High-resolution terahertz digital holography based on frequency-domain diagonal extension imaging

By leveraging frequency-domain diagonal extension (FDDE) imaging in terahertz (THz) in-line digital holography, the resolution of THz digital holography would be dramatically improved. The holography system in this work which utilizes a CO2-pumped 2.52 THz (118.83 μm) continuous-wave laser as the source and a pyroelectric-array camera with a pixel pitch of 100 μm as the detector could attain a resolution of 150 μm (~1.26 λ) near to the resolution limit of the system. FDDE imaging is a simple and effective method for resolution enhancement, where no new detector or other equipment is employed. Samples are imaged twice from different angles and then the high-frequency components of images are stitched to improve the resolution. Different from the traditional FDDE, weight-FDDE (w-FDDE) which can reduce the visibility distinction between different orientations of the synthetic picture by adding weight factors to produce a superior synthetic image is proposed in this work.


Introduction
The terahertz (THz) frequency is 0.1-10 THz (wavelength 3000-30 μm). THz waves have unique properties such as penetrability, being both non-destructive and non-ionizing, which give them far-reaching impacts on fields of imaging and digital holography (Siemion et al. 2021;Zhao et al. 2019;Yamagata et al. 2019), THz beam shaping (Minkevičius et al. 2019;Petrov et al. 2022), non-destructive testing (Xu et al. 2022), terahertz spectroscopy (Zhang et al. 2021), biomedicine (MacPherson et al. 2013), THz microscopy (Blanchard et al. 2018;Tang 2019), food safety detection (Di Girolamo et al. 2021), security inspection (Ma et al. 2018) and more. In THz digital holography (TDH), the resolution is a key criterion for imaging quality. However, the long wavelength of THz wave results in obvious diffraction, which 1 3 235 Page 2 of 20 seriously affects the imaging quality. Thus, improving the resolution of TDH is of great significance. In recent years, many researchers have attempted to enhance the resolution of TDH (Xue et al. 2012;Ahi et al. 2018;Huang et al. 2019;Rong et al. 2014) improved the resolution up to 160 μm (~1.35λ) by using a subpixel shifting and extrapolation method with a THz laser source of 118.83 μm. Huang et al. (2016) leveraged the synthetic aperture method to obtain a resolution of 125 μm (~1.26λ and the laser source is mixed emitted THz waves with wavelengths of 97, 97.6, and 98.9 μm by a ratio of 4:1:2. In Li et al. (2016), a lateral resolution of 150 μm (~1.26λ was obtained by using a subpixel sampling algorithm with a THz laser source of 118.83 μm. Li et al. (2019) achieved a lateral resolution better than 70 μm (~λ) by employing a novel reconstruction algorithm with a laser source of ~70 μm. Balbekin et al. (2019) presented a novel numerical approach for improving the resolution in THz pulse timedomain holography (THz PTDH). Mao et al. (2022) proposed a method based on the physical model and multiscale retinex (MSR) algorithm to improve the quality of THz images and retain sufficient image details. Moreover, a profound approach to improve the resolution is to expand the frequency domain, which has been utilized in various imaging fields. Zheng et al. (2013) proposed Fourier ptychography microscopy (FPM), which efficiently expands the frequency domain and enhances the resolution. Valzania et al. (2018) performed ptychography at THz frequencies and a lateral resolution < 2λ was acquired with a laser source of 96.5 μm (3.1 THz). The methods mentioned before can enhance the resolution, but they may increase the experiment or algorithm complexity. Frequency-domain diagonal extension (FDDE) imaging (Jiang et al. 2020), which has a similar principle to FPM, does not require additional complex experimental devices. This method only needs to adjust the orientations of samples or detectors and the processing algorithm is also simple and efficient.
In TDH, holograms are recorded by a detector, which discretizes continuous signals into digital ones. According to the Nyquist-Shannon sampling theorem, the smaller the pixel pitch is, the higher the sampling frequency is, and the better the result is. However, the image is two-dimensional (2D). The sampling intervals will be different along different directions of the detector because of the sampling anisotropy of grid-like detectors. The sampling interval is 1 pixel along horizontal/vertical directions, while the sampling interval is 0.707 pixels along diagonal directions, which has been proved in the literature (Jiang et al. 2020) proposed FDDE microscopy and the resolution was enhanced by 1.4 times.
As the best knowledge of us, FDDE is first adopted in THz holography to enhance the resolution in this work. A resolution of 150 μm (~1.26λ) is obtained in the THz holography imaging system by using FDDE. Samples are placed at 0° and 45° with respect to the pixel array of the detector chip to obtain two frames of holograms. Then the reconstructed patterns are obtained by using the single-exposure amplitude constrained phase retrieval algorithm T-APRA (Hu et al. 2014). Finally, FDDE algorithm is used to synthesize the two reconstructed images and achieve a higher resolution. The modified algorithm weight-FDDE (w-FDDE) is proposed to reduce the visibility distinction between diagonal orientations and vertical/horizontal orientations of the synthetic image. By adjusting weight factors, the visibility of the synthetic image will be equal in horizontal/vertical and diagonal directions.
Page 3 of 20 235 2 Principle and method

