1 Introduction

The techniques for visualization through scattering or diffuse media such as biological tissues and frosted glass have attracted much attention. Various imaging methods for visualization through scattering or diffuse media have been reported, but most of them specifically examine the reconstruction of intensity or amplitude information. Imaging of phase objects behind scattered objects has attracted attention for cellular imaging in biology, wavefront aberration detection in astronomy, underwater imaging, and airflow visualization. Phase imaging emphasizes the inference of the spatial phase distribution of objects. Particularly for objects that are transparent, reconstructing the phase distribution can lead to visualization of the spatial distribution such as thickness and refractive index. Researchers have also developed the reconstruction of phase information through scattering media [1,2,3,4,5].

Reports have described studies of techniques using holography to reconstruct the information of an object placed behind a diffuser or scattering medium [6,7,8,9]. Studies on techniques using digital holography to reconstruct the complex amplitude of an object placed behind a diffuser or scattering medium have been reported [4, 5, 10,11,12]. In addition, holographic wavefront correction has been adopted in imaging through scattering media [13,14,15,16]. Most of these methods are limited to retrieving amplitude objects. Kodama et al. realized three-dimensional microscopic imaging through scattering media based on in-line digital holography [4, 5]. By correlation holography, the imaging of phase objects hidden behind a scattering medium has been demonstrated [17,18,19]. Nevertheless, few attempts have been undertaken to recover the phase information through diffusers or scattering media with digital holography. We earlier demonstrated phase-shift digital holography for the reconstruction of intensity objects hidden behind a diffuser [16]. In this method, reference beam is combined with object beam through diffusers. Here, we demonstrate the potential of the technique by utilizing digital holography by phase correction of the diffuser. The influences of different diffuser angles of a diffuser are investigated on reconstruction of the phase information of an object behind diffusers. Using diffusers with different diffusion angles of 1, 20, 40, 60, and 80 degrees, the phase distribution of the lens and wedge plate behind the diffuser is calculated. The shape of the visualized spherical plano-convex lens is evaluated by calculating the curvature radius.

2 Experiment procedures

Phase imaging through diffusers is based on phase-shifting lensless digital holography. Digital holography is useful to reconstruct amplitude objects behind a diffuser by acquiring and correcting the diffuser information in advance [14, 16]. Based on the reconstruction algorithm presented in an earlier report [16], we modified a method for reconstructing phase objects behind a diffuser. To visualize the phase object behind the diffuser, the diffuser is placed between the object and the camera, as shown in Fig. 1. The image sensor position is defined as the camera plane, the diffuser position as the diffuser plane, and the object position as the object plane. Figure 2 portrays the concept and a diagram for reconstruction of a phase image of an object behind a diffuser. The procedure for reconstructing the object intensity and phase is the following: (1) first, the complex amplitude of the diffuser at the sensor plane is obtained without insertion of an object. The intensity is captured by blocking the reference beam. Two beams from a reference and object are spatially overlapped using a beam splitter to produce a hologram. The phase image is obtained from a four-step phase-shifting method. (2) We reconstruct the complex amplitude of the diffuser at the diffuser plane by calculation with optical back propagation. Thereby, a priori information of the phase of the diffuser is obtained. (3) After an object is inserted, the complex amplitude of the object through the diffuser at the sensor plane is acquired using digital holography. (4) The phase of the object through the diffuser at the diffuser plane is obtained using optical back propagation. (5) Thereafter, the phase of the diffuser at the diffuser plane is compensated. (6) Furthermore, the complex amplitude of only the object is propagated back to the object plane. Then, the phase image at the object plane is reconstructed.

Fig. 1
figure 1

Schematic showing lensless phase-shift digital holography of an object through diffusers. BS Beam splitter

Full size image
Fig. 2
figure 2

Concept and diagram of reconstruction of a phase image behind a diffuser. BP denotes optical back propagation

Full size image

We denote the amplitude and phase of the object in the camera plane as \({A}_{obj}^{^{\prime\prime} }\) and \({\varphi }_{obj}^{^{\prime\prime} }\). Let \({A}_{d}^{^{\prime\prime} }\) and \({\varphi }_{d}^{^{\prime\prime} }\) be the amplitude and phase of the diffuser in the camera plane. When light transmits through the diffuser, the complex amplitude of the diffuser at the camera plane, \({U}_{d}^{^{\prime\prime} }\) is expressed by Eq. (1).

$${U}_{d}^{^{\prime\prime} }={A}_{d}^{^{\prime\prime} }{\text{exp}}\left(i{\varphi }_{d}^{^{\prime\prime} }\right).$$
(1)

When light transmits through both the object and the diffuser after the insertion of the object, the complex amplitude information \({U}^{^{\prime\prime} }\) at the sensor plane is expressed by Eq. (2).

