optic modulators (AOMs). The AOM in one arm of the interferometer is driven by a sinusoidal voltage with the AOM radio frequency $f_{RF}$ and an additional frequency shift $\Delta f$: $f_{RF}+\Delta f$, while the other arm is modulated only with $f_{RF}$. If $\Delta f\neq 0$, the frequency difference of the interfering waves results in a continuous time variation of the phase shift $\delta$. Given a frame rate (FPS), the phase shift between two successive frames can be controlled by the frequency shift $\Delta f$ according to the formula:
The frequency shift $\Delta f$ is set to obtain $N$ images within one period. Each $\nu$-th image, representing the $\nu$-th phase shifted hologram, is denoted by a symbol $I_\nu$, where $\nu$ runs from 1 to $N$. The sequence of phase-shifted holograms is used as input to the PS algorithm [18], whose output is a complex field given by the formula
Although the distance between the sensor and the object is as small as possible, it is necessary to numerically propagate the complex field from the sensor plane to the sample plane in order to obtain the maximum resolution of the method. The angular spectrum method [17] with the transfer function:
The complex field $U_d$ is stored in the computer’s memory as an array. Assume a measurement at the reference point indicated by index 0. The reference measurement carries information about the optical aberrations of the interferometer, including the WG aberrations. Other measurements made in the presence of the sample (denoted by index 1) can be related to the reference measurement in order to suppress unwanted optical aberrations. The phase change $\Delta \phi$ can be calculated using the formula:
3. Instrumentation
The aim of the DHI method is to calculate the phase of the complex optical wave propagating through the transparent sample from the hologram produced by the interference of the object wave with the reference wave. In the conventional digital holographic interferometer, the interference of the object and reference waves occurs behind the beamsplitter of the interferometer. However, the conventional beamsplitter requires a large space, depending on the diameter of the interfering beam and the mutual angle of the object and reference beams. This poses a challenge to the use of the on-chip approach, as the achievable resolution is highly dependent on the distance between the sample and the camera. The solution is to use a waveguide as the reference arm of the interferometer, as this allows the volume required for the reference beam to be minimised. This section describes the development of the optical waveguide (WG), since its use is a key idea behind the presented method. It should be noted that the WG was designed and fabricated in-house. In order to achieve the best imaging results, we modified the conceptual scheme shown in Fig. 1(b) and designed a WG whose description is presented in the following paragraphs.
The principle of the waveguide is to guide the input beam to a defined output position using the geometry of the waveguide volume as a guide path. The WG therefore consists of three parts: the input, the propagation and the output part. In our case, we need to propagate a sensor-sized input wavefront, about 20 mm$^2$, to a distance of about 50 mm. The propagation part of the WG was made of fused silica with a thickness of 800 $\mathrm{\mu}$m. The infusion part was realised by an optically bonded symmetrical prism (P) with an apex angle of 90$^\circ$ to the WG with glycerol. The extraction part was realised by the developed transmission diffraction grating (tDG). The choice of the material of the WG and its thickness follows the idea of a durable, thermally and mechanically stable, especially to avoid bending of the thin plate due to internal stresses, and easy to manufacture optical element that can withstand regular handling but also maximize the achievable lateral resolution. The purpose of this element is to prove the concept of the imaging approach. Optimization of its parametres is expected to lead to even better results. The wavefront propagates in the WG by the total internal reflection phenomena described by the formula:
where $\theta _c$ is the critical angle for the chosen two media combination with refraction indices $n_0$ for air and $n_{WG} (\lambda )$ material of the waveguide, respectively.We made the WG out of fused silica, so the critical angle for total internal reflection using the 631.52 nm laser is $\theta _c =\arcsin {\left (1/1.4751\right )} = 43.33^\circ$. This condition places demands on the infusion and extraction parts of the WG, since the propagation of the wavefront through the WG is governed by the mutual relationship between the angle of infusion to the WG, the total internal reflection angle and the line density of the extraction grating. The total internal reflection angle was adjusted by adjusting the illumination source to the prism. The wave propagates to the further part of the WG, where it is then extracted from the WG using the transmission diffraction grating, which is denoted by tDG2 in Fig. 1(b) and by tDG in Fig. 4(a). The tDG was prepared by an in-house constructed modified Leith and Upatnieks exposition scheme, see Ref. [19] and see Fig. 2. The recording process of a tDG to the photopolymer material (polycarbonate, Bayfolreg; HX 101) [20] consisted of three stages. First, the pre-exposure by the reference wave through the upper arm of the arrangement to the 50% of the power density. Then the pattern was exposed using both arms of the setup to 100% of its power density, i.e. 24 mW/cm$^2$ at the wavelength of 488 nm. Finally, the exposed pattern was stabilized using UV. The whole process took several minutes to achieve sufficient power density.
