1. Introduction

Holography is well-known for its three-dimensional imaging and phase imaging [1–4], and plays an important role in different applications such as industrial inspection, material science, and biomedicine [5–8]. As a branch of holography, incoherent holography generates holograms based on the self-interference of incoherent light [9–12], which solves the inherent problem of relying on coherent light in holography. Compared with the holography relying on coherent light (coherent holography for short), incoherent holography is not only suitable for objects illuminated by natural light or self-illuminating [13–16], but also has high-resolution and high-quality imaging [17–20], and is a promising three-dimensional imaging technology [21–26].

In incoherent holographic systems, the light emitted from each object point is divided into two parts and passes through two optical systems with equal optical paths but different light modulations, and finally interferes to form a hologram [27–30]. The most concise and classic incoherent holographic system is the FINCH (Fresnel Incoherent Correlation Holography) system, as shown in Fig. 1. The light emitted from original object point ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ splitted by the spatial light modulator (SLM) can be regarded as two sub-light fields emitted from two sub-object points ${u_{01}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ and ${u_{02}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ formed by the polarization decomposition of ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$. These two sub-light fields pass through the horizontal polarization subsystem (marked in blue) and vertical polarization subsystem (marked in red) of the FINCH system respectively. Since the optical path difference between the two subsystem is small enough, and the temporal varying random phases of these two sub-object points are consistent, the two light fields on the camera can interfere with each other. The intensity superposition of the interference patterns formed by each original object point constitutes a hologram. When interference caused by two sub-object points occurs, the temporal varying random phases will cancel each other out, and the stable phase difference between the two sub-object points (termed as the phase difference for short) will be preserved in the interferogram, that is, the phase difference will be preserved in the hologram and can theoretically be recovered from the hologram to achieve the phase-difference imaging of incoherent holography.

 

Fig. 1. The schematic of the FINCH system. P is a polarizer, L is a lens, BS is a beam splitter, SLM is a spatial light modulator, and CCD is a camera.

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According to the above analysis, phase-difference imaging should be an imaging technology suitable for various incoherent holographic systems, that is, like the well-known intensity imaging, it is a general imaging technology for incoherent holography. However, there is no general phase-difference imaging theory due to both diverse incoherent holography systems and the complicated reconstruction process from holograms based on the diffraction theory. To the best of our knowledge, only FINCH achieves phase-difference imaging among all incoherent holographic systems, which is reported in our previous works [31,32]. Since the FINCH system is one special incoherent holographic system, the phase of the reconstructed image only needs to eliminate constant phases and the accurate phase-difference imaging is achieved, as reported in Ref. [31]. For other incoherent holographic systems, there are constant phase, linear phase, and quadratic phase need to be eliminated generally, and the accurate phase-difference imaging can be achieved, which will be discussed in this paper.

In this paper, the general physical model of incoherent holographic systems is summarized and used to reveal the general relationship between the phase difference and the phase of the reconstructed image, then the method of eliminating the additional phase generated by actual holographic systems is introduced, and finally the general method of phase-difference imaging is proposed. This method can be used for most known incoherent holographic systems, such as those in Refs. [12,27,33–35] and others. We call this technique as general Phase-Difference Imaging of Incoherent Digital Holography, or PDI-IDH for short.

With the help of PDI-IDH, incoherent holography may have potential applications in materials science, biomedicine, and polarized optics, because the phase difference is of great significance for studying the optical properties of anisotropic objects such as birefringence materials, biological tissue, and vectorial light converter.

2. Methodology

2.1 Analytical expression of reconstructed image

Incoherent holographic systems decompose the object point ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ on the object plane (x₀, y₀) into two separate points that finally form two coherent light fields ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ and ${U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ in the camera plane (x_c, y_c). Light intensity $|{U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0}) + {U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0}){|^2}$ generated by all object points superimposed with each other to form a hologram

The component $H({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$

Although the optical systems of various on-axis incoherent holograms are different, their physical process of recording holograms is the same. Most on-axis systems decompose one object point into two separate points, and perform lens imaging in succession, then finally form two mutually interfering spherical waves on the camera plane. This is the general physical model of on-axis incoherent holographic systems. Therefore, only the general expression for two spherical waves ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ and ${U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ is needed, and then the expression of $H({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ for most on-axis systems can be calculated, and finally the general expression of the image ${a_i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})\exp [j{\phi _i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})]$ will be obtained, which is the analytical expression of the reconstructed image.

