Conical holographic display to expand the vertical field of view

: Recently, cylindrical holographic display technology as a 360-degree display technology has attracted much attention. However, all the studies are based on the field of view (FOV) in the azimuth direction, and the issue of FOV in the vertical direction has never been discussed. In this paper, a new holographic display is proposed to expand the vertical FOV by a conical holographic diffraction model, in which the object plane is the outer cylinder and the observation plane is a part of the cone. In this proposed method, the proposed diffraction model is firstly established by the Rayleigh-Sommerfeld diffraction formula, and then the convolution and FFT are used for a fast diffraction calculation. The correctness and effectiveness of our proposed method are verified by the simulation of Young’s interference and the numerical reconstructions from the complex amplitude and encoded holograms, respectively. In addition, an accurate relationship between the conical inclination angle and the vertical FOV expansion is analyzed, and the simulation results show that our proposed method can significantly expand the vertical FOV by 0.4 of the original object. Therefore, the issue of the vertical FOV in cylindrical holography is deeply discussed and successfully addressed for the first time.


Introduction
Holographic technology is widely considered as an ideal three-dimensional (3D) display technology because it can provide to human eyes the complete parallax and depth information [1].With the development of computers, computer-generated-hologram (CGH) [2] plays an important role in 3D display.By computer synthesis and coding, the information of complex amplitude of virtual 3D objects can be recorded, which provides convenience for the generation of the hologram.However, the field of view (FOV) [3] of the traditional plane hologram is limited by its spatial frequency and physical size [4], and it has always been a significant issue in the research of CGH.A curved hologram is an effective way to overcome the constraint of FOV [5][6], and the cylindrical computer-generated hologram (CCGH) is considered as an ideal choice to expand FOV because it has a 360-degree look-around property [7][8][9], which makes it possible to observe the image of the object from any direction.
In recent years, many researchers have devoted to the field of cylindrical holography.Yamaguchi et al. proposed a fast calculation method for cylindrical CGH that is viewable in 360-degree look-around [10].Sando et al. proposed a new algorithm based on fast Fourier transform (FFT) to obtain cylindrical holograms, which greatly reduced the computation time [11].In addition, they proposed a calculation method of CCGH based on 3-D Fourier spectrum [12] and used a Bessel function expansion to reduce the calculation time and memory usage for CCGH [13][14].Jackin et al. proposed another fast calculation method based on FFT and summarized the angular spectrum diffraction formula for cylindrical models, which were used to simulate 360-degree reconstruction of 3D objects [15].Zhao et al. accelerated the calculation of CCGH by introducing a wavefront recording plane (WRP) [16], which further reduced the cost of computation.Wang et al. proposed a unified and accurate diffraction calculation method based on FFT, which could satisfy both inside-out propagation and outside-in propagation models [17][18].Kang et al. proposed a method for calculating a curved hologram to increase the view angle in a holographic display [19] and contributed two fast calculation methods for a curved hologram based on WRP.One was the diffraction compensation, and the other was the approximate compensation.More recently, Li et al. explored a curved composite hologram generation method with suppressed speckle noise [20].Li et al. proposed an occlusion culling method based on the horizontal optical-path-limit function, which improved the visibility and quality of cylindrical holography [21].However, in all above these existing researches, they tried to increase computational speed or expand the FOV in the azimuth direction, while the FOV in the vertical direction has never been discussed.This paper focuses on how to expand the vertical FOV based on the 360-degree look-around, which has never been deeply discussed before, as far as we know.
In this paper, a new holographic display is proposed to expand the vertical FOV for the first time.In the proposed method, a conical holographic diffraction model is established, firstly, based on the application scenario that the object and hologram planes are an outer cylinder and a part of a cone with an inclination angle, respectively.In the proposed model, the Rayleigh-Sommerfeld (RS) diffraction formula is applied to establish the point spread function (PSF).Since the calculation formula conforms to the convolution form, FFT can be used to improve the calculation speed.Secondly, the sampling condition in the proposed method is probed, and the calculation method of sampling points is given.Thirdly, by analyzing the diffraction range of the source point, it is obviously that the diffraction region of each source point is not the whole observation plane.Otherwise, it can result in an overlapping reconstructed image.To address this problem, the horizontal optical-path-limit function of the cylindrical diffraction is introduced, which should be re-derived to satisfy our method.Fourthly, the relationship between the conical inclination angle and the expanding of the vertical FOV is deduced by theoretical analysis, and an accurate calculation formula is given.The correctness of the proposed method is demonstrated through the simulation of Young's interference.Furtherly, the effectiveness of the proposed method is verified by the numerical reconstructions from the complex amplitude, the amplitude-truncation and the double-phase method encoded holograms.