Fig. 2. Photograph of the modified Leith and Upatnieks arrangement for diffraction grating exposition. Abbreviations used in the figure: laser source (LS), mirrors (M$_1$, M$_2$, M$_3$, M$_4$), beam-splitter (BS), spatial filters (SF$_1$, SF$_2$), collimating lenses (L$_1$, L$_1$), photopolymer with created trasmission diffraction grating (tDG).
The tDG line density necessary for wave extraction in the first diffraction order was calculated using the grating equation:
where $m = 1$ is the diffraction order, $\theta _{WG} = 45^\circ$ is the chosen total internal reflection angle and $\Lambda$=590 ${\rm n}$m is the grating period. The calculated grating density in line pairs per millimetre (lp/mm) $d_{GD}$ is then equal to $d_{GD} = 1/\Lambda = 1695$ lp/mm.In order to achieve sufficient efficiency of light propagation through the WG to the sensor, it is necessary to ensure that the components are correctly adjusted to each other. Other parameters that can affect wave propagation include vibration, temperature effects, homogeneity, flatness and micro-roughness of the WG. The photopolymer from which the tDG is made also plays an important role. It is known that the substrate of the photopolymer causes spatial non-uniformity of the diffracted beam, which affects the background noise [13]. Spatial nonuniformity can arise due to a variety of factors, such as variations in the thickness or composition of the photopolymer layer, non-uniform exposure, or the presence of defects in the grating. We examined the photopolymer with optical microscopy (OM), which depicts surface imperfections, and the measured topography of the photopolymer with WLI, as shown in Fig. 3(a) and 3(b), respectively. From the intensity image and surface topography it can be concluded that microroughness and imperfections in the photopolymer are one of the sources of noise and speckles in the DHI measurement data. Note that the diffraction pattern is too fine to be resolved and visible in the OM and WLI images.
Fig. 3. (a) Optical microscopy (OM) visualization of the photopolymer showing surface imperfections. (b) Measurement of the surface topography of the photopolymer with WLI.
The intensity of the light propagated by the WG, measured by the Thorlabs PowerMeter PM100D with the S130C detector, corresponds to approximately 1$\%$ of the intensity used for coupling to the WG, which is sufficient for the camera to record. However, the properties of the waveguide are being further optimised.
4. On-chip digital holographic Mach-Zehnder interferometer
The on-chip method (DHI-on-chip) was tested and validated by measuring a stepped transparent object and comparing the results with digital holographic microscopy (DHI-with-lens) [21] and a Zygo NewView 7200 (WLI) coherence scanning interferometer. The schemes of DHI-on-chip and DHI-with-lens are shown in Fig. 4. Differences in the approach to the design of the optical configurations lead to a significant simplification of the arrangement, a reduction in the number of optical components and the size of the measurement arrangement, but still allow comparable measurement results to be achieved.