Based on this physical model of multi-lens imaging, we start to calculate the general expression of spherical wave in the camera plane.

The object-image relationship of lens imaging is shown in Fig. 2, where the ordinates of the object point ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$, the lens L₁ with a focal length of f₁, and the image point ${u_1}\textrm{(}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}\textrm{)}$ are z_o1, 0 and z_i1= (1/f₁+ 1/z_o1)^-1, respectively. When the Fresnel approximation condition holds, the exact relationship between ${u_1}\textrm{(}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}\textrm{)}$ and ${u_0}\textrm{(}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0}\textrm{)}$ is (refer to the appendix of [32])

 

Fig. 2. Schematic diagram of lens imaging

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The subscript “j” corresponds to the jth lens imaging. The image point coordinate is x_n= A_s(n)x₀ and y_n= A_s(n)y₀, where A_s(n) is the magnification of image after the nth lens imaging. The object point ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ with illumination infinitesimal element dx₀dy₀ will generate the image point ${u_n}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n})$ with the district $d{x_n}d{y_n} = A_s^2(n)d{x_0}d{y_0}$. In the camera plane (x_c, y_c), the spherical wave generated by image point ${u_n}\textrm{(}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n}\textrm{)}$ is $A_s^2(n){u_n}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n})S({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n},z)$, where $S({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n},z)$ represents the spherical wave in the (x_c, y_c, 0) plane generated by a unit point source located at (x_n, y_n, z).

In incoherent holographic systems, the light fields ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ and ${U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ in camera plane of image points ${u_{n1}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n1}})$ and ${u_{n2}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n2}})$, which are formed by ${u_{01}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ and ${u_{02}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ decomposed by the same object point ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$, can be written as

To obtain ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})U_2^ \ast ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$, we calculate ${u_{n1}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n1}})u_{n2}^ \ast ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n2}})$ and $S({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n1}},{z_1}){S^ \ast }({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n2}},{z_2})$ separately,

Equation (12) can be written in the form of spherical wave,

Substitute Eq. (11) and (15) into ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})U_2^ \ast ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ to get

Because the amplitude is non-negative, the absolute value sign is added to A_s1A_s2(z₂-z₁)^-1 in Eq. (17). The “sign()” in Eq. (18) is a sign function. The sign[A_s1A_s2(z₂-z₁)] indicates that π needs to be added to the phase when A_s1A_s2(z₂-z₁) is negative, that is A_s1A_s2(z₂-z₁)^-1 is negative. Equation (6) indicates that D_s1 is the distance from the object point ${u_{01}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ to the image point ${u_{n1}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n1}})$. -z₁ is the diffraction distance of the spherical wave defined by Eq. (8), that is, the distance from the image point ${u_{n1}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n1}})$ to the camera plane. Therefore, D_s1-z₁ is the distance from the object point ${u_{01}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ to the camera plane. Similarly, D_s2-z₂ is the distance from the object point ${u_{02}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ to the camera plane. For incoherent holographic systems, the distance between ${u_{01}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ and camera is required to be equal to that between ${u_{02}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ and camera, so that ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ and ${U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ can interfere with each other. Therefore, D_s1-D_s2+ z₂-z₁ is equal to 0 in Eq. (18). Considering -j, the constant phase is exp[j(kZ_i-π/2)].

