1. Introduction

The common-path interferometer (CPI) is an ubiquitous tool for extracting and quantifying phase distributions of semitransparent objects. A CPI applies the same principle of phase-to-intensity transformation as amplitude-dividing interferometers (e.g. Michelson, Mach-Zender) except that the object takes part in synthesizing the reference beam. Another primary advantage offered by a CPI is its insensitivity to mechanical vibrations and ambient air fluctuations, which is characteristic of its optical geometry. Perhaps the most well-known CPI is the phase-contrast microscope (PCM) [1]. The PCM is a powerful imaging tool that enables microscopic observation of weak phase objects at diffraction-limited resolution.

Although most PCM systems employ an annular phase filter it is sufficient to describe its underlying principles under coherent illumination by considering a phase-contrast filter (PCF) placed between two lenses of a standard 4f imaging configuration as shown in Fig. 1. In the Fourier plane, the spatial frequency components of the object under test are laterally distributed. The PCF imparts a certain amount of phase shift between the central portion and the rest of the spectrum. The effect of this spatial filtering procedure is to synthesize a reference field that depends on the object field itself and the PCF’s size and phase shift. Interference between the relayed object field and the reference wave can produce a high-contrast intensity image in the output plane.

It has been common to use PCMs for qualitative visualization of phase-only objects that otherwise appear as faint intensity images under ordinary optical microscopes. Previous efforts, considered the extension of PCM’s use in quantitative phase imaging (QPI) with an aim to apply this technique for non-invasive in vivo studies of microbiological structure and dynamics, for example, of blood cells and HeLa cells to name a few [2-5]. CPI-based QPI techniques have shown superior stability [4, 5] for long-period phase measurements over the field of view without the need for raster scanning. However, previous CPI-QPI schemes described in the literature are constrained to the assumption of a planar reference wave, which limits the accuracy and the operational field of view of QPI systems.

 

Fig. 1. Common-path interferometer – a 4f imaging system with a phase contrast filter (PCF).

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In this work, we describe a simple modification to the CPI optical setup that, in principle, allows for the measurement of the generally spatially varying synthetic reference wave (SRW) associated with an object. We also provide an alternative and accurate model of the SRW based on the generalized phase contrast (GPC) analysis [6-9] of the CPI-QPI scheme. Using numerical simulations, we demonstrate the limitations imposed by the plane wave model of the SRW by highlighting considerable inaccuracies that a plane wave model introduces in the measured phase values as the reference wave departs from a plane wave. In contrast, the GPC-based QPI takes into account the general spatial dependence of the SRW, which results in more than an order of magnitude reduction of peripheral phase errors. Furthermore, we also describe a novel method of using the halo intensity profile of the CPI output interferogram to accurately estimate the amplitude of the zero-frequency component of the object.

2. Common-path interferometer and the synthetic reference wave

Microscopy typically involve optics with high numerical aperture (NA). Using a conventional microscope, quantitative phase imaging can be done by relaying the microscope output into the input plane of a common-path interferometer (CPI), such as was done experimentally in ref. [5]. The relatively lower NA in a CPI can be suitably described using a paraxial propagation model, which we adopt for analyzing a GPC-based CPI. For maximum light throughput, GPC utilizes a phase-only circular or square PCF. The PCF introduces a phase-shift θ between portions of the field that fall inside and outside a circle of radius R ₀ or a square of side-length L ₀ centered at the Fourier plane. This results in an amplitude transfer function given by for circular geometry

where $\rho =\sqrt{f_X^2+f_Y^2}$ , ρ ₀=R ₀/λf and f ₀=L ₀/λf (f is the focal length of the Fourier lens). The corresponding point-spread function (PSF) is

where $r\prime =\sqrt{x\prime +y\prime }$ . The output intensity corresponding to an input transmission function u(x, y)=|u(x, y)|exp[jϕ(x, y)] in a 4f spatial filtering setup with the above impulse response function is described by