$${U}^{^{\prime\prime} }={A}_{\mathit{obj}}^{^{\prime\prime} }{A}_{d}^{^{\prime\prime} }{\text{exp}}\left[i\left({\varphi }_{obj}^{^{\prime\prime} }+{\varphi }_{d}^{^{\prime\prime} }\right)\right].$$
(2)

First, the intensity correction is carried out at the camera plane. Equation (3) shows the complex amplitude of the object light corrected for the intensity of the diffuser at the camera plane, \({U}_{1}^{^{\prime\prime} }\).

$${U}_{1}^{^{\prime\prime} }=\frac{{A}_{\mathit{obj}}^{^{\prime\prime} }{A}_{d}^{^{\prime\prime} }}{{A}_{d}^{^{\prime\prime} }}{\text{exp}}\left[i\left({\varphi }_{obj}^{^{\prime\prime} }+{\varphi }_{d}^{^{\prime\prime} }\right)\right]$$
$$={A}_{\mathit{obj}}^{^{\prime\prime} }{\text{exp}}\left[i\left({\varphi }_{obj}^{^{\prime\prime} }+{\varphi }_{d}^{^{\prime\prime} }\right)\right].$$
(3)

Next, the light back propagation is performed from the camera plane to the diffuser plane. The complex amplitude at the diffuser plane of the object is shown in Eq. (4) as \(U_{2}^{\prime }\).

$${U}_{2}{\prime}={\mathcal{P}}_{c\to d}\left[{U}_{1}^{^{\prime\prime} }\right]={\mathcal{P}}_{c\to d}\left[{A}_{\mathit{obj}}^{^{\prime\prime} }{\text{exp}}\left[i\left({\varphi }_{obj}^{^{\prime\prime} }+{\varphi }_{d}^{^{\prime\prime} }\right)\right]\right]$$
$$= A_{obj}^{\prime } \exp \left[ {i\left( {\varphi_{obj}^{\prime } + \varphi_{d}^{\prime } } \right)} \right].$$
(4)

Here, \({\mathcal{P}}_{c\to d}\) represents the light back propagation calculation from the camera plane to the diffuser plane. The complex amplitude of the diffuser at the diffuser plane \(U_{d}^{\prime }\) is calculated by the light back propagation calculation from the sensor plane to the diffuser plane Eq. (5). Here, we denote the intensity of the object as \(A_{obj}^{\prime }\) and the phase of the object at the diffuser plane as \(\varphi_{obj}^{\prime }\). We also denote the phase of the diffuser at the diffuser plane as \(\varphi_{obj}^{\prime }\)

$${U}_{d}{\prime}={\mathcal{P}}_{c\to d}\left[{U}_{d}^{^{\prime\prime} }\right]={\mathcal{P}}_{c\to d}\left[{A}_{d}^{^{\prime\prime} }{\text{exp}}i\left({\varphi }_{d}^{^{\prime\prime} }\right)\right]$$
$$= A_{d}^{\prime } \exp \left( {i\varphi_{d}^{\prime } } \right).$$
(5)

At the diffuser plane, the phase compensation is carried out. Equation (6) shows the complex amplitude of the object after phase compensated at the diffuser plane, \(U_{3}^{\prime }\).

$$\begin{aligned} U_{3}^{\prime } = & \,A_{obj}^{\prime } \frac{{\exp \left[ {i\left( {\varphi_{obj}^{\prime } + \varphi_{d}^{\prime } } \right)} \right]}}{{\exp \left( {i\varphi_{d}{\prime} } \right)}} \\ = & \,A_{obj}^{\prime } \exp \left( {i\varphi_{obj}^{\prime } } \right). \\ \end{aligned}$$
(6)

Finally, the complex amplitude of the object can be obtained by calculating the optical back propagation of \(U_{3}^{\prime }\) from the diffuser plane to the object plane, which leads to the complex amplitude information \({U}_{obj}\) of the object at the object plane. Let \({A}_{obj}\) be the amplitude information of the object and \({\varphi }_{obj}\) be the phase at the object plane, where \({\mathcal{P}}_{d\to o}\) represents the optical back propagation calculation from the diffuser surface to the object plane.

$$\begin{aligned} U_{obj} = & {\mathcal{P}}_{d \to o} \left[ {U_{3}^{\prime } } \right] = {\mathcal{P}}_{d \to o} \left[ {A_{obj}{\prime} \exp \left( {i\varphi_{obj}^{\prime } } \right)} \right] \\ = & A_{obj} \exp \left( {i\varphi_{obj} } \right). \\ \end{aligned}$$
(7)

The phase information \({\varphi }_{obj}\) can be obtained from the complex amplitude \({U}_{obj}\) obtained in Eq. (7). After phase unwrapping, phase image at the object plane is obtained.