Substituting Eq. (16) into Eq. (2), the diffraction relationship is obtained as

Since ${a_{01}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ and ${a_{02}}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ in amplitude ${a_i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})$ (see Eq. (17)) are the two amplitude components of the object point ${u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ (see Eq. (11)), ${a_i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})$ is proportional to $|{u_0}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0}){|^2}$. So, ${a_i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})\exp [j{\phi _i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})]$ is the reconstructed image we need. By performing diffraction calculation on $H\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_c}} \right)$,

Although Eq. (21) is the well-known traditional reconstruction method, the analytical expression of the image, namely Eqs. (17) and (18), are accurately derived for the first time. The general relationship between the phase ${\phi _i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})$ of the image and the phase difference $\Delta \varphi ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ is shown in Eq. (18).

In the above discussion, we only analyzed the hologram formed by a single object plane. When considering multiple object planes, the hologram ${I_s}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ is the intensity superposition of holograms formed by all object planes. Using Eq. (21), the hologram ${I_s}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ can be focused on different object planes with different reconstruction distances, that is, three-dimensional intensity imaging can be achieved by the amplitude ${a_i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})$ of the reconstructed image. Then, the constant and quadratic phase terms related to the object distance in Eq. (18) can be calculated and eliminated from the phase ${\phi _i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})$ of the reconstructed image. Finally, the phase difference $\Delta \varphi ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ of objects located on different planes can be obtained, that is, three-dimensional phase-difference imaging can be achieved.

Considering the calculation process of parameters such as Z_i, B, A_i in Eqs. (21) and (18) is troublesome, we compiled the above theory into a general algorithm, as we show in Code 1 (Ref. [36]). By inputting the parameters of the holographic system into the algorithm, the user can directly obtain Z_i, B, A_i, and the sign of A_s1A_s2(z₂-z₁), which will be demonstrated in our experimental section.

2.2 Additional phase of actual systems

The above analysis is based on the ideal on-axial system, but there is inevitable deviation of the actual system from the ideal system, which will change the phase of spherical waves ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ and ${U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$. Therefore, the $H({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ in ideal system will become to the $H^{\prime}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ in actual systems, and an additional phase ${\phi _c}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ between $H({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ and $H^{\prime}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ is inevitable. ${\phi _c}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ will be transferred to image after diffraction calculation of Eq. (21), which will distort the relationship between the phase ${\phi _i}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} _i})$ of the image and the phase difference $\Delta \varphi ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ in Eq. (18). Therefore, ${\phi _c}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ needs to be eliminated before reconstruction to make Eq. (18) suitable for actual on-axial systems.

Off-axial systems can be regarded as on-axis systems with additional phases, where the linear phase in the additional phase can achieve the off-axis effect. Therefore, after eliminating the additional phase ${\phi _c}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$, Eq. (18) is also suitable for actual off-axial systems.

We express the additional phase in the form of a series expansion, that is

The coherence length of incoherent light is generally only tens of microns, and the changes of ${U_1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ and ${U_2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c};{x_0},{y_0})$ cannot be too large in a carefully constructed holographic system. Therefore, the values of α_p and β_q should be small. In addition, the photosensitive area of the camera is generally less than 2 × 2 cm², the values of x_c and y_c are small, too. Therefore, the multi-power terms (p, q ≥ 2) of x_c and y_c in Eq. (22) are high-order small quantities and can be ignored. Keeping only the terms of p = 0,1 and q = 0,1 in Eq. (22), ${\phi _c}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c})$ can be simplified to

We compensate the additional phase before image reconstruction and modify the reconstruction formula (21) to

Since φ_c and k(αx_c +βy_c) are formed by actual systems, experiments are required to calibrate them. Therefore, the calibration method will be explained in the experimental section.

Generally, for a stable incoherent holographic system, the constant phase φ_c and the linear phase k(αx_c +βy_c) are stable and only need to be calibrated once.

3 Experiments

3.1 Incoherent digital holographic system

To verify that the phase-difference imaging method is generally suitable for various incoherent holographic systems, the experimental system should not only have multiple lenses, but also need to generate significant additional phase. Therefore, the triangular interference system shown in Fig. 3 was selected, which not only has multiple lenses, but also forms an off-axis hologram with a large additional phase.