Fig. 4. Schemes of two measurement arrangements compared in presented paper. (a) DHI-on-chip approach and (b) DHI-with-lens scheme, depicting differences in optical components used for reference wave introduction as well as arrangement size differences. Abbreviations used in the figures: polarization maintaining fiber (PM Fibre), mirror collimators (MC, MC$_1$, MC$_2$), prism (P), optical waveguide (WG), transmission diffraction grating (tDG), sample (S), camera (Cam), beam-splitters (BS$_1$, BS$_2$), mirrors (M$_1$, M$_2$), microscope objectives (MO$_1$, MO$_2$)
The DHI on-chip interferometric arrangement comprises a mirror collimator (MC) in the reference arm of the interferometer, which illuminates a symmetrical prism with an apex angle of 90$^\circ$. The light from the prism is then transmitted by total internal reflection to the further part of the waveguide (WG) and then extracted from the waveguide by first order diffraction by a diffraction grating. The object arm of the interferometer consists of a second mirror collimator (MC2) and a sample (S). The illumination from the object arm passes through the WG and interferes with the illumination from the reference arm, creating a micro-interference pattern that is detected by the uEye UI-3482LE-M camera (5 Mpix sensor with square pixels of 2.2 $\mu$m arranged in 1920 rows and 2560 columns and 3 FPS). The coherent illumination source is a Toptica DLC pro grating stabilised tunable single mode diode laser with a wavelength range of 631 nm to 635 nm. The wavelength used in the measurement was set to 631.52nm. The frequency shift $\Delta f=0.1875$ Hz between AOMs was set, see Eq. (2). The arrangement was placed on an optical table to avoid vibrations that would introduce errors into the phase shift technique. Furthermore, the arrangement can be easily modified (by tilting the WG) to a single-shot off-axis arrangement, which is inherently much more resistant to vibration. Given the short measurement time (a few seconds), temperature drifts are negligible. One of the main objectives in the design of the on-chip arrangement was to minimize the distance between the sample and the camera, as this is an important feature in achieving high resolution. The physical constraint on the camera-to-sample distance is the thickness of the waveguide. Since certain precautions must be taken to avoid damage to the waveguide and camera, the distance of the sample from the camera sensor during the measurement was approximately 2.5 mm, as shown in Fig. 5. Assuming the diffraction limit formula governing the lateral resolution $R = \lambda d/a_s$, where $a_s \approx$ 4.4 mm stands for the sensor size, the distance of $d=2.5$ mm allows to achieve the theoretical lateral resolution of $R \approx \lambda$/2. A better lateral resolution cannot be achieved due to the condition for wavefront propagation in Eq. (4). Thus, this configuration could achieve the maximum possible resolution of $\lambda$/2 using super-resolution techniques. Without the application of super-resolution techniques, the theoretical lateral resolution is driven by the pixel size of 2.2 $\mu$m.
Fig. 5. Photograph of the measurement setup showing the proximity of the sample to the camera, with the components of the setup highlighted. The abbreviation used in the figure stands for sample (S), camera (Cam) and optical waveguide (WG) with its components: prism (P) and transmission diffraction grating (tDG).
A model glass sample (MGS) with different layer thicknesses was prepared to test the performance of our measurement setup. The structure was created by depositing (SiO$_2$) layers of different thicknesses on the parallel plate glass substrate. The pattern of the sample is shown in Fig. 6(a). The target thicknesses during (SiO$_2$) deposition were approximately 190 and 320 nm for layers $L_1$ and $L_2$ respectively.
Fig. 6. (a) Scheme of the model glass sample (MGS) consisting of SiO$_2$ layers, n$_{SiO_2}$ = 1.46, with different thicknesses $L_1 = 190$ nm and $L_2 = 320$ nm deposited on the glass substrate (GS), n$_{GS}$ = 1.52, of thickness 2.3 mm. (b) Scheme of the periodically poled lithium niobate (PPLN) single crystal sample, n$_{PPLN}$ = 2.27, with thickness $h = 0.5$ mm and periodic domain pattern with half period $a = 100~\mu$m. Both sides of the PPLN have conductive ITO electrodes through which a voltage $V$ can be applied to the sample. The linear electro-optic effect produces a difference in the values of the refractive index in the alternating domains, denoted by the symbols $n_1$ and $n_2$.