3 Experiment setup

The optical setup used for our experiments is presented schematically in Fig. 3. The experiment setup of such an apparatus was described in an earlier report [16]. The laser we used was a linearly polarized He–Ne laser irradiating at 632.8 nm. A collimated and expanded laser beam was separated using a beam splitter (2 cm cube-type) into two beams: a reference beam and a beam passing through the sample, i.e., the object beam. Two beams from a reference and object were spatially overlapped through a beam splitter (2 cm cube-type). Phase shifting was performed using a piezo-transducer (PZT, P-752.21C; PI Polytec Inc.). The diffused light was captured as a speckle image using a 12-bit monochrome charge-coupled device (CCD) camera (ORCA2.0; Hamamatsu Photonics KK) with 1920 × 1440 pixels and pixel pitch of 3.63 µm. The object beam intensity information was recorded by blocking the reference beam. It should be noted that it is also possible to numerically obtain intensity information from the object intensity I0, I1, I2, and I3. Phase information was retrieved using phase-shifting method [20]. Four digital holograms, I0, I1, I2, and I3 were produced, respectively, with the reference beam phase-shifted by 0, π/2, π, and 3π/2. From the four images, we calculated the quantitative phase using the following equation:

$$\Delta \varphi (x,\,y)\, = \,\tan^{ - 1} \left[ {\frac{{I_{3} - I_{1} }}{{I_{0} - I_{2} }}} \right].$$
(8)
Fig. 3
figure 3

Experiment setup for imaging through scattering media: BS Beam splitter; PZT piezo-transducer

Full size image

Because the phase has a period of 2π, phase is defined in the range of − π < φ <  + π by use of atan2 function. The size of the captured original image was 1920 pixel × 1440 pixel. The discrete phase of the wrapped phase was unwrapped using a phase-unwrapping algorithm using software (Image J). The size of the image was cropped to be 1024 pixel × 1024 pixel. The area around the original image was filled with zero to form a 2048 pixel × 2048 pixel image by padding zeros around the recorded hologram. The images were analyzed using software (Matlab; The Mathworks Inc.). Optical back propagation was conducted by the angular spectrum approach [21]. Holographic diffusers with different diffusion angles were inserted into the object beam path. Details of the holographic diffusers are presented in Table 1.

Table 1 Diffusers used for experiments
Full size table

4 Experiment results and discussion

4.1 Reconstruction of phase distribution of plano-convex lens

We investigated the image quality of reconstructed images of the phase object behind the diffuser. For the experiments, the distance between the diffuser and the sample was 9.5 cm. The distance between the image sensor and the diffuser was 10.5 cm. Experiment results obtained using a spherical plano-convex lens (BK7, SLB-30-1000P; Sigmakoki Co., Ltd.) with 1000 mm focal length as a phase object are described.

We reconstructed phase images of the lens using the method described in the preceding section. Figure 4 portrays the reconstructed phase distribution images without a diffuser and with five diffusers of 1°, 20°, 40°, 60°, and 80° diffusion angles. The bottom panels of Fig. 4 depict the phase cross section on the red line of the phase distribution image in the top panels. The spherical plano-convex lens shape can be visualized from the phase cross section of the phase images. In the case with a diffuser, the reconstructed phase does not completely remove the effect of the diffuser because the phase of the diffuser in priori has ambiguity. Artifacts are seen at the rim of the reconstructed phase images. This is because of the failure of phase unwrapping since the estimated phase of the diffuser is not continuous and phase difference is large.

Fig. 4
figure 4

(Top column) Reconstructed wrapped phase images at the object plane. (Middle column) reconstructed unwrapped phase images at the object plane. (Bottom column) Phase cross section on the red line: a without diffuser, b diffusion angle, 1 degree; c diffusion angle, 20 degrees; d diffusion angle, 40 degrees; e diffusion angle, 60 degrees; and f diffusion angle, 80 degrees

Full size image

It is possible to optimize the reconstruction of amplitude images by varying the distance [16]. However, in the case of phase objects, the distance cannot be specified in the reconstruction. Reconstruction of phase objects requires a priori information on the distance between the camera and the diffuser as well as the position of the phase object. Depth discrimination of the phase object without prior information is a future issue.

