Digital holography has emerged as a powerful tool for 3D shape measurement, finding extensive applications across various scientific and engineering disciplines due to its ability to capture detailed spatial information of an object in a non-destructive manner[1-4]. By recording the phase modulation of light reflected or transmitted from an object onto a charge-coupled device (CCD) or complementary metal–oxide–semiconductor (CMOS) sensor, digital holography facilitates the retrieval of both amplitude and phase information through numerical reconstruction of the hologram. This process yields a high-resolution, wide-field reconstruction of the object[5, 6]. However, the phase values obtained in digital holography are inherently wrapped within the range of [-π,π], posing challenges when the optical path difference (OPD) exceeds the wavelength of the light source, leading to 2π phase ambiguities.

To eliminate these ambiguities, dual-wavelength holography has been introduced, extending the depth range of 3D profile measurements beyond that of single-wavelength holography[7]. Although this technique allows for deeper profiling, it still requires phase unwrapping to accurately reconstruct the true 3D morphology of the sample. Additionally, digital holography typically employs laser sources, which are characterized by high coherence[8-10]. While coherent sources are beneficial for creating high-contrast interference patterns, they are also highly susceptible to imperfections in the optical path, resulting in coherence noise that significantly degrades the quality of the reconstructed images. This issue is further compounded in dual-wavelength holography, where speckle noise from coherent sources can amplify the coherence noise, thereby diminishing the optical quality of the final 3D profile.

To mitigate coherence noise, partial coherence sources have been employed in digital holography systems[11, 12]. While these sources reduce the impact of coherence noise, they do not fully resolve the challenges associated with 2π phase ambiguities and the necessity for phase unwrapping. In this study, we propose a novel 3D measurement method that utilizes the reconstructed amplitude image in a partial coherence digital holography microscope (DHM) system, bypassing the need for phase unwrapping to obtain accurate 3D information. By leveraging the degree of coherence modulated by the OPD of the object, our method enables the direct calculation of 3D information from the reconstructed amplitude image. We demonstrate the efficacy of this approach experimentally, highlighting its potential for high-precision 3D measurements in applications where conventional methods are limited by coherence noise and phase ambiguities.

A hologram results from the interference between object wave EO emanating from the object and a reference wave ER in laser based digital holography, given as Eq. (1).

where λ is the wavelength of light source, OPD is the optical path difference between reference wave and object wave[3, 4] and γ(x,y) is the degree of the coherence[13]. The degree of the coherence is defined as

The visibility of the interference patterns is conventionally defined, V=Imax-IminImax+Imin, which turns out be V=|γ| if the intensity of the object wave is equal to intensity of the reference wave[13, 14]. In digital holography employing a coherent light source, the coherence length typically exceeds the optical path difference (OPD) across the hologram plane, resulting in a uniform degree of coherence, denoted as γ(x,y)=γ, over the entire hologram plane. Under these conditions, the degree of coherence remains constant, ensuring consistent visibility of the interference pattern across the hologram. However, when a partial coherence light source, such as an LED, is utilized in digital holography, the degree of coherence becomes dependent on the OPD, potentially altering the visibility of the interference pattern across the hologram plane.

In digital holography microscopy (DHM) with a partial coherence source, the visibility of the interference pattern is directly influenced by the degree of coherence, which in turn affects the reconstructed amplitude image. This relationship arises because the diffraction efficiency of the wave is linked to the visibility of the interference patterns[13-18]. Consequently, the intensity of the reconstructed amplitude image becomes a function of the degree of coherence, and this dependency can be exploited to extract 3D information from the reconstructed amplitude image directly. This approach offers a distinct advantage, as it allows for the calculation of 3D information without the need for phase unwrapping, thereby simplifying the measurement process and potentially improving the accuracy of 3D reconstructions in DHM systems using partial coherence light sources.

If a reconstruction plane is placed at a distance d behind the hologram, a real image is formed on it. Mathematically, the amplitude and phase distributions in the plane of the image can be found using the Fresnel–Kirchhoff integral[3, 4],

where (ξ,η) are the coordinate of reconstruction plane and pixel sizes are Δξ=λdNΔx and Δη=λdNΔy[1, 2].

Because Ψ(ξ,η) is an array of complex numbers, one can obtain the phase image by calculating the argument

The phase values are limited in [-π,π]. The amplitude image was calculated using the intensity

The reconstructed amplitude image in digital holography using partially coherent light sources inherently contains information about the degree of coherence of the hologram. In low coherence digital holography, the intensity of the reconstructed amplitude image is a function of the degree of coherence and the optical path difference (OPD). As a result, it is possible to extract the three-dimensional (3D) information of the sample directly from the reconstructed amplitude image, without relying on phase information. This method eliminates the need for phase unwrapping and allows for accurate 3D measurements in environments with varying coherence, thereby enhancing the versatility and application range of digital holography.

Figure 1 illustrates a schematic of a slightly off-axis reflection-type partial coherence digital holographic microscope (DHM). The experimental setup is fundamentally similar to that of a Michelson-type interferometer. A light-emitting diode (LED) serves as the illumination source, characterized by a center wavelength of 630 nm and a full width at half maximum (FWHM) of 20 nm. The phase-shifting method was employed for image reconstruction, which allows for high-quality reconstruction of holographic images. However, phase-shifting digital holography is not well-suited for real-time measurements of dynamic objects due to its sequential acquisition process. This sequential acquisition makes the system highly susceptible to external environmental changes, complicating phase measurement and reducing its robustness.