where ⊗ represents the convolution operation. The second term in Eq. (3), given by [exp(jθ)-1]{u(x′, y′)⊗[ρ ₀ J ₁(2πρ ₀ r′)/r′]} for circular geometry, represents the SRW, which is evidently dependent on the PCF phase shift and size, and on the object. We propose a novel optical scheme on how to extract the actual SRW of a particular complex object field as illustrated in Fig. 2 (dashed beam path). As in the work by Kadono et. al. [3], a half-mirror (HM2) is inserted between lens (L1) and the dynamic PCF to deflect a measurable fraction of the focused beam to a pinhole of the same radius R ₀ as that of the PCF’s phase-shifting center. Along this secondary optical axis, we place an identical Fourier lens (L2) at a distance f from the pinhole, which results in an additional 4f setup whose output field offers an accurate measure of the convolution term u(x′, y′)⊗[ρ ₀ J ₁(2πρ ₀ r′)/r′] of the SRW. For a large class of objects, this convolution term, which is a slowly varying complex field, can be accurately measured by a Shack-Hartmann wave front sensor.

 

Fig. 2. Modified common-path interferometer with a dynamic PCF implemented with a spatial light modulator (SLM). Additional half mirror (HM2), pinhole (PH), lens (L2) enables the measurement of the complex synthetic reference wave by a Shark-Hartmann sensor (SH).

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3. GPC-based quantitative phase imaging

In a practical implementation, u(x′, y′) vanishes outside a region in the image plane defined by the magnified finite extent of the incident beam or a circular (square) input aperture of radius R (side-length L). Within this region, J ₁(2πρ ₀ r′)/r′ (or sinc(f ₀ x′)sinc(f ₀ y′)) is slowly varying in our implementation with a small PCF and approaches a constant as the PCF size is made to shrink. The convolution term in Eq. (3) corresponds to a multiplication in the Fourier plane, which represents truncation of the field within a small PCF aperture centered on the zero-frequency component. For PCF aperture sizes that are smaller than the central lobe of the Airy or sinc function generated by the input aperture on the Fourier plane, it is sufficient to approximate the field within the PCF aperture as having an Airy or sinc variation that is scaled by the complex zero-frequency value. Thus, GPC approximates the phase-contrast intensity image as [6-8]

where the zero-frequency complex coefficient is

where the surface of integration, Γ, is taken across the clear area of the input aperture and the SRW profiles, g_C(r′) and g_S(r′), for circular and square geometry respectively, are given by

and

Note that the intensity distribution outside the image aperture (or the “halo intensity”) is approximated by

In comparison to Eq. (3), Eqs. (4) and (8) ignore the strict dependence of the SRW profile on the input object u(x, y)=|u(x, y)|exp[jϕ(x, y)]. However, this model is satisfactory for most objects including those having weak variance in amplitude |u(x, y)|, i.e. weakly absorbing objects like biological cells and micro-optical elements, especially for small PCF radius R ₁.

For brevity, we limit our discussion to the case of circular geometry from this point. With our approximation, we note from Eq. (8) that the halo intensity is simply a scaled version of the squared modulus of the SRW in the region r′<R′. Experimentally, this suggests that using the same detector array already in place, an accurate and real-time averaged measurement of the zero-frequency amplitude may be obtained by considering the energy ratio

The calculation involved in Eq. (9) can be performed rapidly since data points of $g_C\left(\sqrt{{x\prime }_m+{y\prime }_n}>R\prime \right)$ obtained from Eq. (6) can be stored in a lookup table. Interestingly, the halo intensity that is habitually ignored in CPI measurements now becomes a significant part of the detected interferogram. The manner in which |U(0,0)| is determined above is a better alternative to the currently known approach [3, 5], which requires a cumbersome insertion of additional beam splitter and point detector. Equation (4) can now be rewritten as

Unambiguous measurement of the relevant phase, $\tilde{\varphi }$(x′, y′)=ϕ(x′, y′)-ϕ_U, with an unconstrained dynamic range is possible with the use of three interferograms I ₀, I ₁ and I ₂ corresponding to θ=θ ₀=0, θ ₁=π/2 and θ ₂=π, respectively. From Eq. (10), these three intensity distributions are given by

and

where the relevant phase, $\tilde{\varphi }$(x′, y′), can then be extracted from

In practical experiments, determination of |U(0,0)| by Eq. (9) is best carried out using I ₂ due to the strong halo light that accompanies this interferogram. Note that the phase image of a generally complex-valued object (i.e. those that modulate both amplitude and phase of the incident field) is obtained which can be combined with Eq. (11) to completely map the complex object.