4.2 Evaluation of phase distribution

To evaluate the phase distribution quantitatively, the curvature radius was calculated from the experimentally obtained data and was compared with the catalog value. The catalog value for a spherical plano-convex lens (1000 mm focal length) was 519 mm. Figure 5 presents a schematic showing the curvature radius for spherical plano-convex lenses. A spherical plano-convex lens can be regarded as a part of a sphere with curvature radius R. The curvature radius R is calculable using the following equation:

$$R = \frac{{d^{2} + h^{2} }}{2h} .$$
(9)
Fig. 5
figure 5

Image of curvature radius in a spherical plano-convex lens

Full size image

Here, 2d denotes the corresponding diameter of the lens in the CCD camera. The phase difference ∆φ(rad) symbolizes the phase difference in the measurement region of the lens in Fig. 5. These values are related to the lens thickness h according to the following equation.

$$h = \frac{\Delta \varphi \lambda }{{2\pi \left( {n_{1} - n_{2} } \right)}} .$$
(10)

Therein, λ stands for the He–Ne laser wavelength (632.8 nm); \({n}_{1}\) denotes the refractive index of the spherical plano-convex lens (\({n}_{1}=1\).515); \({n}_{2}\) denotes the refractive index of air (\({n}_{2}=\) 1.0). First, we investigated the performance with standard digital holography without a diffuser. When phase imaging was performed without a diffuser, phase difference of 17.1 rad yields lens height h = 3.35 × 10–3 mm. The number of pixels from the apex to the left edge of the graph in Fig. 4 was 513 pixels, leading to 1.86 mm by d = 513 × 3.63 mm: the pixels of a CMOS camera. The curvature radius R is obtained from Eq. (9) as 517.4 mm, resulting in an error of -1.6 mm by comparison to the catalog value, 519 mm.

Then, the performance of estimation curvature is compared to standard digital holography without a diffuser. The curvature radius was calculated from the procedure described above. Table 2 presents the curvature radius without a diffuser and at diffusion angles of 1, 20, 40, 60, and 80 degrees, as well as the error and error rate compared to the catalog value of 519 mm. The error rate is the error value divided by the catalog value of 519 mm. The curvature radius error ranges from − 2.6 mm to + 1.8 mm. The phase image can be reconstructed with error rates of + 0.3% to − 0.5%.

Table 2 Estimated curvature radius. The catalog value of the curvature radius is 519 mm
Full size table

To verify the effectiveness of our method, we present the results obtained for spherical plano-convex lenses (SLB-30-300P, SLB-30-500P; Sigmakoki Co., Ltd.) having different curvature radii. For the spherical plano-convex lens (focal length 300 mm, SLB-30-300P; Sigmakoki Co., Ltd.), the curvature radius was calculated as 157.4 mm. Because the catalog value is 155.7 mm, the error is + 1.7 mm diffuser with a diffuse angle of 80 degrees. The experimentally obtained results are presented for a spherical plano-convex lens (SLB-30-500P, 500 mm focal length; Sigmakoki Co., Ltd.). The curvature radius was calculated as 262.0 mm. Because the catalog value is 259.5 mm, the error is + 2.5 mm. The errors of the curvature radius were + 1.1% and + 1.0% in these cases as well, indicating that the method is useful for spherical plano-convex lenses with different focal lengths.

We used a wedge substrate (WSB-25C05-10–1; Sigmakoki Co., Ltd.) with a wedge angle of 1 degree. The wedge angle was calculated to be 0.967°. This value is within the catalog value of 1° ± 5 min. The wedge substrate can visualize the shape at a diffusion angle of 1 degree, but it becomes difficult to reconstruct the shape at a diffusion angle of 10 degrees or more. This difficult might be attributable to the larger phase change of the object than that of a spherical plano-convex lens.

Imaging of phase objects behind diffusers has been verified through experiments using a plano-convex lens and a wedge substrate as phase objects. In addition, we confirmed the visualization of intensity and phase objects at different depth behind diffusers by used of this method [22].

The experimental method requires obtaining information on the diffuser in advance and uses a four-step phase-shift method, which is not applicable when the scatterer or target object is moving. In this manuscript, we confirmed the effectiveness using phase-shifting method. At present, the visualization of atmospheric distribution behind frosted glass, such as for security, where prior information is can be obtained is considered. The latter problem can be solved using a parallel phase-shift method [23] or fast phase shifter [4].

5 Conclusion

We demonstrated the visualization of phase objects behind diffusers with different diffusion angles by acquiring and correcting the diffuser phase in advance using digital holography. Lensless digital holography was used to reconstruct the phase of an object located behind a holographic diffuser. Spherical plano-convex lenses with different focal lengths and wedge substrates were used as phase objects. Then, the phase reconstruction results through diffusers with diffusion angles of 1°, 20°, 40°, 60°, and 80° are demonstrated. The curvature radius of the spherical plano-convex lenses was calculated from the phase cross sections to demonstrate the feasibility of the method. Presented method requires prior information of diffusers; however, this method leads to applications such as the visualization of atmospheric distribution behind diffusers, where prior information can be obtained.