Modulating Phase Encoding With Amplitude Compensation for Hologram Reconstruction

In holographic three-dimensional (3D) display, phase-only spatial light modulators (SLMs) can only accept pure phase modulation signals, often leading to amplitude distortion in the reconstructed image and inaccurate object reconstruction. This paper presents a new modulating phase encoding method with amplitude compensation. Based on the imaging characteristics of phase digital holograms, the object light amplitude variation is extended to the nonlinear region of the first-order Bessel function to maximize the intensity of the reconstructed 3D image. The amplitude distortion of the reconstructed image is effectively eliminated by pre-distortion treatment of the light amplitude. Numerical simulations and optical experiments validate the presented method.

expression with the actual light amplitude as its parameter.To reduce distortion, the variation interval of the Bessel function must be compressed to achieve an approximately linear change, but this reduces the intensity of the reconstructed image.In this paper, we study the imaging characteristics of phase digital holograms and present two improvements to address the distortion.First, the object light amplitude is changed to the nonlinear region of the first-order Bessel function to maximize the intensity of the reconstructed 3D image.Second, the amplitude distortion of the reconstructed image is effectively eliminated by pre-distortion processing of the light amplitude.The results demonstrate that this modulating phase encoding method with amplitude compensation can improve the intensity of the reconstructed image and accurately reconstruct the 3D image with both amplitude and phase information.

II. HOLOGRAPHIC IMAGING BASED ON PHASE SPATIAL LIGHT MODULATOR
In the Cartesian coordinate system, o − xyz, z = 0 represents the hologram plane, j = √ −1 , the object wave in the hologram plane is denoted by O (x, y) = o(x, y)exp[jϕ(x, y)], and the reference wave is expressed as R (x, y) = A r exp[jϕ r (x, y)].Let the reference light be a light wave that propagates parallel to the o-xz plane and has an angle θ with the z-axis, where ϕ r (x, y) = kθ x (k = 2π/λ, and λ is the wavelength).The digital hologram can be expressed by: Since the photosensitive amount of the phase hologram recorded by the photosensitive material is proportional to the above equation, the phase digital hologram can be expressed as: Where, g is an undetermined constant called the phase hologram modulation parameter [12].Let: Then (2) can be rewritten as: Due to the variation range of sin(ψ(x, y)) function between ±1, to ensure that the transmission function value is singlevalued when forming a phase-type digital hologram, the selection of the composition parameter g must satisfy: According to the properties of the integer-order Bessel function J n (α), (6) can be expanded as: The above formula indicates that when a phase hologram is illuminated with a unit amplitude plane wave, there are zeroorder diffracted waves (n = 0) propagating along the optical axis z in the transmitted light, and diffracted waves of n = 1, 2, … are symmetrically distributed on both sides.Due to the ability to select light waves that contribute to imaging through a gating filter composed of a lens system [13], [14], for simplicity, only diffraction waves with n = 0, ±1 will be discussed.
When the phase hologram is loaded onto the SLM (Spatial Light Modulator) and illuminated by the original reference light, the complex amplitude of the transmitted wave modulated by the SLM is: Here, The results reveal that the transmitted light waves become three beams of light that are tilted in the direction of the reconstruction light.Where, U H0 (x, y) represents a zero-order diffracted light wave, U H+ (x, y) represents the conjugate light wave, and the last U H− (x, y) represents the object light that can form real images.Because of α = 2gA r o(x, y), to obtain strong light, the parameter g should be suitably designed so that the value of −J −1 (α) has a wide interval with the variety of the object light amplitude.Because of J 1 (α) = −J −1 (α), the curve of the Bessel function J 1 (α) is shown in Fig. 1, and the first maximum position of J 1 (α) is marked.The function φ(α) is the straight-line equation that connects origin and the point (α max , J 1 (α max )).
To analyze the properties of the reconstructed image, U H− (x, y) can be expressed as: The propagation direction of this light wave is the same as the optical axis; the diffracted wave U H− (x, y) can reconstruct the real image of the object light field.However, since α = 2gA r o(x, y), A r J −1 (α) is not proportional to the change in the magnitude of the object light amplitude o(x, y), resulting in both amplitude and phase distortion in the reconstructed object image.An improvement is presented in [13], where the phase hologram is formed using the last item in (1): When the SLM loaded with this phase hologram is irradiated with the original reference light, the transmitted light becomes: It is evident that the effect of the object light phase with the complex function K is eliminated, which is an important improvement.