4. Results and discussion

4.1 Limitations of the plane wave model of the SRW

Previous analyses of CPI-based phase-imaging instruments [2-5] have assumed a planar reference wave. In general, the SRW profile described here clearly indicates a radial dependence by noting Eq. (6). Using θ=θ ₂=π, Fig. 3 compares the SRW profiles obtained by FFT-based simulations and by the GPC-model, i.e. 2|U(0,0)|g_C(r′), through numerical integration of Eq. (6) and use of Eq. (9). These plots are for similar PCF sizes R ₁=15.96 µm, and for the case without an input object, u(x′, y′)=1. Here we treat different aperture radii R or different values of the dimensionless parameter η=RR ₁/0.61λf, which serves as a measure of the PCF size relative to that of the Airy disc,λ=532 nm and focal length f=30 cm. The assumption of a planar SRW is only valid if the aperture size (blue solid curve) is greatly reduced to a limited field of view [Fig. 3(a)]. If a larger field of view needs to be utilized, the radial variation can no longer be neglected [Figs. 3(b) and 3(c)]. Note that the GPC-model of the SRW (red solid curve) is in good agreement with the FFT-based result (open squares). A careful inspection of the minimal error shows that the GPC-model slightly overestimates the SRW produced by the numerical experiments. This observation can be traced to the use of a discrete grid in the FFT simulation, which produces discrete jumps in the available PCF aperture sizes that may not coincide with the optimal size. We choose the available size that is closest to the optimal value in the simulations. The results presented used a PCF size that is slightly smaller than the optimal size and, thus, corresponds to a slightly smaller η. In real experiments, such quantization errors are eliminated by fabricating high-quality input and PCF apertures. Intensity linescans of the corresponding interferograms are also shown in Figs. 3(d)-3(f) to illustrate the effect of the spatially varying reference wave. Further reduction of the aperture size makes the SRW flatter at the expense of diminished imaging field of view. Alternatively, a flatter SRW can be achieved either by decreasing the PCF radius instead of the size of the operational field of view or by using Fourier lenses with larger f. The former may be difficult to achieve due to PCF fabrication constraints, especially if an SLM is used as a dynamic PCF, and the latter may detrimentally increase the form factor and decrease the resolution of the imaging system. Another major drawback associated with the flattening and thus weakening of the SRW is the decrease in interferogram contrast, i.e. decrease in signal-to-noise ratio (SNR), which can consequently degrade phase measurement accuracy.

In the GPC-QPI scheme, calculation of the unknown phase distribution becomes more exact over the entire field of view as it accounts for the SRW profile in Eq. (14) for a range of η values. To maintain high SNR, it is also advisable to make the SRW amplitude-matched with the input object. This suggests the use of η≥0.41 [see Fig. 3(b)] with preference for the smaller value for weaker phase objects. In the succeeding section, we investigate the robustness of the GPC-based SRW description to determine categories of input objects in which the model is considered adequate.

 

Fig. 3. Amplitude profiles of the SRW for θ=π obtained by FFT-based simulation (squares) and by GPC model (red curve) for aperture sizes corresponding to (a) η=0.20, (b) η=0.41 and (c) η=0.64. Corresponding aperture-truncated input fields are also plotted (blue curve). FFT-calculated output interferograms for (d) η=0.20, (e) η=0.41 and (f) η=0.64

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4.2 Robustness of the GPC model of the SRW

A simple test object we have considered is a circular π-phase disc concentric with the aperture. Figure 4 shows the effect of changing the aperture size, proportional to η, and fillfactor F of the π-phase disc on the SRW. An increase in F causes a decrease in the SRW strength, which is due to the decrease in |U(0,0)| as expected from Eq. (5). We note that as η increases, the GPC-representation of the SRW becomes less accurate, especially for larger F. A good compromise between interferometric contrast and SRW accuracy is therefore found by choosing η≈0.41.

 

Fig. 4. SRW amplitude profiles for θ=π obtained by FFT-based simulation (squares) and by GPC model (red curve) for an input π-phase disc of different fill factor and aperture size combinations (a) η=0.41, F=0.1, (b) η=0.51, F=0.1, (c) η=0.64, F=0.1, (d) η=0.41, F=0.2, (e) η=0.51, F=0.2, and (f) η=0.64, F=0.2. Corresponding aperture-truncated input fields are also plotted (blue curve).