Figure 1. (Color online) Schematic experimental set-up for slightly off-axis reflection-type digital holographic microscope.

To address these limitations, it is necessary to capture multiple phase-shifted holograms (PSHs) simultaneously. This can be achieved using a polarization imaging camera (PIC) and circular polarization techniques. Specifically, by right-circularly polarizing the reference beam and left-circularly polarizing the object beam, and then combining these beams, phase-shifted holograms can be captured using the PIC. The PIC is capable of simultaneously measuring four polarization states, enabling the simultaneous acquisition of four phase-shifted hologram datasets[19, 20]. This approach significantly enhances the capability of digital holography systems to perform real-time 3D measurements, even in dynamic environments.

We investigated the relationship between the optical path difference (OPD) and the intensity of the reconstructed image in a partial coherence digital holography system. To explore this dependence, holograms were recorded while varying the OPD by adjusting the position of the reference mirror in the experimental setup. The results are presented in Fig. 2. Figure 2(a) shows the hologram of a USAF resolution target, Fig. 2(b) provides an enlarged view of the area marked by the white box in Fig. 2(a), and Fig. 2(c) depicts the fringe profile along the dotted line in Fig. 2(b). From the fringe profile in Fig. 2(c), we calculated the visibility of the interference patterns, which was found to change with the OPD. The OPD was systematically varied by translating the reference mirror.

Figure 2. OPD and the intensity value of the reconstructed amplitude image. (a) hologram created by 630 nm LED; (b) expanded hologram image; (c) profiles along the line in (b); (d) intensity value of the reconstructed amplitude image with OPD.

Figure 2(d) illustrates the gray level of the reconstructed amplitude image corresponding to the region within the white box in Fig. 2(a) as a function of OPD. The reconstructed amplitude images were computed using Eq. (5) based on the recorded holograms. As predicted by the theoretical framework, Fig. 2(d) demonstrates that the gray level of the reconstructed amplitude image varies with the OPD in a partial coherence digital holography system. Notably, the degree of coherence reaches its maximum value when the OPD is approximately 35 μm for this system. In Fig. 2(d), the filled circles (●) represent the gray level obtained directly from the hologram, while the solid line represents the results of a Gaussian fit to these data.

These findings confirm that the intensity of the reconstructed amplitude image is indeed dependent on the OPD. The strong agreement between the measured data and the fitted model suggests that this approach can be effectively used to determine the height of a sample by comparing the measured gray level values with the theoretical model.

The three-dimensional (3D) measurement of a stepwise sample was experimentally conducted using a partial coherence digital holography system, with the results presented in Fig. 3. Figure 3(a) displays the hologram of the sample, while Figs. 3(b) and 3(c) show the reconstructed phase and amplitude images, respectively. Figure 3(d) illustrates the profile along the white dotted line in Fig. 3(c), where the annotated values indicate the average gray values of the reconstructed amplitude image.

Figure 3. (Color online) Reconstructed image and profile. (a) hologram; (b) reconstructed phase image; (c) reconstructed amplitude image; (d) profiles along the dotted line in (c).

The measured gray values of the reconstructed amplitude images are 188, 150, 138, 129, and 126, with differences between successive steps of 38, 12, 9, and 3, respectively. By comparing these gray level differences with the calibration data presented in Fig. 2(d), a gray level difference of 38 corresponds to a step height of approximately 5.3 micrometers. Similarly, gray level differences of 12, 9, and 3 correspond to step heights of approximately 2.0, 1.14, and 0.38 micrometers, respectively.

Figure 4 provides the sample shape and the corresponding measurements obtained using a confocal microscope (NS-3800, Nanoscopes). Figure 4(a) shows the designed sample structure, while Fig. 4(b) presents the measured profiles from the confocal microscope, with the step values indicated. A slight discrepancy is observed between the measurements obtained from the partial coherence digital holography system and the confocal microscopy, which may be attributed to noise in the DHM.

Figure 4. Sample structure and profile. (a) sample structure; (b) confocal microscope measurement data.

In this study, we proposed a partial coherence digital holographic microscope (DHM) for three-dimensional (3D) shape measurement without the need for phase unwrapping. The main objective was to investigate the effects introduced by using partial coherent illumination in a digital holographic microscope and to explore its advantages for 3D imaging. A key feature of the partial coherence DHM is its ability to adjust the degree of coherence based on the characteristics of the sample, enabling more versatile and precise measurements.

Our approach utilizes the reconstructed amplitude image to measure 3D shapes, leveraging the dependency of the gray level of the reconstructed amplitude image on the degree of coherence rather than on the phase information. By establishing a direct relationship between the optical path difference (OPD) and the intensity of the reconstructed amplitude image, we successfully obtained 3D information without encountering 2π phase ambiguity or the need for phase unwrapping. This method enables the measurement of sample heights with an extended OPD range, exceeding ten times the wavelength, making it particularly suitable for steps larger than the visible wavelength.

While this technique may have limitations in measuring step heights much smaller than the wavelength, it proves highly effective for larger steps. Furthermore, by integrating this method with conventional phase measurement techniques, it is possible to accurately measure even minute steps. Overall, the proposed partial coherence DHM method represents a significant advancement in 3D microscopy, offering an effective solution for a wide range of applications where traditional methods may be limited by coherence noise and phase ambiguities.

This research was supported by the 2024 scientific promotion program funded by Jeju National University.