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Abstract
Most optical characterization methods rely on measuring the complex optical fields emerging from the interaction between light and material systems. Nevertheless, inherent scattering and absorption cause ambiguities in both interferometric and noninterferometric attempts to measure phase. Here we demonstrate that the complete information about a probe optical field can be encoded into the states of polarization, and develop a topography measurement method by blindly varying the ambient refractive index surrounding the sample in a wedged cuvette, which is capable of simultaneously measuring the thickness and the ambient refractive index of the sample in real time, as well as extending the measurement range of the sample thickness. With the method, we have successfully measured the topography of a 136.7 µm thick coverslip by blindly changing the ambient refractive index by 0.001246, resulting in the thickest sample characterization ever achieved by quantitative phase imaging, to the best of our knowledge. An efficient and complete characterization of optical fields is critical for any high-resolution imaging approach and the technique demonstrated here should prove attractive for applications ranging from microscopy to remote sensing. Thanks to the high precision and fast response speed, this method may pave a new way for measuring the topography of the thick samples, such as biological tissues.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical imaging generally involves detecting the changes in amplitude, phase, or both caused by the interaction of complex optical electrical field with a target. However, because radiation detectors are usually phase insensitive, most common techniques are capable of reproducing only the amplitude variations. Measuring phase is more involved because it is a relative measure that characterizes an oscillation with respect to a reference point in time or space. A common use of this concept is in describing the wavefront of electromagnetic waves, which designates the geometric locus of points of equal phase. This implies that the phase ${\varphi _p}({\boldsymbol \rho } )$ across a wavefront can only be measured relative to a reference phase distribution ${\varphi _r}({\boldsymbol \rho } )$. In other words, one always measures a phase difference $\varphi ({\boldsymbol \rho } )= {\varphi _p}({\boldsymbol \rho } )- {\varphi _r}({\boldsymbol \rho } )$ with respect to a pre-determined, reference wave ${\mathrm{{\cal E}}_r}({\boldsymbol \rho } )= |{{\mathrm{{\cal E}}_r}({\boldsymbol \rho } )} |{e^{i{\varphi _r}({\boldsymbol \rho } )}}$. Regardless of the actual implementations, the measurement provides qualitative or quantitative information about this phase difference between the probe and the reference fields.
In general, quantitative phase measurements across a spatially extended wavefront can be implemented using either non-interferometric [1–7] or interferometric techniques [8–12]. Non-interferometric methods do not require a physical reference field. Instead, they rely on complex phase retrieval algorithms that exploit the relation between the phase and the intensity of a propagating wave. For instance, one may iteratively recover a phase image by measuring intensity at multiple axial distances [1,2] or by measuring multiple phase-coded diffraction patterns at a known axial distance [5]. Even though easy to implement, these methods are less efficient for targets that introduce both amplitude and phase variations. One may also use intensity-based wavefront sensors that quantify the phase variation across a wavefront at the cost of sacrificing the spatial resolution [6]. In all these methods, the phase is measured with respect to an ad-hoc reference provided by the object field itself.
Interferometric techniques can be integrated within imaging systems and can be further classified either as off-axis [8,9,11] or phase shifting methods. The former exploits a reference beam with slightly different propagation wave vector to generate interference fringes that, subsequently, are used to acquire the desired phase information [8,9,11]. Unfortunately, one needs several fringes for determining the phase, which limits the achievable spatial resolution. On the other hand, the phase shifting approaches rely on additional elements with adjustable phase distributions such as spatial light modulators (SLM) or programmable phase modulators (PPM) to control the phase across the reference beam [10,12–14]. As a result, the optical systems are usually rather complex and their spatial resolution is limited by the ability of these externally controlled elements to modify the phase across the wavefront. Recently, a scheme of encoding phase into polarization was proposed, which introduced the polarization dimension instead of phase modulation [15,16].
On the other hand, since phase is influenced by both thickness and refractive index, one needs the whole refractive index distribution to get a topography map from quantitative phase imaging (QPI). To solve this conundrum, Mohanty demonstrated a method for decoupling geometric thickness from refractive index by changing the ambient medium [17], which makes topography measurements possible without knowing the refractive index of sample. Unfortunately, in practical applications, it is difficult to rapidly change the medium without influencing other conditions especially in biomedical measurements. To improve the practicability, in this paper, we derive the process of quantitative phase imaging (QPI) by phase to polarization encoding (P2P), as well as demonstrate a method of topography measurements by blindly varying ambient refractive index in a wedged cuvette. This method is dubbed P2P-QPI, which is capable of simultaneously measuring the thickness and ambient refractive index of the sample in real-time and effectively extending the axial height to hundreds micrometers. Using this method, one can get the topography of an object with arbitrary axial heights from an ordinary QPI system.