Phase retrieval algorithm
We use the single-exposure amplitude constrained phase retrieval algorithm (T-APRA) to reconstruct the wavefront from the hologram (Hu et al. 2014). The flow chart of the algorithm is shown in Fig. 1a. The final retrieved hologram h final is the average of 40 original holograms: where n is the number of original holograms (n = 40 here), H i is the i-th original hologram. By averaging multiple holograms, the noise could be suppressed to some extent. In T-APRA, the reconstruction of the wavefront is accomplished by iterating the light field distribution pattern between the recording (detector) plane and the object (sample) plane. The sequence number of iterations is indicated by the superscript. The recording plane is indicated by subscript 1 and the object plane is indicated by subscript 0. Firstly, the light field distribution U 1 1 on the recording plane is taken as the initial value of iteration. The amplitude A 1 of U 1 1 can be known by squaring the normalized hologram which is obtained by dividing h final and the background intensity: h final ∕B background , whereh final is the final hologram, B background is the intensity distribution when samples are removed. According to the literature (Hu et al. 2014), the initial phase φ 1 of U 1 1 can be set to a random number. Suppose that the initial phase φ 1 is a constant c 0 , which is set to π/4 in this paper: where, U 1 1 is the starting data of the iteration part, (x 1 , y 1 ) is coordinates of the recording plane, A 1 and 1 (x 1 , y 1 ) are amplitude and phase, respectively. U 1 1 (x 1 , y 1 ) represents the light field distribution of the first iteration on the recording plane.
Firstly, the angular spectrum is used to get the light field distribution from the recording plane to the object plane. The distribution of light field on the object plane U n 0 is obtained: is the inverse angular spectrum transfer function, (x 1 , y 1 ) is the coordinates of the object plane,(f x , f y )represent the frequencydomain coordinates, λ is the wavelength, k is the wave vector, z is the distance between the object plane and the recording plane, F represents fast Fourier transform, and F − 1 represents inverse fast Fourier transform.
After obtaining U n 0 , then the amplitudeA 0 (x 0 , y 0 ) of U n 0 will be constrained in T-APRA algorithm. The amplitude A 0 (x 0 , y 0 )should not be greater than 1 because of the energy loss in propagation, while the phase 0 (x 0 , y 0 )remains unchanged: whereA � 0 (x 0 , y 0 )and � 0 (x 0 , y 0 )will replace A 0 (x 0 , y 0 )and 0 (x 0 , y 0 ) , respectively. Then we obtain a new U n 0 : The new light field distribution on the recording plane is obtained by angular spectrum: is the forward angular spectrum transfer function. The phase n+1 1 is extracted from the light field U n+1 1 , while the amplitude of U n+1 1 is replaced by initial amplitudeA 1 (x 1 , y 1 ) . Thus, a new light field on the recording plane is obtained: An iteration is completed here and U n+1 1 is used as the starting data for the next iteration. The wavefront can be reconstructed after multiple iterations.