Conical holographic diffraction model
To expand the FOV, a conical holographic diffraction model is established.In the proposed model, the hologram plane is set to be a part of a cone instead of a cylinder.The schematic of the proposed model is shown in Fig. 1(a), where the outer cylinder is the object plane, and the observation plane is a part of the cone.For convenience of calculation, the origin "O" is located at the middle of the truncated cone.P is a point on the inner cone, and Q is a point on the outer cylinder.In cylindrical coordinates, they can be described as P(θ 1 , R z , z 1 ) and Q(θ 0 , R, z 0 ).For P point, the relationship between radius R z and height z 1 is given by: where r 0 and r 1 denote the radius of the lower and upper circle of the cone, respectively, and H is the height of the hologram as shown in Fig. 1(b).Besides, the inclination α of the cone can be described as: Based on the assumption that an object is a collection of multiple point light sources, each point on the object surface can be regarded as a point light source.In cylindrical coordinates, the RS diffraction formula [22] is expressed as: where H(θ 1 , z 1 ) and U(θ 0 , z 0 ) are complex amplitudes of diffraction field and object plane, respectively.The integral range should be the entire cylindrical object plane.When the object plane radius is much larger than the hologram plane radius, the obliquity factor K(θ 0 ) can be approximated to 1, and d(θ 0 , z 0 , z 1 ) defines the propagation distance from the source point (θ 0 , R, z 0 ) to the destination point (θ 1 , R z ,z 1 ).According to the formula for distance between two points, d(θ 0 , z 0 , z 1 ) is calculated by: where R is the radius of cylinder, and R z is the radius of the cone at height z 1 .Therefore, the PSF h(θ, z 0 , z 1 ) is defined as: For each specific height z 1 , R z is a constant, the Eq. ( 3) can be rewritten as: Obviously, Eq. ( 6) represents the convolution calculation: where the symbol * θ indicates the one-dimensional convolution integral with respect to the azimuthal direction.It can be calculated by FFT: where the FFT used for the vertical direction, after calculating every z 1 in this way, the diffraction field at different heights can be obtained.

Sampling conditions
In numerical simulation, the Nyquist theorem must be satisfied in both the azimuthal and the vertical directions.Since the spatial frequency of the object function U(θ 0 , z 0 ) is much less than the PSF h(θ, z 0 , z 1 ), the maximum value of the spatial frequency only depends on h(θ, z 0 , z 1 ).
Here, z 1 can be regarded as a constant since each specific each height z 1 corresponds to a specific one-dimensional PSF.The frequency of change in the azimuthal and the vertical directions can be calculated as follows: In the calculation, due to the speed of spatial change in exp(ikd(θ, z 0 )) is much faster than d(θ, z 0 ), the maximums value of f θ (θ, z 0 ) and f z (θ, z 0 ) can be simplified to: When R z is set to maximum value, the f θ and f z can get the highest spatial frequency.Besides, the Nyquist theorem requires the sampling frequency to be twice the maximum frequency.The spectral band width W θ and W z are given by: thus, sampling points N θ and N z are as: where the |R z | max is the maximum value of R z .

Occlusion culling in a conical diffraction model
According to Eq. ( 6), the integral range is the entire cylindrical object plane, which means the diffraction waves from the source point can act on all points on the cone.However, because the recording plane frequently is opaque, the occlusion culling in conical diffraction model should be considered as that in cylindrical diffraction model [21].As shown in Fig. 2(a), the light from a source point F cannot reach the point T on the observing plane, which should be filtered.Here, the occlusion culling can be achieved by a filtering function of the horizontal optical-path-limit function [21].As shown in Fig. 2(a), the points of L and P as well as the points N and S are the points on the cone with different radiuses.If FL and FP as well as FN and FS are tangent to the cone on different radiuses, which means that the angles of ∠FLE, ∠FPE, ∠FNE, and ∠FSE are right angles, the corresponding filtering function is summarized as: where r is the distance between the source point and a point on the concentric circle, so the Eq. ( 5) should be written as: Besides, for a cone, the larger radius leads to the smaller angle of the effective area after the occlusion culling by filtering function.The angle of the effective area γ can be exactly calculated as: As shown in Fig. 2(a), the angle ∠FEN is larger than angle ∠FEL as an example, which are 68.58°and59.76°, repetitively.In order to further show this finding, a simple simulation is carried out.The object is a line light source located at θ 0 =0 in the cylindrical surface and is just F in top view as shown in Fig. 2(a).And the values of R and H are both fixed to 100 mm, r 0 and r 1 are set to 120 mm and 80 mm, respectively, the wavelength is set to 300 µm.In this case, a sampling number of 400×800 can satisfy the sampling conditions.When the inclination α is 6°, the conical hologram in 2D view is shown as Fig. 2(b).Here, the black areas in Fig. 2(b) are filtered parts, there are two sloping edges between the black areas and diffraction patterns.The sloping edges show that the larger radius leads to the smaller angle of the effective area, which agrees with the mentioned finding.