While the stepped (SiO$_2$) sample is primarily used for method verification, the on-chip DHI has also been used to measure the domain structure within a ferroelectric crystal. The second sample was a 5% MgO doped periodically poled lithium niobate (PPLN) single crystal from OXIDE Corporation (Japan), see Fig. 6(b). The change in refractive index $\Delta n$ was produced by the electro-optic effect due to the external electric field $E = 800$ V/m applied to the sample through transparent electrodes (ITO) deposited on the sample surface. The refractive indices in adjacent domains in Fig. 6(b) are denoted as $n_1$ and $n_2$, where $n_1 = n-\Delta n$ and $n_2 = n+\Delta n$. The mechanism of inducing the refractive index contrast for optical measurements of the PPLN sample is described in detail in [21,22].
5. Measurement procedure and the results
In the first measurement step, a set of phase-shifted reference holograms is recorded. Next, the phase-shifted holograms with the sample are also captured. Both sets of holograms are input to the data evaluation procedure described by the scheme in Fig. 7.
First, the lateral resolution of the method was investigated by measuring the USAF 1951 resolution target. The USAF 1951 was placed at the sample position marked S in Fig. 4(a). The reconstructed intensity map is shown in Fig. 8(a). The sixth element of the sixth group on the USAF 1951, corresponding to 114 lp/mm, was resolved with a normalized modulation (contrast) of about $M\sim 0.75$. The modulation was related to the first element of the second group (denoted as 2-1 area in Fig. 8(a)) which was assigned the value $M=1$. Intersections through the sixth element of the sixth group with spline fitting are shown in Fig. 8(b) up. The second element in the seventh group (144 lp/mm) was still resolvable, but its modulation dropped sharply to 0.4, see Fig. 8(b) down. Smaller elements with lower modulation could not be resolved due to noise in the intensity image. This noise was largely due to imperfections in the diffraction grating, as the sixth element of the seventh group (203 lp/mm) could be resolved with certainty when the diffraction grating was removed. It is already close to the cut-off frequency of the sensor (227 lp/mm). In summary, the present arrangement achieves a spatial resolution of about 140 lp/mm, which corresponds to details of about 3.5 $\mu$m. Improving the quality of the diffraction grating could lead to a resolution of over 200 lp/mm.
Fig. 8. (a) The intensity image of the USAF 1951 resolution target and its central part. The rectangles denote the selected group-element areas. (b) Intersections of the sixth element of the sixth group (up) and second element of the seventh group (down). The blue markers are the measured values, while the solid lines are the data interpolated by the spline.
Once the lateral resolution of the method was established, it was necessary to verify the measurement of the optical path difference (OPD) change and to estimate the sensitivity of the method. A well-characterised stepped object (MGS) was used for this purpose, see Fig. 6(a). The MGS was placed in the sample position and measured according to the procedure described in section 2.. First, a part of the MGS sample without stepped structure was measured to obtain a reference complex field $U_{d0}(x,y)$. In the next step, the MGS region with steps was measured and the complex field $U_{d1}(x,y)$ was obtained. Hologram depicting area of the MGS sample with glass steps is displayed in the Fig. 9(a). Due to the FPS of the camera, it takes about 6 seconds to acquire the hologram sequence and the time delay between the reference measurement and the sample measurement is in the order of a few seconds. The phase change due to the sample topography was measured relative to the reference state, see (6). The obtained phase difference $\Delta \phi$ allows the calculation of OPD according to the formula:
and hence the topography mapIn Eq. (10), $n_\textrm{layer}$, $n_{0}$ are the indeces of refraction of the SiO$_2$ layer and background, respectively.
Fig. 9. (a) Full FOV hologram of MGS sample measurement by DHI-on-chip method with marked Region of Interest (ROI) denoted in yellow rectangle, profile along the stepped sample consisting of substrate (GS) and SiO$_2$ layers (L$_1$, L$_2$) denoted by green line and a Marker at the bottom of the image denoted in blue. (b) shows full FOV hologram of the DHI-on-chip method measurement of the PPLN sample denoted in yellow rectangle.
