1. Introduction

Digital holography (DH), also known as wide-filed interferometry, has attracted much attention from the research community as it can be used for non-destructive, high precision, full filed and especially quantitative phase measurements [1–4]. DH has been widely used to investigate a variety of unlabeled specimens, such as biological specimens, surface profile, and micro structures. In terms of setup, DH can be divided into a series of groups, including separated-path geometries [1, 2] or common-path geometries [4–8], on-axis geometries [1, 9, 10] or off-axis geometries [6–8]. Compared to the separated-path geometries, the common-path geometries provide simplicity, robustness and even compactness in the setup due to the same path between the reference and sample beams. On the other hand, the on-axis geometries can make full use of the space-bandwidth product of a CCD camera. But to eliminate the zero-order and conjugate image terms, phase-shifting algorithm [9–12] is applied, which penalizes the acquisition rate [1, 9] or the utilization of the field of view of the CCD camera [10–12]. To overcome this, the off-axis geometries filter out the unwanted image terms in the Fourier domain with a single shot [6–8], and then allow for much faster acquisition rates. So the off-axis common-path DHs are of growing interest in recent years. However, most of the off-axis common-path DHs just utilize a single wavelength [5–12], in which if a specimen is optically thicker than the illumination wavelength, its imaging phase is ambiguous and produces 2π discontinuities. A digital phase unwrapping algorithm [13, 14] can enable continuous phase retrieval. However, these algorithms are often computational intricate at the expense of retrieval time, and especially cannot identify high aspect-ratio phase discontinuities of over one wavelength. As a normal solution, dual-wavelength DH (DWDH) [15–22] is proposed to avoid the 2π discontinuities. This method records two interferograms with different wavelengths to extract each wrapped phase map separately, and then utilizes a subtraction procedure between the two wrapped phases to yield an unwrapped phase map with a large synthetic wavelength, which improves the unambiguous phase range. Combining the significant advantages of the off-axis common-path DH and DWDH presented above, the off-axis common-path DWDH has being received great scientific interest in recent years [20–22].

In 2014, Kim [20] et al. utilized a pinhole array to select a different diffraction order of a grating for each wavelength so that they can distinguish the dual-wavelength interferogram obtained in a single shot. However, to avoid the cross-talk between the two wavelengths, a smaller period of the interferogram is required at the cost of either spatial resolution or field of view. In 2016, Zhao [21] et al. employed a parallel glass plate to reflect two lasers from its front and back surfaces, and then create a lateral shear to obtain a multiplexed interferogram with the two-wavelength information. However, the size of the measured specimens must be smaller than the shear, and a cross-talk also exits between the two wavelengths. To reduce the cross talk, Shaked [22] et al. adopted a modified Michelson interferometer using transmission pinhole array to multiplex two wavelength information into one interferogram with orthogonal spatial frequencies. Their setup used a dichroic mirror to separate the two wavelengths and two mirrors to adjust each wavelength to create an orthogonal off-axis interference fringe direction. However, the illumination wavelengths are restricted by the commercial dichroic mirror.

Therefore, inspired by previous work [16–19, 23–25], we present an off-axis nearly common-path DWDH using polarization-multiplexing to yield a multiplexed interferogram with orthogonal spatial frequencies for each wavelength in a single shot. A division algorithm is also used to perform the dual-wavelength unwrapping. In terms of simplicity in procedure and setup, we would like to emphasize our advancement. Compared with [16–19], our method is more stable and compact due to the quasi common path. While compared with [22], our method may be difficult to multiplex more than two wavelengths, and slightly harder to align the angles between the sample and reference beams. However, our method can make the two wavelengths very close to each other, and optimize both the illumination wavelengths and synthetic wavelength to fit application requirements but not be limited by the commercial dichroic mirror. To some extent, our method can be more practical and compact in application.