Expanding of the FOV
The advantage of using the conical plane as a hologram plane lies in the expanding of FOV in the vertical directions.In this section, the expanding of vertical FOV is explained theoretically.According to the diffraction theory [23], when the pixel shape of the hologram is square, the diffraction angle β should be calculated as: where d is the pixel pitch, λ is the wavelength of the reference beam in free space.The diffraction angle β of a cylindrical hologram is shown as Fig. 3.When a conical plane is used as a hologram plane, the angle of view is equal to the diffraction angle plus the inclination α which can be calculated by Eq. ( 2).
As shown in Fig. 3, in two kinds of practical applications, the FOV can be divided into three parts: the expanding of FOV, the overlap of FOV, and the useless part of FOV.The expanding of FOV is the FOV that we can expand with our proposed model, and the useless part of FOV is the part that is difficult to observe in practical application.In order to get an expanded FOV, the angle α and the angle β should satisfy: Therefore, in the vertical direction, there is a geometric relationship that shows the size of the expansion of FOV, ∆h(R, α), which can be calculated as:

Correctness verification of the proposed conical diffraction model
To verify the correctness of the proposed diffraction model, the simplest experiment is Young's interference experiment.There are two source points M and N in the object surface located at (-π/32, 0) and (π/32, 0), respectively.The radius R of the outer cylinder, which is the object plane, is set to 50 mm, and the height of the cylinder and the cone is 50 mm.According to Eq. ( 11), in the terahertz wavelength range, a considerable number of sampling points is obtained.Therefore, the wavelength λ is set to 250 µm, and an image size of 512×1024 is chosen as the object, which satisfies the sampling conditions.In the case of different cone inclination angles, different holograms are obtained as shown in Fig. 4. From the experimental results, it is found that the symmetry of the hologram is less obvious with the gradual increase of the inclination.The interference is related to the optical path difference.There is a good symmetry in the azimuthal direction, which is consistent with the simulation results.Besides, to better explain the asymmetry in vertical direction of the simulation results, the optical path difference between the source point and the corresponding height is calculated and normalized by selecting the positions θ=π/2 and θ=-π/2, which are shown in Fig. 5.It is found that the extreme point in Fig. 5 gradually moves down as the inclination α gradually increases, corresponding to a central point in Fig. 4(a)-(d) moves down.
In addition, it is found from Fig. 5 that the symmetry of the vertical results will decrease as the inclination α increases.From Fig. 4, it is obviously that the symmetric part in Fig. 4(d) is much less than that in Fig. 4(b), in which the inclination angles are 8°and 2°, respectively.Therefore, the correctness of the proposed diffraction model is verified.

Results of the conical holographic display
In this sub-section, the holograms are generated by the point source (PS) method [3] or the proposed method and reconstructed by the proposed method or the PS method, respectively.The image size of object is 128×256.The results are shown in Fig. 6.
From Fig. 6, it shows that the proposed method can be used for both generation and reconstruction of conical holograms.In order to obtain an intuitive reconstruction, gray image with simple distribution is used to generate and reconstruct holograms.The Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) are employed to evaluate the quality of the reconstructed results [24].The radius of the outer cylinder is 150 mm, the wavelength is 150 µm, and r 1 is 9 mm.Here, the object is shown in Fig. 7   In the above results, the images are flat view.In order to simulate the scene realistically, we present the reconstructed images from different perspectives.As shown in Fig. 8, the reconstructed image is observed at 0°, 120°, and 240°.It shows that the reconstructed image of the conical hologram can be seen at all angles of 360°.Since the reconstruction quality is slightly different at the inclination angle of 4°and 14°as shown in Figs.7(c) and (e), we investigate effect of the inclination angle to the reconstruction quality.The results are shown in Fig. 9.It shows that the effect to reconstruction quality becomes weaker with the increase of inclination angle.The reason is that the paraxial condition will no longer be satisfied with the increase of inclination angle, which means the obliquity factor K(θ) in Eq. ( 3) cannot be approximated to 1.In addition, when the inclination angle is between 5°and 12°, the proposed method can not only expand the perspective vertical FOV but also achieve a high quality of reconstructed image.Since the common spatial light modulators can only modulate amplitude-only or phase-only holograms, the amplitude-truncation (AT) or double-phase method (DP) [25] can be applied to generate phase-only hologram.Meanwhile, a random phase can be added to the object to achieve diffuse reflection.The time multiplexing method (TM) [26] can be applied to reduce speckle noise in reconstructed images.Reconstructions with different methods are applied for comparison, and the reconstructed results are shown in Fig. 10.