2. Phase to the polarization encoding (P2P)
In transversal electromagnetic fields, the state of the polarization is determined by the relative amplitude and the phase difference between components of the oscillating field along two orthogonal directions ${\hat{e}_\parallel }$ and ${\hat{e}_ \bot }$. Therefore, the state of polarization (SOP) can be changed by introducing an additional phase between the two orthogonal field components. Here we exploit this property to encode phase information into changes of the SOP. Thanks to the introduction of a new dimension, this phase encoding can be implemented independently across the field.
Let us analyze more rigorously this phase encoding process. First, one creates two replicas of a coherent beam, which are linearly polarized along the two orthogonal directions. One of the two beams is used as a reference ${\varepsilon _r}(\boldsymbol{\rho } )$ and the other one, ${\varepsilon _p}(\boldsymbol{\rho } )$, as a probe that interacts with the target to acquire the sought-after phase (φ) and amplitude (a). These two beams can then be overlapped to create a superposition field
As can be seen, the field contains two terms, which correspond to the linear and circular states of polarization (SOPs). Figure 1(a) shows the SOP as the superposition of two orthogonal linear polarizations, while Fig. 1(b) shows the SOP as the superposition of two orthogonal circular polarizations. Because both the amplitude and phase affect the ellipticity and the orientation θ of the polarization ellipse, by using a simple polarimetric measurement, it is difficult to distinguish between the phase or the amplitude variations encoded in the probe beam.
Fig. 1. Representing an arbitrary SOP as the superposition of two orthogonal polarizations. (a) superposition by two orthogonal linear polarizations; (b) superposition by two orthogonal circular polarizations.
This limitation can be circumvented by passing the light through a quarter wave plate (QWP) oriented at ±45° from the reference and probe SOPs, respectively, which converts the reference and the probe fields into the circular SOP with opposite handedness. In this case, the superposition field becomes
As it is clear from Eq. (2), the first part represents linear polarization, while the second is circular. As shown in Fig. 1(b), due to the components in all direction of circular polarization are equal, the direction of long axis only depends on the phase. Moreover, the ellipticity of the SOP is only affected by the amplitude modulation of the target. Therefore, the phase and amplitude of probe field are encoded into orientation and ellipticity of SOP, respectively.
In principle, measurements corresponding to three different orientations of a linear polarizer should suffice to evaluate unambiguously the desired phase [18]. However, any measurement is affected by noise or systematic errors [19,20]. To address this issue, one can use four intensity measurements corresponding to four separate polarizer orientations angled at 0, π/4, π/2, 3π/4 to recover both the amplitude and the phase information as
The schematic diagram of information retrieving procedure is shown in Fig. 2, for a generic target with phase distribution of ‘peppers’ and amplitude of ‘baboon’, as shown in (a) and (b) respectively, which are standard grayscale test images with size of 512 × 512 pixels and 8 bits per pixel (BPP). Without interference, we only get the amplitude information as shown in (b). To realize polarization encoding, as discussed in Eq. (2), this complex field is superposing with a coherent plane wave with two orthogonal circular polarizations, with the interferograms of different linear polarization components shown in (c1-c4), which can also be understood as four phase-contrast images with different phase shifts. It is easy to see that phase-contrast images enhance the phase information as well as reduce the influence of amplitude. Therefore, as a classical type of microscopy, phase-contrast microscopy has been widely used in the last half century. (d1-d4) are the Stokes parameters S0, S1, S2, and |S3| of the superposition field. (e) and (f) are resolved phase and amplitude with the same resolution of pixels and BPP. Theoretically, phase recovery Eq. (4) is not affected by intensity variation, however in simulation and experiment, the precision of phase recovery is indeed affected due to the limitation of BPP of image and CCD. (g), (h) are the 2D phase error map and histogram, respectively. The resolved complex field shows that although the amplitude modulation has a very complex distribution as well as a high absorption coefficient, the QPI result shows a high accuracy of λ/467.
Fig. 2. The simulation of complex field retrieving procedure. (a) phase; (b) amplitude; (c1-c4) intensity maps at χ=0, π/4, π/2, 3π/4; (d1-d4) Stokes parameters S0, S1, S2, and |S3|; (e) recovered phase; (f) recovered amplitude; (g) phase error; (h) histogram of the phase error.