The principle and algorithm of FDDE
FDDE is a simpler method in terms of experiments and processing algorithms compared with FPM, synthetic aperture, and other frequency-expanding methods. The principle of FDDE is straightforward: the sampling frequency of diagonal directions is higher than vertical/horizontal directions in grid-like detectors and it will cause an anisotropic resolution when imaging. It can be explained from the frequency domain, which has been proved in detail in the literature Jiang et al. 2020). Under the condition of under-sampling, the frequency domain of digital imaging is smaller than the theoretical optical transfer function (OTF) in diffraction-limited imaging systems, as shown in Fig. 1 (b). OTF and frequencydomain pattern of the digital image can be represented: where r OTF is the boundary of OTF, M and NA are the magnification and numerical aperture of the optical imaging system, respectively, and Δ is the pixel pitch of the detector. Frequency-domain coordinates in diagonal directions should meet the following relationship: where f diagonal represents the maximum frequency in diagonal orientations. The sampling frequency of diagonal orientations is √ 2times higher than vertical/horizontal orientations. The reason is that the sampling interval of diagonal directions is smaller-the interval is only1∕ √ 2 pixels, approximately 0.707 pixels (Fig. 1b) in diagonal directions. Diagonal directions have a higher sampling frequency. That means we can get a higher resolution in diagonal directions, but not in horizontal/vertical directions. We can get multiple reconstructed images from different imaging angles and these original images have high resolution only in diagonal directions. But FDDE algorithm can make the synthetic images have a high resolution not only in diagonal directions but also in horizontal/vertical directions. Referring to Jiang et al. 2020), the flow chart of THz digital holography FDDE algorithm is shown in Fig. 1c. First, the wavefront is reconstructed from holograms by using T-APRA. Then the image interpolated method FFT interpolation with a zero-padding method is conducted on the reconstructed images. FFT interpolation is implemented in three steps: i. FFT of the image, ii. Pad the spectrum with zero, iii. IFFT of the spectrum. Then rotate the image in the same direction as the other and crop out the interesting areas. A subpixel registration algorithm is utilized to correct the position of two images and ensure that the imaging object locates at the same position in the two original images (Guizar et al. 2008). Then FFT is performed on the image to obtain the frequency spectrum. The high sampling frequency directions of two reconstructed images are selected, respectively, and then they are stitched together, as shown in Fig. 1d. One part of the final spectrum is selected in four diagonal directions from one reconstructed image, while the other is selected in four horizontal/vertical directions. This is because the other one is rotated. High sampling frequency directions are selected from the two images. Finally, IFFT is performed on the final spectrum to obtain the synthetic image.

The proposed algorithm weight-FDDE (w-FDDE)
According to FDDE algorithm, as shown in Fig. 1 (c), the spectrum of the synthetic image consists of two parts that come from two reconstructed images, respectively. The final spectrum of the synthetic image is: where f final is the final spectrum, f 1 and f 2 are spectrums from two reconstructed images, respectively. The final synthetic image I final is given by IFFT: where I 1 and I 2 are variables that are related to two original images' visibility and intensity. However, it is hard to get two images that have the same visibility due to the samples' rotations, the instability of the laser source, and other factors in the experiment. Usually, one image has high visibility, corresponding to the 'strong' frequency spectrum, while the other has the 'weak' one. If we add the spectrums directly, the final synthetic image I final will be affected by I 1 and I 2 related to two the original images' visibility and intensity. The final synthetic image can have distinct visibility at different orientations. The visibility can be expressed as: where I max and I min are the maximum and minimum values of cross-sections. Three steps of calculating visibility: i. Plot the cross-section curves of diagonal directions or horizontal/vertical directions. ii. Find the maximum I max and minimum I min values from the curves.
iii. Calculate the visibility using the above format. We can acquire three pairs of peakvalley values according to our samples utilized in the work and three visibility values will be calculated from each cross-section of the image, as shown in below tables. In the proposed weight-FDDE (w-FDDE) algorithm, we put weight factors on the spectrum to cut down the 'strong' frequency spectrum or enhance the 'weak' frequency spectrum. We can get a synthetic image that has the same visibility at different orientations. The final spectrum is: where K 1 and K 2 are weight factors. The final synthetic image is given by IFFT: Compared with FDDE algorithm, the final image synthesized by w-FDDE algorithm is not only determined by I 1 and I 2 which are related to two original images' visibility and intensity but also weight factors. We can get a synthetic image that has the same visibility at different orientations by adjusting factors K 1 / K 2 . The flow chart of w-FDDE is shown in Fig. 2. The w-FDDE algorithm implementation is as follows: 1. Adjust weight factors K 1 /K 2 to get different synthetic images; (9) The synthetic image's visibility of diagonal directions V d , and the visibility of horizontal/ vertical directions V h are calculated, respectively; 3. Compare V d and V h . The synthetic image is obtained when they are equal.