 

Fig. 3. (a) The schematic diagram of the triangle interferometer system. (b) The simplified schematic diagram for analysis. The green solid line represents the horizontally polarized component, and the red dashed line represents the vertically polarized component. L₁, L₂ and L₃ are lenses, P₁ and P₂ are polarizers, PBS is a polarizing beam splitter, and M₁ and M₂ are mirrors.

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Figure 3(a) is a schematic diagram of the triangular interferometric system, and Fig. 3(b) is a simplified schematic diagram for analysis. LED (GCI-060403, Daheng Optics) with small light-emitting area and stationary ground glass construct an incoherent light source with large light-emitting area for improving imaging quality. The illumination light illuminates the object after passing through the polarizer P₁ whose polarization direction is 45° from the horizontal. Object beams are converged by the lens L₁, and continues to propagate through a triangle interferometer system, which is composed by polarizing beam splitter (PBS), lens L₂ and L₃, and mirrors M₁ and M₂. The transmission and reflection beams are shown as green solid and red dashed line, respectively. The two beams are interfering with each other after passing through polarizer P₂, and the hologram are captured by the CCD (Grasshopper3 GS3-U3-51S5 M, Point Grey).

The distances and focal lengths of lenses in Fig. 3(b) are shown in Table 1, where Δd is the distance correction due to the refractive index difference between PBS and air to ensure that the imaging position can be calculated correctly (refer to the Supplement 1 of [31]). The Δd is equal to [(1/n_p)-1]h, where n_p= 1.7174 and h = 25.4 mm are the refractive index and thickness of PBS, respectively.

Table 1. The distance parameters in Fig. 3(b)

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3.2 Calibration of φ_c and k(αx_c+βy_c)

To determine the additional phase generated by the actual holographic system, the light-transmitting circular hole is used as the object. Since the illumination light is a linearly polarized light inclined at 45°, the phase difference between the horizontal and vertical polarization components of the object light is zero, and the distribution of the phase difference is uniform.

Let α=β=φ_c= 0, we use Eqs. (24) and (18) to reconstruct the object and calculate the phase difference, where the parameters required in Eqs. (24) and (18) can be obtained by our general algorithm [36]. The parameters input to the algorithm include the object distance d_o= 37 mm, all distances and focal lengths in the holographic system (see Table 1). The output of the algorithm is Z_i= 0.289 m, B = 1.873 m^-1, A_i= -0.780 and the sign[A_s1A_s2(z₂-z₁)]=+1. Figures 4(a) and 4(b) are the amplitude and phase of the reconstructed image respectively, where zero-order frequency spectrum is suppressed. The messy phase distribution in Fig. 4(b) indicates that the phase of the reconstructed image is not suitable for imaging objects. Figure 4(c) is the distribution of phase difference. Affected by the additional phase (including constant phase and linear phase) generated by the actual system, the phase difference changes linearly.

 

Fig. 4. (a) and (b) are the amplitude and phase of the reconstructed image. (c) is the phase difference of the object, where the additional phases (constant phase and linear phase generated by the actual system) are not calibrated.

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We keep β=φ_c= 0, and set α equal to −0.001, −0.002 and −0.003 in sequence, then the phase difference is obtained as shown in Figs. 5(a)–5(c). From Figs. 5(a)–5(c), the relationship between α and the change direction of the phase difference is monotonic. Therefore, we can easily find α=-0.0043 that makes the phase difference only change in the vertical direction as shown in Fig. 5(d). In the same way, we can obtain β=0.0059 where the phase difference only changes in the horizontal direction as shown in Fig. 5(e). Substituting α=-0.0043, β=0.0059 and φ_c= 0 into Eqs. (24) and (18), the obtained phase difference is shown in Fig. 5(f).

 

Fig. 5. (a), (b) and (c) are the phase differences of the object obtained when α is −0.001, −0.002 and −0.003 respectively, where β=φ_c= 0. (d) is the phase difference of the object obtained when α=−0.0043 and β=φ_c= 0. (e) is the phase difference of the object obtained when β=0.0059 and α=φ_c= 0. (f) is the phase difference of the object obtained when α=-0.0043, β=0.0059 and φ_c= 0.