3. Experimental demonstration
We demonstrate the concept of encoding the phase into polarization in a wide field transmission microscopy where the phase target is placed in one arm of a Mach-Zehnder interferometer as schematically illustrated in Fig. 3. In this geometry, an expanded beam of He-Ne laser with wavelength λ=632.8 nm is used as the light source. To generate the probe and the reference fields with two orthogonal polarizations, the linear polarized laser beam is first rotated using a half wave plate (HWP) with π/4 radians and then divided by a polarized beam splitter (PBS). After interaction with the target, the reference and probe fields are recombined using a non-polarizing beam splitter (NPBS), and pass through a quarter-wave plate (QWP) that projects the two field into circular states of the polarization with opposite handednesse. Finally, the intensity distributions are measured by a Pol-CCD in which each sensing element has for four polarizing pixels in different orientations. In particular, the Pol-CCD deploys Sony polarization image sensor IMX250MZR, which allows single-shot with four linear polarization images of 0°, 45°, 90° and 135°. In our experiment, the probe field is collected through an objective lens OL with 16X magnification and NA = 0.4.
Fig. 3. Experimental setup for the proposed quantitative phase imaging. HWP is for adjusting the powers in the reference (upper) and the probe (lower) arms of the interferometer and QWP is oriented ±45° from the SOPs of the reference and probe beams, respectively. HWP: half-wave plate; QWP: Quarter-wave plate; PBS: polarization beam splitter; NPBS: non-polarization beam splitter; Pol-CCD: polarization CCD with a polarizer array shown in the inset.
To test the sensitivity of the scheme, we measured a micro-wedged cuvette, which was handcrafted with a bare single-mode fiber with the buffer coating removed and a pair of glass slides. Figure 4(a) shows a SM fiber with a 125 μm-cladding without the buffer coating for making a micro-wedged cuvette with two glass slides of high flatness, as shown in (b). The length (L) between the fiber and wedge is 7 cm, therefore the wedge angle θ is 1.7857 mrad. Next, slowly inject deionized water with a refractive index ${n_0}$ of 1.3317 into the wedged cuvette. Due to the tiny wedge angle and small volume, the water is very stable inside the wedged cuvette. When a light beam passes through the wedge, a phase difference φ will be produced across the beam width, which can be expressed as
where n0 and nair are the refractive indices of water and air, respectively. l is the vertical distance from the bottom of beam. Note that Eq. (5) is actually the differential phase image of the wedge between two measurements, one with the wedge filled with air and the other filled with water for extracting out the background imperfections in the imaging system. Figure 4(c) shows the measured phase as a function of l, from which the wedge angle is derived to be 1.7737 mrad, which is 0.012 mrad smaller than the actual value with a remarkable accuracy of 99.33%. The range of l is chosen to be 80 µm, which is the range limit determined by the field-of-view on the image plane, and the corresponding cuvette thickness as a function of l can also be derived from the phase measurement data, which is presented by the second vertical axis on the right-hand side of the figure. Moreover, the standard deviations of repeated measurements of the phase and thickness are 10.96 mrad and 3.33 nm, respectively. Comparing with traditional phase-shifting methods [10,12–14], our proposed phase encoding method not only simplifies the experimental construction, but also enables the realization of single-shot on-axis quantitative phase imaging with high stability and accuracy.Fig. 4. Schematic diagram of using a single-mode (SM) fiber to make a micro-wedged cuvette. (a) A fiber with a cladding diameter of 125µm was obtained by stripping off the coating layer of a SM fiber; (b) a micro-wedged cuvette was formed by using two glass slides and the SM fiber with the buffer removed; (c) QPI measurement result of the micro-wedged cuvette.