Experimental process
The experimental setup is described in detail in the literature (Xue et al. 2012), as shown in Fig. 3a. Here, we need to rotate the samples for FDDE imaging. A CO 2 -pumped continuous-wave THz laser SIFIR-50 with an operating frequency of 2.52 THz is used as the imaging source. Two gold-coated off-axis parabolic mirrors (PMs) with focal lengths of 50.8 mm are utilized to compose a collimated lens combination. A gold-coated mirror M 1 with a diameter of 50 mm is used to change the beam direction. The distance between PM 1 and the laser source is about 180 mm. The distance between PM 2 and M 1 as well as the distance between M 1 and the Pyrocam III detector, are almost 95 and 170 mm, respectively. The pyroelectric camera Pyrocam III with 124 × 124 pixels is utilized as a detector. The pixel pitch is 0.1 mm. The samples are placed close to the detector. Rotating the samples will satisfy the requirement of different imaging angles.
Moreover, a commonplace issue unsolved is that THz imaging suffers from beam inhomogeneity and singular point which will seriously influence the picture captured by the detector. Various THz sources such as a quantum cascade laser (Chopard et al. 2022), diode laser (Petrov et al. 2020;Agour et al. 2022), or free electron laser (Choporova et al. 2022) have to face the thorny problem. Happily, FDDE is possible to alleviate the problem of THz beam inhomogeneity and singular point by rotating the object. Rotating the object or detector makes no difference in resolution enhancement in FDDE imaging. But it's way better to rotate the object in the situation of beam inhomogeneity or singular point of THz source. The reason is straightforward. Two pictures captured by the detector by rotating objects share different information about the object because the singular point can come to different places in the image. And the two pictures are complementary to each other. The object can be retrieved from the two pictures by using a synthesizing algorithm, as shown in Fig. 3 (b). However, two pictures got by rotating the detector contain almost the same information about the object because the singular point affects the same part of the object. We cannot reconstruct the extra information about the object and the final image will lose part of the information about the object because of the beam inhomogeneity or singular point.
Referring to (Jiang et al. 2020), in order to observe the improvement of the resolution, we designed one object similar to a ring and the other similar to an octagonal ring, which are hereinafter referred to as the ring and octagonal ring, as shown in Fig. 4 (a) and (b). The smallest feature sizes of samples are 200 μm and 150 μm, respectively, corresponding to 2 pixels and 1.5 pixels of the detector. There is a connecting line on one side of the ring and the octagonal ring to identify the orientation. The octagonal ring is different from the ring in the materials. The material of the rings is copper foil stripe based on polytetrafluoroethylene (PTFE) printed board and the stripe spacing is 0.15 and 0.2 mm, respectively, as shown in Fig. 4 (c) and (d). The total thickness of the ring is about 1.5 mm and the overall transmittance is about 80%. The material of the octagonal rings is PTFE glass fiber cloth copper clad sheet F4BM220. The stripe material is copper and the stripe spacing is 0.2 mm, as shown in Fig. 4 (e). The thickness of the copper coating is about several microns. The total thickness of the material is 0.25 mm and the overall transmittance is about 70%.