A. Maximize the Intensity of the Reconstructed Image
Observe (15) and the curve of function J 1 (α), to obtain the maximum intensity of the reconstructed image, the parameter α should vary from zero to the point α max , as shown in Fig. 1.The maximum value of the light amplitude o(x, y) is o max .To ensure o max meet α max , let α max = 2go max A r .According to (4), the constituent parameter g is obtained:

B. Eliminate the Amplitude Distortion of the Reconstructed Image
Since the imaging object light described by (15) has amplitude distortion when α is in the range of 0-α max , the 3D image of the object cannot be accurately displayed in theory.To solve this problem, the following improvements are presented: First, let φ (α) = αJ 1 (α max )/α max .Substituting with (4), we can obtain the expression of the pre-distortion function: Then, the object light amplitude arriving at the plane of the phase hologram is subjected to pre-distortion treatment Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.according to the following formula: Finally, replace o(x, y) in ( 13) with function ô(x, y).It can be readily demonstrated that the phase-type digital hologram produced using the described method encapsulates the object light information.This information is directly proportional to the actual complex amplitude of the object light, thereby allowing for an accurate reconstruction of the object's 3D image.

A. Optical System With Filter
To eliminate other order reconstructive beams, an optical system with a filter is presented in Fig. 2. A plane light beam illuminates the liquid crystal on silicon (LCOS) where a phase hologram is loaded.The −1 st order of the reflective wave is selected by passing through the filter located on the focal plane of the lens.Since the diffraction calculation of a 3D object surface can be obtained by calculating a series of planar light sources perpendicular to the optical axis, we can study the quality of the reconstructed image of different spatial planes perpendicular to the optical axis to evaluate the quality of the reconstructed image of 3D objects [15].The object is a two-dimensional image in the z = −d plane, a lens is placed on the z = d 1 plane, and an aperture is placed on the back focal plane of the lens.The distance between the aperture and the observation screen is d i : The lateral magnification of the image is expressed as:

B. Numerical Simulation
We can calculate the optical diffraction field behind the optical system using scalar diffraction theory.To verify that the amplitude distortion can be eliminated by the improved method, the object is designed as a two-dimensional transmission screen on the z = −d plane, with the transmittance varying in the horizontal direction from 0 to 255.By illuminating the transmission screen with a unit amplitude plane wave and comparing the amplitude of the reconstructed image with the transmitted light, we can observe the amplitude of the reconstructed image is distorted or not.The simulation steps are as follows: 1) Calculate the object light on the z = 0 plane, which is diffracted along the z-axis by a distance d.The width of the object plane aperture is ΔL 0 and the amplitude transmittance is A 0 (x 0 , y 0 ).Illuminating the aperture with a unit amplitude uniform plane wave, the object wave field after the aperture can be written as: 2) Find the maximum value o max of the object light amplitude at the z = 0 plane.3) Let the reference light amplitude A r = o max , and select the composition parameter g according to (16).4) Obtain ô(x, y) by applying the object light amplitude predistortion processing according to (18).5) Because the pixel size of LCOS is Δx = 6.4 μm, to satisfy the sampling theorem in the simulation calculation, the angle between the reference light and the optical axis must satisfy λ/θ ≥ 2Δx.The light source is YAG laser with a wavelength of 532 nm; the calculation yields θ = 1.2 • < λ/(2Δx).Replace o(x, y) with ô(x, y).The phase digital hologram is obtained according to (14): 6) Illuminating the phase digital hologram with reference light R (x, y) = o max exp[jϕ r (x, y)], we can calculate the light wave field at the focal plane.The result is the Fourier transform of the input planar light wave field multiplied by a quadratic phase factor [16]: ) Suppose the transmittance of the selective filter is Θ(x, y), and the light wave field at the image plane is: In the simulation, the lens focal length is f = 300 mm, and the number of samples is N = 1024, ΔL = N Δx = 6.5536 mm.Fig. 3 depicts the reconstructed images using three different coding methods.
Fig. 4 displays the curves along the x-axis of the reconstructed images (b), (c), and (d) from Fig. 3.It is evident that the reconstructed images using the coding method referenced in    [12].(b) The method in [13].(c) The method in this article.[12], [13] exhibit amplitude distortion.In contrast, the proposed method more effectively eliminates this distortion, resulting in higher quality reconstructed images.
It is easy to observe from the image of the Bessel function J 1 (α) that if a smaller constituent parameter g is chosen at the cost of reducing the intensity of the reconstructed image, the encoding methods in [12] and [13] can also reconstruct images with smaller amplitude distortion.To confirm this analysis, let the constituent parameters determined in (16) be g max , and g = 0.2g max , 0.4g max , 0.6g max , 0.8g max , 1.0g max .Fig. 5 shows the axial direction amplitude distribution curves of the reconstructed images using the three methods (each curve from bottom to top corresponds to the added value of g).
Evidently, the choice of a smaller constituent parameter g can improve the quality of the reconstructed image in the coding methods proposed in references [12] and [13].However, this improvement comes at the expense of reducing the intensity of the reconstructed image.In contrast, the improved coding method of amplitude compensation modulation phase proposed in this paper can reconstruct the image without amplitude distortion, regardless of the composition parameter g, as long as it does not exceed g max .