Figure 10(b) shows the topography map $L(x,y)$. The lower part of the figure shows a reference mark on the sample surface. This is used to facilitate orientation of the fine structure of the transparent MGS. For verification, the results of on-chip DHI were compared with digital holographic microscopy in transmission mode using the microscope objective presented in [21] (DHI-with-lens) and the commercially available optical profilometer Zygo NewView 7200 (WLI). It is important to note that the compared methods have different fields of view (see Fig. 10(a)) and therefore it was necessary to select appropriate areas from the measured data. The DHI-on-chip and DHI-with-lens methods had a large enough field of view to capture the entire MGS structure, whereas the WLI method required sub-aperture measurements that were subsequently stitched together.
Figures 11(a), 11(b), 11(c) display the comparison of the DHI-on-chip, DHI-with-lens, and WLI methods measurements of nearly the same MGS sample area. The thicknesses of the SiO$_2$ layers $L_1$ and $L_2$ are measured relatively to a glass substrate (GS) chosen as a reference. The sample profiles shown in 11(d) from almost the same location were calculated from the lines (blue, green, red) highlighted in the figures corresponding to the DHI-on-chip, DHI-with-lens, and WLI methods. The visualized data of the DHI-on-chip method were filtered with the average filter of dimensions 9$\times$9 pixels.
Fig. 10. (a) Comparison scheme of the FOVs of the DHI-on-chip, DHI-with-lens (with MO 4x) and WLI (with MO 5x) methods. Note that MO stands for microscope objective. (b) shows the whole area measurement of an MGS sample using the DHI-on-chip method with the pen marker in the lower part of the image.
Fig. 11. Comparison of MGS sample measurement with (a) DHI-on-chip, (b) DHI-with-lens, (c) WLI methods and (d) corresponding measurement profiles (blue, green, red) of the methods, respectively.
In order to asses the accuracy of the measurements, the measured topographic maps in Fig. 11 were used for further analysis. The histogram in Fig. 12(a) is a representation of the distribution of the measured shape data. The values are normally distributed around three significant peaks whose mean values correspond to the height of each step. The shape of the histogram indicates that this is a normal (Gaussian) distribution. Approximately 68$\%$ of the measured values lie within one standard deviation, 95$\%$ of the measured values lie within two standard deviations and 99.7$\%$ of the measured values lie within three standard deviations. The standard deviation can therefore be used to estimate the accuracy in Table 1. The results of the measurement methods (DHI-on-chip, DHI-with-lens, and WLI) are summarised in Table 1.
Fig. 12. (a) Histogram comparison of the measured values of the MGS sample of the DHI-on-chip, DHI-with-lens, WLI methods. The number of measurements corresponding to $L_1$ shown in the WLI histogram has been reduced by approximately half to emphasise the comparison of the method distributions. (b) The values of the histograms correspond to the regions GS, $L_1$ and $L_2$ marked in the figure.
Table 1. Comparison of MGS sample measurements (mean value $\pm$ standard deviation) by DHI-on-chip, DHI-with-lens and WLI methods.
As the WLI measurement is in reflective mode and the beam does not pass through coatings or float glass substrate, it cannot be used to directly compare the accuracy of the measurement. DHI with lens is more suitable for this purpose. However, the results show that the individual step heights of all methods are in good agreement. Slight differences in the measured heights may also be due to the ambiguous position of the sample or uncertainties in the refractive index values in Eq. (10). The standard deviation in the measured on-chip DHI data, which expresses the noise level, is about twice that of the DHI. This higher noise is due to the presence of the WG and in particular the photopolymer film, the homogeneity of which is not ideal (see Fig. 3), as discussed in section 3. However, it can be summarized that the on-chip DHI gives correct results with an estimated uncertainty in the OPD measurement of about 26 nm, see Table 1. The results also suggest that further development and optimisation of the WG will be able to reduce this uncertainty.