Simulation of the anisotropic resolution of FDDE imaging
Grid-like detectors have distinct sampling intervals along different directions and it will result in an anisotropic resolution when imaging. In this section, we would like to demonstrate the anisotropic resolution of FDDE imaging. The simulation will also prove that FDDE imaging still works in such THz imaging system. The simulation steps are as follows: 1. Get a reconstructed image of the sample; 2. Get a reconstructed image of the sample rotated 45° against step i; 3. Compare the two reconstructed images.
The simulation wavelength is 118.83 μm and the interval of detector pixels is 0.1 mm. The recording distance (distance between the object plane and the screen plane) is 15 mm. The sample with a size of 124 × 124 pixels corresponding to 12.4 × 12.4 mm 2 is shown in Fig. 5a. Figure 5b is the hologram of Fig. 5a. Figure 5c is the reconstruction of Fig. 5b. Figure 5d is the interpolated image of Fig. 5c by employing the FFT interpolation method and we can get the cross-section in the image. Then, we rotate the sample 45°, as shown in Fig. 5e. Figure 5f is the hologram of Fig. 5e. Figure 5g is the reconstruction of Fig. 5f. Figure 5h is the interpolated image of Fig. 5g and the cross-section is also shown in the image. It can be concluded that the reconstruction of the sample rotated 45° has a better cross-section which means a higher resolution and better visibility by comparing the

Simulation of the proposed w-FDDE
The w-FDDE algorithm is utilized to decrease the visibility distinction between synthetic images' different orientations. In the simulation, we will know how the synthetic image and its corresponding cross-sections change when we change the weight factors K of w-FDDE. The distinctions will be highlighted by comparing FDDE and w-FDDE. It will also show the advantages of w-FDDE in the simulation. The sample is a ring with a size of 124 × 124 pixels corresponding to 12.4 × 12.4 mm 2 , as shown in Fig. 6a. Figure 6b is the hologram of Fig. 6a after down-sampling and FFT interpolation. The recording distance is 2 mm. Figure 6c is the reconstruction of Fig. 6b. Figure 6d is the interpolated image of Fig. 6c. The cross-sections of diagonal and horizontal/vertical orientations are shown in Fig. 6d. The anisotropic resolution of FDDE imaging can be demonstrated from the cross-sections because it can be distinguished from diagonal direction cross-sections but vertical/horizontal direction cross-sections. Figure 6e is the hologram of the sample rotated 45°. Figure 6f is the reconstruction of Fig. 6e. Figure 6g is the interpolated image of Fig. 6f. Then Fig. 6g is rotated back to the original position, as shown in Fig. 6h. Figure 6i is the spectrum of Fig. 6d. Figure 6j is the high-frequency component f 2 that has been cropped out from Fig. 6i and it has been marked with white lines. Figure 6k is the spectrum of Fig. 6 (h). Figure 6l is the high-frequency component f 1 cropped out from Fig. 6 (k) and it has been marked with white dot lines. Figure 6 (m) has shown how the synthetic image and its corresponding cross-sections change when we change the weight K 1 . The visibility of vertical/horizontal directions becomes better and better when we increase K 1 gradually and keep K 2 = 1, as shown in Fig. 6m. The visibility of different directions can be adjusted by changing weight factors in the w-FDDE algorithm, while it cannot be achieved in the FDDE algorithm. It can be concluded that the visibility distinction can be well addressed by w-FDDE algorithm.