2. Experimental setup

Figure 1 shows the proposed experimental setup. The two laser sources are a red laser with a wavelength of λ₁ and a green laser with a wavelength of λ₂. The red beam is linearly polarized by a polarizer P1 through 0° and the green beam is done by a polarizer P2 through 90°, both with respect to the horizontal axis. The two orthogonal polarized beams are combined into one beams by a beam splitter BS1 and then collimated and expanded by a collimator & expander (BE). After passing through a specimen S, the combined beam is Fourier-transformed by a lens L1, and then split into two copies using another beam splitter BS2. One copy, referred to as the reference beam, is low-pass filtered and reflected by a pinhole-mirror PM [24]. The other copy, referred to as the sample beam, is separated into two wavelength channels again using a polarizing beam splitter PBS. Each of the channels is reflected by retro-reflectors, RR1 and RR2, to yield a full multiplexed off-axis interferogram on a CCD camera [7, 23]. The two retro-reflectors are fixed perpendicular to each other [23] to create orthogonal Fourier-plane shifts between the two wavelengths. The three beams are merged again by BS2 and Fourier-transformed back to the CCD camera plane by a lens L2. To compensate chromatic aberration, all the lens are achromatic. In this case, two orthogonal off-axis interference fringe directions are created for each wavelength, and thus a single multiplexed off-axis interferogram for the two wavelengths can be acquired by the CCD camera in a single shot. Noted that each wavelength interferograms fringes can be independently adjust by RR1 and RR2 both in spatial frequency and orientation, respectively. But to minimize the cross-talk between the two wavelengths, the better choice is to enable orthogonal spatial frequencies orientation for each wavelength, which can ensure optimum overlaps of the image terms in the interferogram Fourier domain.

 

Fig. 1 (a).Experimental setup: P1 and P2, polarizers; BS1 and BS2, beam-splitters; BE, beam expander; S, sample; L1 and L2, lenses; PBS, polarizing beam splitter; PM, pinhole mirror; RR1 and RR2, retro-reflector mirrors ;(b) and (c) three-dimensional positions of RR1 and RR2.

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3. Division reconstruction algorithm for dual wavelengths

Assuming that the two-wavelength beams illuminated the specimen are both uniform plane wave, and the multiplexed interferogram is an incoherent superposition of the interferograms λ₁ and λ₂, the intensity of the multiplexed interferogram can be then expressed as

To extract the specimen phase φ_i and remove the background phase aberration φ_bgi, a double exposure method [21] can be applied with the help of one offline prior acquisition of a multiplex interferogram without any specimen. However, this method requires to ensure the spatial frequencies accurately. Similar to the double exposure method, a division method [24, 25] was proposed by us to relieve the background phase aberration and spatial frequency errors, but it is only suitable for single-wavelength reconstruction. So we modify the division method to make it to fit dual-wavelength reconstruction as following.

After applying Fourier-transform operation, the zero-order term and the two couples of cross-correlation terms can be fully separated in the spectral domain. The real image terms for each wavelength can be chosen through band pass filters, and then transferred to two complex results using inverse Fourier-transform operation, which become

When there is no specimen in the position of S, Eqs. (2) and (3) can be rewritten as

After applying division operations, we can obtain

Equations (6) and (7) show that with the help of a multiplexed specimen-free interferogram, we can retrieve the specimen phases for each wavelength but remove the phase disturbance caused by the spatial frequency and background phase. Noted that we can prior record and analyze the specimen-free interferogram only once, and then use their results in the subsequent process.

According to the dual-wavelength unwrapping method [21, 22, 26], we can achieve

From Eq. (8), we can extract the height map h(x, y) of the specimen. But since our method operates in a transmission mode, in which the phase information associates with the height and the refractive index, the refractive index must be taken into consideration during the extracted process.

We start with the definition of the specimen phase, which can be expressed as

As a result, we can obtain

4. Results and discussion

In order to illustrate our approach, the experimental setup is constructed by using a He-Ne laser with red wavelength of λ₁ = 632.8 nm, a solid-state laser with green wavelength of λ₂ = 532 nm, two lenses L1 and L2 with the same focal length of 200 mm, a pinhole-mirror PM with a pinhole diameter of 30 μm, and a monochrome CCD camera with pixel size 4.8 μm × 4.8 μm and 1280 × 1024 pixels.