Once the on-chip DHI method was validated by measuring the stepped transparent object, it was used to visualise the domain structures of the PPLN sample, see Fig. 6(b). The large field of view of the on-chip DHI allowed visualization of the entire sample with a lateral resolution of about 3.5 $\mu$m. The reference hologram was taken at no voltage and the second hologram was taken at 400 V. The hologram of the PPLN sample with voltage applied is displayed in the Fig. 9(b). The phase change map $\Delta \phi$ was then obtained using Eq. (6). The phase change is related to the change in refractive index. Assuming a linear Pockels electro-optic effect, where the piezoelectric contribution to the refractive index change is neglected because it is assumed to be small, we can calculate the change induced by the electro-optic effect as
The full field of view of the refractive index change measurement using the DHI on-chip method is shown in Fig. 13(a). The red line then corresponds to the selected domain structure profile shown in Fig. 13(b). The visualised PPLN data has been filtered with an average filter of 9$\times$9 pixels. The measured changes in refractive index $\Delta n \approx$ 8$\times 10^{-5}$ correspond to the electro-optical effect induced in the sample by applying an external electric field of magnitude 800 V/m. Due to the large field of view, it was possible to measure the domain width and its standard deviation as $a$ = (100.1 $\pm$ 12.9) $\mu$m. The measured values agree with the nominal values specified by the crystal manufacturer.
Fig. 13. (a) The full field of view image of the refractive index variation $\Delta n_E$ in the ferroelectric domain pattern in the PPLN produced by the linear electro-optic effect by applying roughly the 400 V on the electrodes visualized by DHI-on-chip. (b) Measured profile of the refractive index variation along the red line.
6. Conclusion
In conclusion, a new method for measuring wavefront deformation has been developed based on a combination of digital holographic interferometry in a Mach-Zehnder configuration and an on-chip measurement approach. The performance of the measurements was demonstrated on a model glass object (MGO) with SiO$_2$ layers of different thicknesses and the visualization of ferroelectric domains in a periodically poled lithium niobate (PPLN) single crystal sample using the refractive index change induced by the electro-optical effect. The measurement results of the MGS sample were compared with both a traditional DHI with lens arrangement and a commercial WLI (Zygo NewView 7200) and showed good agreement. In addition, we demonstrated the ability to visualise the domain structure of the PPLN in almost the entire sample area using DHI-on-chip. Although the measured refractive index changes are affected by noise, the values show agreement with theory. In addition, the measured domain thickness is in agreement with the manufacturer’s design. The current measurement setup allows measurements over an area of about 20 mm$^2$ with a lateral and axial resolution of about 3.5 $\mu$m and 26 nm respectively. The measurement noise in transmission geometry methods is significantly affected by inhomogeneities in the sample material, internal mechanical deformations and temperature distribution compared to WLI where light only propagates through the air and interacts with the sample in reflection geometry. In addition, the difference in noise between DHI-on-chip and DHI-with-lens measurements is most likely due to imperfections in the in-house fabricated extraction grating and the quality of the photopolymer material itself. Further optimization of the method can lead to a significant reduction in measurement noise, reducing the size of the apparatus, creating the potential for the development of portable measurement devices at an acceptable low cost.
We have shown that it is possible to achieve comparable measurement results with a much simpler measurement geometry using two collimators, a waveguide and a camera, without the need for expensive microscope lenses and in a much more compact measurement setup. In addition, our measurement and data acquisition approach allows the use of computational techniques such as super-resolution algorithms to further increase lateral resolution, leading to sub-micron resolution. By illuminating the object from multiple angles, the principles of tomography can be used to evaluate the 3D internal structure of transparent objects to measure the refractive index distribution in a large volume.
We believe that a device based on the presented geometry could also be used in the field of biology and medicine for tasks such as tissue pathology, cell counting, etc. In the field of medicine, this approach has the potential to enable the development of compact, portable control devices for applications such as cell counting, visualization of pathology samples, but also, for example, the measurement of concentrations and other variables of completely transparent samples, which would allow the improvement of the quality of medical care in developing countries. The development of DHI-on-chip can accelerate research into internal structures in domain engineering, which have high application potential in optical switches, frequency conversion applications, data storage, etc.
Funding
Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/16_026/0008390); Grantová Agentura České Republiky (GACR 19-22000S).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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