In the demonstration above, the refractive index inside the wedged cuvette is known such that the thickness information of the cuvette can be extracted easily. However, in practice, the measured phase generally includes the contributions from both the refractive index and the thickness of the sample to be measured. In order to decouple them, the method described in [17] can be used. In particular, two measurements using two different media of known refractive indices surrounding the sample inside a cuvette can be conducted. Without prior knowledge of the sample, the thickness can be decoupled from the refractive index of the sample using the equation
where, n1, n2 are the refractive indices of the two media, while $\Delta {\phi _1}$, $\Delta {\phi _2}$ are the measured phases in the corresponding media, respectively. To verify this, a standard photolithography and subsequent radio frequency magnetron sputtering were used to deposit a nanoscale HfO2 thin film on a glass substrate with a small square step dip at the center, as shown in Fig. 5(a). A ceramic HfO2 target with 99.99% purity was used for the deposition of about 120 nm-thick HfO2 film using 30 sccm Ar flow at 2 Pa. The sputter power was 100 W and a 100 V DC bias was applied. As shown in Figs. 5(b1) and 5(b2), by placing the HfO2 thin film with a square step dip inside a cuvette as the sample for measurement and using two different ambient media surrounding the sample, air (${n_1} = 1$) and water (${n_2} = 1.3317$), we have acquired the quantitative phase images of the sample. The average phases of the thin film sample are 1.1939rad and 0.7799 rad in the media of air and water, respectively, as shown in Figs. 5(c) and 5(d). Without knowing the refractive index of the thin film sample, the geometric thickness of 125.71 nm is obtained with Eq. (6), which is highly consistent with the average film thickness of 128.96 nm measured with an atomic force microscopy (AFM), as shown in Fig. 5(e) and 5(f). Note that Figs. 5(d) and 5(f) correspond to the areas (d) and (f) marked in Fig. 5(a), while Figs. 5(c) and 5(e) are the curves obtained by averaging the data in the dashed box areas in Figs. 5(d) and 5(f), respectively, in which the location of the thin film step set to be zero. One may notice that there are many elliptically shaped defects present near the edge of the thin film step, which might be generated during the preparation of the film step. The sizes of the defects are less than 500 nm and the average height is less than 20 nm. Moreover, because the sizes of these defects and distances between them are smaller than the spatial resolution of QPI, they can be barely observed individually in the quantitative phase images.Fig. 5. Thickness measurements of thin film by using QPI and AFM. (a) A nanoscale HfO2 film on a glass substrate with a square step dip; (b1, b2) the HfO2 thin film of thickness d with a step dip placed inside a cuvette surrounding by air and water, respectively; (c) phases measured in air and water, respectively, in which the solid lines are obtained by averaging the corresponding data in the marked area shown in (d), with the step height for each curve obtained by subtracting the average phase in the range of 10 to 25 µm from that in the range of -10 to -25 µm, the bands on the curves are the standard deviations; (d) 3D map of the thin film in the blue box in (a) measured with QPI method, in which the data presented in (c) is obtained by averaging the data in the marked area; (e) thickness distribution measured with an AFM, obtained by averaging data in the marked area in (f), with the step height obtained by subtracting the average thickness in the range of 2 to 6 µm from that in the range of -2 to -6 µm; (f) AFM image of the thin film in the vicinity of the step edge, as shown in the red box in (a).
Unfortunately, when the real phase is larger than 2π, a wrapped phase instead of real phase is retrieved using Eq. (6). The relationship between the retrieved phase φi and the real phase $\Delta {\phi _i}$ (i = 1, 2) can be expressed as $\Delta {\phi _1}\textrm{ = }{\varphi _1}\textrm{ + }2p\pi$, $\Delta {\phi _2}\textrm{ = }{\varphi _2}\textrm{ + }2q\pi$. From these equations, if one wants to get the thickness d without ambiguity, p = q must be satisfied by fine-tuning the refractive indexes n1 and n2. Combining our wedged cuvette technique with the decoupling method, we designed an approach for measuring the phase of the sample and the ambient refractive index around the sample simultaneously, as shown in Fig. 6. In particular, with a plane wave transmitting through the sample in a wedged cuvette, we obtain the phase of the sample as well as the tilted background phase with a slope of α, which contains the information of the ambient refractive index. By mixing the liquid medium with another liquid medium of a different refractive index, within a short time, the phase difference between the sample and background changes. Meanwhile, a series of phase images corresponding to different ambient refractive indices due to the dynamic mixing process can be obtained. From the series of quantitative phase images, one can always identify a pair of phase images with suitable ambient refractive indices such that not only is the phase difference between them sufficiently large to enable good images, but also sufficiently small to satisfy the condition of p = q. Because the field-of-view of the sample only covers a small fraction of the cuvette such that a locally uniform ambient refractive index can be assured.
Fig. 6. Schematic diagram for measuring the phase of the sample and the ambient refractive index simultaneously with a wedged cuvette. (a) A plane wave transmits through the sample in a wedged cuvette in which the refractive index surrounding the sample changes with time; (b) the sample phase φ and the tilted background phase with a slope of α.