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4.3. GPC-based quantitative phase imaging

Using 2D FFT-based numerical simulations that implement η=0.41, we demonstrate below the procedure for reconstruction of different 2D phase objects. Let us consider the centered π-phase disc of fill factor F=0.1 as a test object. According to Eq. (14), three interferograms obtained for three different PCF phase shifts θ=θ ₀=0, θ ₁=π/2 and θ ₂=π are required to reconstruct the phase information. The corresponding interferograms are illustrated in Figs. 5(a)-5(c). The halo region typically associated with the interferometric output is separately illustrated in Fig. 5(d) for the case θ ₂=π. The integrated intensity over this halo region allows us to obtain a good estimate for |U(0,0)| based on Eq. (9). Figure 6(a) shows the obtained accurate reconstruction of the phase object and Fig. 6(b) shows the residual phase error, ε=|$\tilde{\varphi }$ _measured-$\tilde{\varphi }$ _actual|.

 

Fig. 5. The three interferograms obtained with PCF shifts (a) θ=θ ₀=0, (b) θ ₁=π/2, (c)θ ₂=π, and (d) the halo intensity I ₂(r′>R′). η=0.4 is used.

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The planar model of the SRW is treated in the numerical simulations by replacing |U(0,0)|²[g_C(r′)]² in Eq. (14) by a constant t ² _dc I_in representing the fraction of incident light that passed through a pinhole with size similar to that of the PCF. This is a numerical equivalent of the measurement by a pinhole-detector tandem described by Kadono, et al. [3]. In the planar model, we also obtain values at the level of machine precision of the simulation platform or virtually zero phase errors (i.e. 10⁻¹⁵ of a wavelength). This is expected since the term |U(0,0)|²[g_C(r′)]² (or t ² _dc I_in) in the denominator of the right hand side of Eq. (14) has insignificant effect on the calculated phase when the numerator approaches zero, 2I ₁(x′, y′)-I ₂(x′, y′)-I ₀(x′, y′)→0, i.e. for phase values that is an integer multiple of π such as in the considered example.

 

Fig. 6. Surface plots showing (a) the phase reconstruction and (b) the residual error.

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Next, to illustrate that large errors in the measured phase indeed occur when using a plane wave model of the SRW, we consider a test object comprised of alternating π/2 and -π/2 phase discs with total fill factor F=0.2 positioned close to the edge of the aperture in a circular arrangement. The three interferograms for this particular case are given in Fig. 7(a)-7(c). When assuming a planar SRW, a considerable error is found in the calculated phase as evidently seen in Fig. 7(d). In terms of percent deviation from the true phase value of ±π/2, a maximum phase error of about 37.4% is observed, or about λ/12. On the other hand, the GPC-based QPI scheme considerably suppresses the error to a minimal amount of 1.3%, or ~λ/306.

 

Fig. 7. Interferograms for an object consisting of alternating π/2 and -π/2 phase discs obtained with PCF shifts (a) θ ₀=0, (b) θ ₁=π/2, (c) θ ₂=π and plots comparing the residual phase error obtained when the (d) planar and (e) GPC model of SRW are assumed. η=0.4 is used.

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Another interesting phase object we have considered is a helical phase front with a circular obstruction at its center. Motivations for this choice of test object include the fact that a helical or vortex phase contains phase values over the full 2π cycle. Secondly, with this object’s tractable Fourier transform, which consists of a zero-order beam surrounded by an annular beam [10] whose radius increases with the topological charge ℓ, we are able to perform a test on the accuracy of the GPC-QPI method as a function of the object’s spatial frequency content. Figures 8(a)-8(c) depict the output intensities corresponding to the three interferometric measurements. Following the procedures we have outlined above, we obtain the residual error measurements corresponding to the planar SRW assumption and the GPC-QPI method as shown in Figs. 8(d) and 8(e) for the case where the unobstructed annular region of the helical phase with ℓ=10 covers a fill factor F=0.2. In this particular case, we noted maximum peripheral errors of ~λ/9 and ~λ/505 for the planar and the GPC model, respectively. This corresponds to an impressive accuracy improvement in excess of a factor of 50. Furthermore, we investigated the effect of varying the topological charge ℓ on the observed accuracy, represented by ε. We noted that both the planar and the GPC-based SRW representation produce larger values of ε for smaller ℓ as shown in Fig. 8. This is anticipated from the fact that the tails from the higher spatial frequency component (in this case, from the annular beam) get closer to the central region of the PCF as ℓ decreases but remain unaccounted for by either model. Nevertheless, the GPC-model still results in about an order of magnitude improvement even for ℓ=1.