The ring synthesized by FDDE
The resolution disadvantage in horizontal/vertical directions of a single image can be made up by the advantage of the other image. If we first implement imaging on the sample, we can get a higher resolution in diagonal directions but a lower resolution in vertical/horizontal directions. Then we rotate the sample 45°. It can be found that the original diagonal orientations in the first imaging will become vertical/horizontal orientations in the second imaging, while the original vertical/horizontal orientations of the first imaging will become diagonal orientations in the second imaging. In the second imaging, we can get a higher resolution in diagonal directions, corresponding to the vertical/horizontal orientations in the first imaging, but a lower resolution in vertical/horizontal directions, corresponding to  the diagonal orientations in the first imaging. So, when we synthesize the high-resolution directions, the disadvantage of the first imaging can be made up by the advantage of the second imaging. We implement imaging on the ring twice at 0° and 45° and then synthesize them by using FDDE algorithm. The two holograms with a size of 124 × 124 are shown in Fig. 7a and b. Figure 7c and d are reconstructed images of Fig. 7a and b, respectively. The recording distance is 15.35 and 15.40 mm, respectively. In the holographic system, the resolution criterion is R = ∕(2NA) (Latychevskaia 2019), while NA=L/2Z . So, the limited resolution is R = * z∕L , where L is the size of the detector. The limit resolution is about 0.14 mm in the holographic system. The number of iterations for both two samples is 10 in T-APRA. Then we perform the FFT interpolation on reconstructed images with an up-sampling factor of 4, as shown in Fig. 7e and f. Then the center parts of the two images are cropped out, with the size of 150 × 150, as shown in Fig. 7g and h. The sub-pixel matching algorithm is used to correct the position of the center parts. Finally, Fig. 7i is the synthetic image of the ring. We mark 8 scanning lines at the same positions of the synthetic image Fig. 7i Table 1.
Combining reconstructed images and Table 1, the ring with a minimum feature size of 0.15 mm can be distinguished in diagonal directions of reconstructed images, but in horizontal/vertical directions. It well proves the sampling frequency of diagonal directions is higher than horizontal/vertical directions in grid-like detectors. Moreover, there are only two valleys in cross-sections of line 2, line 3, and line 4 in the original image, so stripes cannot be distinguished in the original image, as shown in Fig. 7k, l, and m. However, there are three valleys in cross-sections of the synthetic image synthesized by FDDE, as shown in Fig. 7j, k, l, and m. The stripes in the synthetic image can be distinguished in all directions after synthesizing by FDDE. The resolution is enhanced in horizontal/vertical directions after utilizing FDDE. In the experiment, we successfully distinguish the 0.15 mm ring by using FDDE imaging. However, heavily impacted by the sample materials which have a fine sin-structure around the object, as indicated in the background of Fig. 7c and d, the synthesized image has lower visibility at diagonal directions compared with the original image Fig. 7h, as shown in Table 1. The high-frequency component f 2 contains the frequency information of the fine structure and contributes to the synthesized spectrum, finally influencing the synthesized image after IFFT. In all, the final image synthesized by FDDE makes up the resolution disadvantage caused by the anisotropic sampling frequency of grid-like detectors. But it pays the price of reduced visibility due to the fine structure of the materials.

The ring synthesized by w-FDDE
Two reconstructed images of the ring of 0.15 mm are synthesized by the proposed w-FDDE algorithm. The lines in Fig. 8a and b are cross-sections of the synthetic image of the ring shown in Fig. 7i. We adjust K 2 to get different cross-sections. For each K 2 , cross-sections of the diagonal orientation and the vertical/horizontal orientation are plotted in Fig. 8a and b, respectively. And visibility of each cross-section in Fig. 8a and b is calculated according to Eq. (11). For each K 2 , visibility of both the diagonal orientation and the vertical/horizontal orientation can be obtained. Then two lines in Fig. 8c are plotted. It can be found that with the increase of K 2 , the synthetic image's visibility in diagonal directions increases but decreases in horizontal/vertical directions. This is because f 2 represents the frequency domain in diagonal directions, and increasing K 2 increases visibility in diagonal directions. When visibility in horizontal/vertical and diagonal directions is equal, K 2 = 0.8, and the synthetic image is shown in Fig. 8d. The image synthesized by FDDE is shown in Fig. 8e. The visibility of w-FDDE's image is higher in horizontal/vertical directions than FDDE's image because of the weight factor K 2 < 1.