The first experiment was carried out on a square step target made of fused silica, which has a refractive index of n₁ = 1.457 for λ₁ = 632.8 nm and n₂ = 1.460 for λ₂ = 532 nm. The height of the step target is provided by BRUKER Atomic Force Microscopy (AFM) and the result is about 2500 nm. In the case, a synthetic equivalent wavelength of λ_s = 7018.8 nm can be achieved, which is larger than the target height. Figure 2(a) shows the multiplexed interferogram without any specimen, and Fig. 2(b) shows the multiplexed interferogram with the square step target. When the two images are Fourier transformed, the resulting power spectrums are obtained as illustrated in Figs. 2(c) and 2(d), respectively. We can see that due to the orthogonal spatial frequencies, the zero-order term and the two pairs of cross-correlation terms can be fully separated with low cross-talk in the spectral domain. In other words, the real terms for each wavelength can be selected independently as marked with color circles in Figs. 2(c) and 2(d). Noted that since the two real terms have spectral widths proportional to the frequency band of the specimen profile, the area of these markers must be chosen as large as possible, but avoid the overlap between interference terms [17–21, 27], in which the higher frequency components of these terms can be achieved. Of course, such smoothed filtering maskers [20, 27] as Gaussian and Hamming can be used to avoid the occurrence of high-frequency fluctuations in the extracted images, and automatic spatial filtering approaches [28, 29] can be attempted to optimize the shape of real terms to improve the extracted quality. Using Eqs. (6)-(8), we can obtain the wrapped phase maps for each original wavelength as shown in Figs. 3(a) and 3(b), respectively, and the unwrapped phase map for the synthetic equivalent wavelength as shown in Fig. 3(c). Noted that the wrapped phase maps for each original wavelength should be reversed and matched to each other due to the image flipping introduced by the two retro-reflectors [23]. Figure 4 shows the comparison of 1D height profiles, which are yielded using Eqs. (9) and (10) by the phase maps indicated by the dash lines in Figs. 3(a)-3(c), respectively. Obviously, the conventional single-wavelength method cannot work well but the dual-wavelength method can in the condition of shaper jump on the specimen. The step height can be evaluated as 2590.9 nm from the difference of both averages [22] of the black line in Fig. 4. Although the experimental result is slightly larger than the nominal result, the two results still agree well with each other. Furthermore, the standard deviation is 16.8 nm for λ₁, while 16.6 nm for λ₂ and 172.1 nm for λ_s. As many others [21, 22, 26, 30] pointed out that the oscillations in the results for λ_s are mainly produced by the noise amplification using a synthetic wavelength, which is inherent to DWDH in general. But in fact, the increased noise level can be reduced by such noise-reduction methods as linear programming phase unwrapping [30], and the result is illustrated by the blue line denoted by λ_s′ in Fig. 4. Figure 5 illustrates the 3D height map for the square step target.

 

Fig. 2 Multiplexed interferograms (a) without and (b) with the specimen; (c) power spectrum of (a); (d) power spectrum of (b).

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Fig. 3 Reconstructed phase maps for (a) λ₁, (b) λ₂, and (c) λ_s, with the color bar indicates phase in rad.

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Fig. 4 1D height profiles obtained from phase maps along dash lines marked in Figs. 3.

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Fig. 5 An unwrapped 3D height map of the step target.

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To further access the feasibility of our approach, a circular step target made of fused silica was also measured. Figure 6(a) illustrates the multiplexed interferogram, and Fig. 6(b) illustrates the corresponding power spectrums. After applying the division reconstruction algorithm, we obtain the 3D height map of the circular step target, and the result is shown in Fig. 6(c). The step height of the circular step target is about 2026.3 nm, which also coincides with the measured result of 2000 nm from AFM.

 

Fig. 6 Experimental results for the circular step target. (a) Multiplexed interferogram; (b) power spectrum; (c) 3D height map.

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To demonstrate the stability of our approach, 100 sequential measurements were also performed without any specimen at interval of half minute. The variances of the optical path difference (OPD) [24] are obtained at a randomly selected point for each wavelength, respectively, and the results are shown in Fig. 7. From the Fig. 7, we can calculate that the standard deviation is 3.54 nm for λ₁, while 3.56 nm for λ₂ and 32.54 nm for λ_s. These means that our approach can achieve a long-term stability even the noise level is increased due to the synthetic wavelength.

 

Fig. 7 Stability test for the proposed method.

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5. Summary

In conclusion, we developed an off-axis quasi common-path DWDH using polarization-multiplexing to achieve a single interferogram acquisition. In contrast to the off-axis quasi-common-path DWDH in [22], our approach can enable two-wavelength single-shot imaging using a more practical and compact configuration. A division method for dual wavelengths is also presented to reconstruct the phase of the specimen but remove the phase disturbance caused by the spatial frequency and background phase. Such phase specimens as a square step target and a circular step target can be simply extracted using the division method. However, it should be noted that the two interferograms of specimen and free-specimen share the same gray scale levels of the camera. The amplitude modulation should be then taken into consideration in absorptive specimens, as reported in [31] and [32], which can be resolved from the real part of the complex results of C_i(x, y). However, in all our specimens with phase distribution, amplitude modulation can be negligible, which fits our approximation in the division form. In addition, as many others did [17–21], we used the same sized digital filters to select real spectral terms for red and green wavelengths. However, these may lead to some errors in the reconstructed results to some extent for the corresponding extent of spatial frequencies is different for each wavelength. In fact, the errors can be avoided by keeping the filter sizes for the two wavelengths in appropriate ratio. We hope that our approach provides a powerful tool for quantitative phase imaging of specimens thicker than the illumination wavelength.