Using the method described above, we put a defective coverslip in an unknown wedged cuvette vertically, with one side against one of the cuvette walls. In order to minimize cuvette’s impediment to the liquid diffusion inside the cuvette, one of the coverslip corners is oriented toward the apex of the cuvette without touching the opposite wall. The top corner of the coverslip is kept above the liquid with a small piece of adhesive tape affixing it to the cuvette wall to keep the coverslip stationary during the measurement process, especially when injecting liquid into the cuvette.
First, we obtain the angle of the wedge by injecting water into the cuvette following the procedure of Fig. 4, with the phase image of the coverslip shown in Fig. 7(a). Next, in order to prevent the refractive index from changing abruptly and keep the refractive index uniform in the field-of-view, we slowly inject a small amount of diluted ethanol solution in the cuvette using a syringe with a fine needle. The refractive index of pure ethanol is 1.3604, which is slightly higher than that of water. The diluted ethanol gradually diffuses into the water and increases the refractive index at the location of the sample. A series of quantitative phase images are taken every 10 seconds and analyzed sequentially to obtain the phases of the refractive indices around the coverslip as the refractive index of the mixed solution changes with time. Finally, using the eighth phase image Fig. 7(b) with the ambient refractive index difference of 0.001246, we recover the topography of the coverslip, as shown in Fig. 7(c). The phase jitter of the boundary is due to the diffraction of light. Figure 7(d) shows that the thickness of the coverslip obtained from the phase measurement is 136.7 µm, which is about 200 times larger than the wavelength. In addition, as shown in Fig. 7(e), the 3D topography map clearly shows that the defect of the coverslip appears to be a slope shape, which can also be measured quantitatively. Finally, Fig. 7(f) shows that the cross-section thickness of the coverslip is 141.51 µm, obtained by directly taking a photo of the edge of the coverslip. The corresponding relative error between the phase imaging and direct measurements is 3.4%. The results demonstrate that our P2P-QPI method described in this paper is quick and simple to measure the topography of thick samples with high accuracy.
Fig. 7. Thickness measurement of a defective coverslip. (a) One of the real time phase images of a coverslip in the wedged cuvette in water taking by the pol-CCD; (b) one of the phase images after injecting diluted ethanol solution; (c) 2D thickness map obtained from the phase images of (a) and (b); (d) 2D plot of thickness vs. position along the dashed line in (c); (e) 3D thickness map showing the flat surface and the defective edge, with the dashed line showing the relative location of (d); (f) thickness of the coverslip measured by directly taking a photo of the coverslip’s edge.
4. Conclusion
In conclusion, we have demonstrated that the phase and amplitude information of a sample can be translated effectively to the SOP of a light field. This polarization encoding method requires neither spatial interrogation of interference firings, nor an external controllable component such as a SLM or PPM. The measurement can be implemented using different approaches for determining the SOP, which provides the complete information of the phase and the amplitude distributions of the sample across the probe field. In addition, we have demonstrated a novel method of topography measurements by blindly varying ambient refractive index in a wedged cuvette, capable of simultaneously measuring the thickness and the ambient refractive index of the sample in real-time, which enables the effective extension of the measurable sample thickness to hundreds of micrometers. It is important to note that the spatial non-uniformity of the refractive index may cause image distortions, which can be minimized by using the diluted ethanol solution with much smaller index difference from the water to change the ambient refractive index. Finally, we have successfully demonstrated the measurement of the topography of a 136.7 µm thick coverslip by blindly changing the ambient refractive index surrounding the sample by 0.001246. Sample thickness measurement of more than 1 mm may be achieved if the ambient refractive index can be more finely varied, which is being verified in our continued investigation.
Note that it is also possible to measure the exact refractive index of the sample by injecting a solution with a refractive index higher than that of the sample such that as the solution mixes with the water in the cuvetter, at some movement in time, the surrounding refractive index exactly matches that of the sample, which can be identified by a null differential phase image. This matching surrounding refractive index can be accurately obtained with the method described in this manuscript. Further experiments are under the way to verify this capability. Due to the high accuracy and fast response speed, this P2P-QPI method may pave a new way to measure the topography of the thick samples, such as biological tissues.
Funding
Natural Science Foundation of Hebei Province (A2019201347, F2021201016); National Natural Science Foundation of China (62105091).
Acknowledgment
Shengwei Cui acknowledges Prof. Airstide Dogariu’s guidance and discussion while studying at the University of Central Florida.
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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