5.2
The octagonal ring with a minimum feature size of 0.2 mm

The octagonal ring synthesized by FDDE
We implement imaging on the 0.2 mm feature size octagonal ring twice at 0° and 45°.
The two holograms are shown in Fig. 9a and b. Then the wavefront is reconstructed by T-APRA and the number of iterations for both two samples is 10. Recording distances are 16.98 mm and 16.50 mm, respectively. Final reconstructed images are shown in Fig. 9c and d, corresponding to imaging angles 0° and 45° respectively. According to FDDE, we perform the FFT interpolation on reconstructed images with an up-sampling factor of 2, as shown in Fig. 9e and f. Then the interesting parts of the two images are cropped out, with a size of 100 × 100 pixels, corresponding to the size of 50 × 50 pixels in the original image (because of the up-sampling factor 2), as shown in Fig. 9g and h. Then the sub-pixel matching algorithm is used to correct the position of two images. Figure 9i is the synthetic image of the octagonal ring. To judge whether the resolution is enhanced or not, three lines are marked in the same position as the synthetic image Fig. 9i as the original image Fig. 9h, such as line 1, line 2, and line 3. Figure 9j, k, and l are cross-sections of lines marked in the octagonal ring, respectively. The blue and red curves are cross-sections of the synthetic and original images, respectively. The visibility of images at the marked lines is calculated, as shown in Table 2. According to reconstructed images, the 0.2 mm stripes can be distinguished in both diagonal and horizontal/vertical directions. Combined with cross-sections and visibility in Table 2, it can be seen that the visibility of the image synthesized by FDDE algorithm is higher, so it is easier to distinguish targets, and the resolution in horizontal/ vertical directions of the image is enhanced. However, it can be found that the synthetic image is more blurred in diagonal directions than the original image. The reason is that one of the two reconstructed images has high visibility in horizontal/vertical directions, while the other has low visibility in diagonal directions. So, if adding two parts of the frequency domain directly, the synthetic image will have high visibility in horizontal/ vertical directions but low visibility in diagonal directions. It needs weight factors to reduce the visibility difference between the horizontal/vertical and diagonal directions of the synthetic image.

The octagonal ring synthesized by w-FDDE
The two reconstructed images are synthesized by w-FDDE algorithm. Keep K 1 = 1, adjust K 2 , and cross-sections of diagonal and horizontal/vertical orientations are shown in Fig. 10 (a) and (b), respectively. Then, visibility is calculated according to crosssections, as shown in Fig. 10 (c). When the visibility in horizontal/vertical and diagonal directions is equal, K 2 ≈ 1.6. w-FDDE's synthetic image is shown in Fig. 10 (d) and FDDE's synthetic image is shown in Fig. 10 (e). Compared with the image synthesized by FDDE, w-FDDE's image has higher visibility in diagonal directions because of the weight factor K 2 > 1. And the resolution is enhanced in diagonal directions.

The ring with a minimum feature size of 0.2 mm
We also experiment with a 0.2 mm feature-size ring. The two holograms are shown in Fig. 11a and b. Figure 11c and d are reconstructed images of Fig. 11a and b, respectively. The recording distances are 15.10 and 15.45 mm. The number of iterations for both two samples is 10 in the T-APRA. Figure 11e is the synthetic image of the ring using FDDE. Figure 11f is the synthetic image of the ring by employing w-FDDE when K 2 = 1.8. We mark the lines in Fig. 11e and draw cross-sections of original and synthetic images, as shown in Fig. 11g, h, and i. Visibility is shown in Table 3. In the experiment of a 0.2 mm resolution ring, combined with cross-sections and   holography system achieves a lateral resolution of 0.15 mm (~1.26λ), which is close to the resolution limit. Moreover, we propose weight-FDDE (w-FDDE) to decrease the visibility difference between the horizontal/vertical and diagonal orientations in the synthetic image and it achieves a better synthetic result.