Expanding of the FOV
In this section, the changing trend of the FOV expansion ∆h(R, α) with the change of parameters is discussed, furtherly.In simulation, the sampling points is chosen to 400×1600 according to Eq. ( 12), the diffraction angle β is 38°calculated by Eq. ( 16), then ∆h(R, α) is calculated by Eq. ( 18).The FOV expanding ratio is defined by the ratio of FOV expanding ∆h to object height H, which is ∆h/H, to investigate the effect of FOV expanding.And the simulation results of FOV expanding ratio are shown in Fig. 11.
From Fig. 11, it is found that the FOV expanding ratio increases significantly as the inclination α increases.Besides, the FOV expanding ratio is positively correlated with the change in radius, and it can reach more than 0.6 without considering the reconstruction quality.Although the FOV expanding ratio decreases with the increase of H, it can also be kept above 0.4.The results show that our proposed method can significantly expand the vertical FOV.

Computational cost of the conical hologram
Here, the computational cost of the conical hologram is discussed, furtherly.We performed simulations on images of different sizes and the time results of one convolution and the generation of complete holograms since the hologram is generated by multiple FFT and then superimposed.In simulations, the calculation platform includes Python 3.8, Windows 10 operation system, and AMD Ryzen 5 2500U (2.0 GHz).The results are present below: From Table 1, it can be found the conical hologram requires more computational cost compared with the cylindrical hologram.However, since the generation of holograms is obtained by the accumulation of multiple convolution calculations, and each convolution is independent, the computational cost can be reduced by parallel computing with the Graphics Processing Unit.

Conclusion
In this paper, a new holographic display is proposed to expand the vertical FOV by a conical holographic diffraction model, in which the object plane is a cylinder and the holographic plane is a part of a cone.In the proposed method, the proposed diffraction model is firstly derived from the RS diffraction formula, and the convolution and FFT are used for the fast diffraction calculation.The simulation of Young's interference is adopted to show the correctness of the proposed method.At the same time, the complex amplitude and the encoded holograms are used to reconstruct different object images.And the simulation results further demonstrate the effectiveness of the proposed method.Besides, the expansion of FOV in the vertical direction is discussed theoretically, and the vertical expansion ratio of FOV reaches more than 40%.The results show that our proposed method can significantly expand the vertical FOV.Therefore, the issue of the vertical FOV in cylindrical holography is deeply discussed and successfully addressed for the first time.With the development of a curved screen [27] and flexible display material [28], the proposed conical holography could be a promising technology in the future.

Fig. 1 .
Fig. 1.(a) Schematic of conical holographic diffraction model, and (b) side view of the cone.
(a).The conical hologram is complex amplitude as shown Figs.7(b) and 7(d).Reconstruction results are shown in Figs.7(c) and 7(e), in which the inclination α is 4°and 14°, and their PSNR are 28.456dB and 28.246 dB, respectively.It shown that a small amount of quality is sacrificed in order to expand the vertical FOV.

Fig. 6 .
Fig.6.Verification of the proposed method.The first and third columns are diffraction fields, and the second and fourth columns are reconstruction results.The first line is generated by PS method and reconstructed by the proposed method.The second line is generated by the proposed method and reconstructed by PS method.The third line is generated and reconstructed by PS method.

Fig. 9 .
Fig. 9. Reconstruction of image quality as a function of inclination angle.

Fig. 10 .
Fig. 10.(a) the object, (b) the reconstructed image of AT, (c) (d) reconstructed results of AT with a random phase and TM with 15 multiplexing, respectively, (e) (f) reconstructed results of object with and without random phase by DP.As shown in Fig. 10(b), the reconstructed image of phase-only hologram with AT only contains profile of the object and is low quality.Figure 10(c) shows the reconstructed image of phase-only hologram by AT with adding random phase to the object.There is obvious speckle noise in the reconstructed image, which greatly reduces the quality.Therefore, the TM with 15 multiplexing is use to reduce speckle noise in reconstructed images and the result is shown in Fig. 10(d).When the DP is applied to encode the hologram with and without random phase and the reconstructed images are shown in Figs.10(e) and (f).The PSNRs of Figs.10(b), (c), (d), (e) and (f) are 14.673 dB, 15.237 dB, 17.468 dB, 16.138 dB and 24.316 dB, respectively.It shows that the quality of conical holographic reconstruction can be improved by the time multiplexing method and the double-phase method.

Fig. 11 .
Fig. 11.FOV expanding ratio varies with different (a) radius R when H is 40 mm, and (b) vertical height H when R is 180 mm.