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The phase average of the region (20 × 20 pixels) marked with the white rectangle in Fig. 5(f) is −1.55 rad. Since the phase difference $\Delta \varphi ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _0})$ between the horizontal and vertical polarization components of the linearly polarized light should be 0, −1.55 rad is the constant phase φ_c of the actual system.

The above process is the calibration method of φ_c and k(αx_c+βy_c) for our holographic system. In subsequent experiments, phase-difference imaging will all use φ_c= -1.55 rad, α=-0.0043 and β=0.0059.

3.3 Phase-difference imaging

To construct the object light with non-uniform phase difference, a radial polarization converter (RPC-515-08-515, Workshop of Photonics) and a quarter-wave plate are placed behind the light-transmitting circular hole. The linearly polarized illumination light passes through the radial polarization converter to form radially polarized light. After the radially polarized light passes through a quarter-wave plate whose optical axis is 45° from the horizontal, elliptically polarized light is formed. The intensity distribution of the elliptically polarized light is shown in Fig. 6(a), and the phase difference between the horizontal and vertical polarization directions is shown in Fig. 6(b). Figures 6(c) and 6(d) are the amplitude and phase difference of the object reconstructed by Eqs. (24) and (18). Figure 6(d) is basically the same as that in Fig. 6(b), which verify that the proposed PDI-IDH can achieve phase-difference imaging of anisotropic objects.

 

Fig. 6. (a) and (b) are the theoretical distributions of the intensity and phase difference of the object. (c) and (d) are the amplitude and phase difference of the object obtained experimentally.

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To demonstrate the characteristics of three-dimensional imaging, in addition to the circular hole with non-uniform phase difference, the light-transmitting rectangular hole is also added in the experiment. The linearly polarized light passes through the rectangular hole to form an object light with uniform phase difference. When focusing on the circular hole, the amplitude and phase difference distributions of the objects are shown in Figs. 7(a) and 7(b) respectively. When focusing on the rectangular hole, the amplitude and phase difference distributions of the objects are shown in Figs. 7(c) and 7(d) respectively. The results in Fig. 7 demonstrate that the phase-difference imaging has the same three-dimensional imaging capabilities as intensity imaging.

 

Fig. 7. (a) and (b) are the intensity and phase difference distributions of the objects when focusing on the circular object. (c) and (d) are the intensity and phase difference distributions of the objects when focusing on the rectangular object.

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The above experimental results verify that the proposed PDI-IDH can achieve three-dimensional phase-difference imaging and is suitable for both anisotropic objects (the circular hole) and isotropic objects (the rectangular hole). In addition, the phase-difference imaging can be used to measure the polarization distribution of polarized light, and the method can refer to our previous work [32].

In actual experiments, there are two reasons that may cause a slight distortion of phase difference. First, the effective numerical aperture of the actual recording system is not large enough. Because the numerical aperture is infinite in theory, but it is limited in experiments. Second, the constant phase φ_c in Eq. (24) generated by the actual system may change due to changes in the experimental environment. Because environmental factors such as temperature and air flow will change the refractive index distribution of the air, thereby changing the optical path length and causing phase changes.

4. Conclusion

We summarize a physical model according to the commonality of incoherent holographic systems based on the self-interference, and derive a general relationship between the phase of the reconstructed image and the phase difference of the object. Considering that the additional phase generated by actual holographic systems will distort this relationship, the additional phase is analyzed and the elimination method is introduced to make the relationship in actual systems consistent with that in ideal systems. Finally, the phase-difference imaging suitable for most incoherent holographic systems is proposed, and a general algorithm for quickly calculating the parameters required for the phase-difference imaging is provided [36]. To verify our theory, an off-axis triangulation interferometry system was selected, calibrated only once, and then phase-difference imaging was achieved with a single exposure. The anisotropic properties of objects are related to the phase difference between two orthogonal polarization components, which can be obtained by the phase-difference imaging. With the help of phase-difference imaging, incoherent holography should have potential applications in materials science, biomedicine, polarized optics and other fields.