 

Fig. 8. Interferograms for an obstructed helical phase of charge ℓ=10 obtained with PCF shifts (a) θ ₀=0, (b) θ ₁=π/2, (c) θ ₂=π and plots comparing the residual phase error obtained when the (d) planar and (e) GPC model of SRW are assumed. η=0.4 is used.

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Fig. 9. Maximum peripheral phase error as a function of the topological charge of a centrally obstructed vortex phase object.

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Our observation that phase errors for the plane wave SRW model can reach up to several tens of nanometers for optical wavelengths substantiate those described in previous experimental and numerical investigations [11, 12]. In those works, experimental verification of the accuracy of QPI schemes was carried out using a well-known topometry measurement technique with an atomic force microscope (AFM). In Ref. 10, a new type of CPI was used called the spiral phase contrast microscopy (SPCM) that employs a spiral phase filter. Like in the conventional Zernike phase contrast approximation, SPCM assumes that the reference wave emanating from the transmissive center of the spiral filter is a plane wave. It is interesting to note that phase values measured experimentally under this approximation underestimate those from AFM measurements by up to 40%. This underestimation in the measured phase is also observed with the numerical results for planar SRW we have presented above. The work of Wofling, et. al. [12] also showed that errors can be significantly suppressed if the object-dependent and spatially varying SRW profile can be somehow extracted from the interferometric information. At the expense of computational cost, it was shown that the SRW profile can be better approximated as an n-th degree polynomial through nonlinear optimization algorithms. Higher integer n results in better accuracy but longer computational time. In contrast, the proposed GPC-QPI scheme prescribes a semi-analytic representation of the SRW that thereby enables rapid extraction of the phase image through Eq. (14), especially for most objects that weakly perturb the phase, such as microorganisms [2] and erythrocytes [4, 5]. Our numerical results have also revealed that the GPC model of the SRW can produce QPI accuracy down to ~1 nm level without resorting to computationally intensive iterative methods, for objects with sufficient separation between low and high spatial frequency components.

5. Summary and outlook

The foregoing analysis and the supporting numerical results point to bounds on the achievable accuracy that are imposed by the theoretical models used as basis for processing the phase contrast output to achieve quantitative phase imaging. In an experimental implementation, this model-based error will persist and can become a limiting factor even after accounting for and correcting the other sources of error, such as noise and detector constraints (e.g., gray-level quantization and nonlinearities), among others. Through FFT-based numerical experiments, we demonstrated a new procedure for quantitative phase imaging based on the GPC method that significantly reduces the model-based error. We primarily showed the ramifications brought about by the conventional plane wave approximation to the synthetic reference wave in common path interferometry. It was shown that the widely used planar SRW model tends to limit the effective field of view for accurate QPI using CPIs as it can lead to considerable amounts of error in phase depth measurements near the aperture periphery. Conversely, we showed, for several test objects, that the GPC-based SRW model can lead to a superior phase measurement accuracy of ~1 nm without the need for iterative calculations. Thus, GPC-based QPI extends the nanometer-accuracy over the entire aperture while maintaining rapid acquisition rates [5] – making it possible to accurately and simultaneously probe dynamics of multiple biological specimen in a colony over the entire field of view. Other variants of the common path interferometer recently described in the literature, including SPCM and diffraction phase microscopy (DPM) [13] have also assumed a plane reference wave when reconstructing a phase image. We believe that similar extensions to the model of the reference wave in SPCM and DPM, which both employ a finite-sized region in spatially filtering the zero-order beam, can also result in considerable enhancement in the overall accuracy of these methods. We also showed that, by virtue of the typical phase reconstruction formula of Eq. (14), the sensitivity of phase error on the accuracy of the SRW estimation depends on the actual phase stroke being measured.