C. Experimental
To obtain experimental proof of the theoretical analysis and intuitively understand the effect of amplitude distortion on the quality of reconstructed images, three different methods were implemented to generate the corresponding phase holograms using MATLAB on a Windows 10 system.The hologram resolution was set to 1024 × 1024, with a pixel size of 0.0045 mm, a light wavelength of 532 nm, and a propagation distance of 200 mm.The generated phase holograms were then reconstructed sequentially.During reconstruction, the filtering system shown in Fig. 2    (Charge-coupled Device) and displayed on computer 2, which is connected to the CCD.
The SLM used in the experiment is a phase type with a resolution of 1920x1080 and a pixel size of 6.4 um.The laser wavelength is 532 nm, the SLM is placed 200 mm from lens2, which has a focal length of 200 mm, and the CCD is 200 mm from the filter.Fig. 8 shows the original image and the reconstructed images of phase holograms generated by the three methods.
It can be seen that the experimental results demonstrate good agreement with the theoretical simulations, confirming the superiority of the improved coding method in terms of image quality.
It is worth noting that although the improved method enhances image quality, it necessitates pre-distortion processing of the image amplitude, which increases computational complexity.From the perspective of human eye observation of monochromatic  light imaging quality, the method proposed in reference [13], despite containing amplitude changes, remains a simple and suitable coding approach.However, when displaying color images, since the imaging points are formed by stacking different color dots, distortion of the dot color components leads to distortion of the synthesized color.Accurately displaying the object's color 3D image with high brightness and quality is always the desired goal.
To further verify the effectiveness of the improved encoding method in encoding three dimensional object holograms, we use the three dimensional model shown in Fig. 9 to generate holograms, and the encoding method is consistent with the method used to generate planar object holograms.After calculating the diffraction light field of a three dimensional object on the hologram plane, a plane reference light is added to form the hologram.The amplitude is pre distorted according to the steps in Section III (c), and then encoded into a phase hologram.The encoded phase hologram is then simulated and reproduced, and the result is shown in Fig. 10.From the figure, it can be seen that the reconstructed images at different distances focus on different parts of the object, and the intensity and quality of the reconstructed images have good effects, verifying the effectiveness of the proposed method for encoding three dimensional object holograms.

V. CONCLUSION
In the study of holographic 3D display based on the phaseonly SLM, reconstructing object images with high brightness and quality remains an active research topic.This paper presents improvements to two recently proposed SLM control coding methods, resulting in a coding approach that achieves high diffraction efficiency and accurate reproduction of 3D object images, supported by theoretical simulations and experimental evidence.It is hoped that the work presented in this paper will serve as a valuable reference for the research and application of digital holographic 3D displays.

Fig. 4 .
Fig.4.X-axis amplitude curves of the reconstructed images using three different methods.(a) The method in[12].(b) The method in[13].(c) The method in this article.

Fig. 5 .
Fig. 5.The influence of modulation parameter g on the reconstructed image amplitude distribution of three methods.(a) The method in [12].(b) The method in [13].(c) The method in this article.

Fig. 6 .
Fig. 6.Comparative Analysis of Reconstructed and Original Images Using Three Different Methods.(a) Original image.(b) Method described in reference [12].(c) Method described in reference [13].(d) Method developed in this article.
was simulated in MATLAB to eliminate the other orders of reconstructed beams, with parameters d = 200, d 1 = 200, and f = 200 set accordingly.Fig. 6 illustrates a comparison between the original image and the reconstructed images simulated using the three different reconstruction methods.An optical path based on the filtering system of Fig.2was built, as schematized in Fig.7.Computer 1, connected to the SLM, controls the loading of phase holograms onto the SLM.The reconstructed image is received through a CCD Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 7 .
Fig. 7. Schematic Representation of the Optical Reconstruction Light Path.

Fig. 8 .
Fig. 8. Comparative Display of Optical Reconstructions of Holograms Produced by Three Distinct Methods.(a) Original image.(b)The method described in reference[12].(c) The method described in reference[13].(d) The described in this article.

Fig. 9 .
Fig. 9. Schematic Overview of the Optical Reconstruction Light Path.

Fig. 10 .
Fig. 10.Numerical Reconstructions from Holograms Created Using the Proposed Method.(a) Focused numerical reconstruction on the tiger's hind legs.(b) Focused numerical reconstruction on the tiger's forelimbs.(c), (d), (e), (f) Enlargements of specific areas in (a) and (b) respectively.