Abstract

This paper presents a method for the implementation of speckle reduced lensless holographic projection based on phase-only computer-generated hologram (CGH). The CGH is calculated from the image by double-step Fresnel diffraction. A virtual convergence light is imposed to the image to ensure the focusing of its wavefront to the virtual plane, which is established between the image and the hologram plane. The speckle noise is reduced due to the reconstruction of the complex amplitude of the image via a lensless optical filtering system. Both simulation and optical experiments are carried out to confirm the feasibility of the proposed method. Furthermore, the size of the projected image can reach to the maximum diffraction bandwidth of the spatial light modulator (SLM) at a given distance. The method is effective for improving the image quality as well as the image size at the same time in compact lensless holographic projection system.

© 2017 Optical Society of America

1. Introduction

Pico-projection has gained great popularity and under active development toward the targets of miniaturization and portable type. One of the major and attractive methods for pico-projection is the laser based holographic projection. Holographic projection has attracted considerable attention for its high-contrast, low power consumption, wide color gamut and potentials for 3D display [1, 2].

In recent years, lensless holographic projection technique has been developed [3], which paves a way for versatile and portable projection. In lensless holographic projection, by replacing the optical lenses with the lens phase factors in the numerical calculation of phase-only computer-generated holograms (CGH), the image can be projected from the SLM without using lenses. Moreover, zoomable pixel size of the projected image is obtained in lensless holographic projection by using ARSS-Fresnel diffraction algorithm [4, 5]. However, random phase distribution is usually assigned to the image to calculate phase-only CGH, inevitably leading to serious speckle noise in the reconstruction. A considerable amount of research has been dedicated to the suppression of speckle noise problem, including the iterative CGH algorithm [6–8] and time-integrating method [9–11]. However, the iterative CGH algorithm is time-consuming in the CGH calculation. Besides, the time-integrating method require very high frame rate SLM to display sequentially loaded CGHs, resulting in the inconvenience of real-time dynamic holographic projection. More recently, T. Shimobaba introduced a “random phase-free (RPF)” method [12, 13] that can improve the image quality and reduce the speckle noise by multiplying the image with an initial spherical phase. Despite usefulness in terms of speckle problems in lensless holographic projection applications, the RPF method suffers limitation of image size due to existence of zero order and conjugated component.

Complex amplitude modulation based on phase-only CGH is an appealing method for high quality holographic image reconstruction in recent years, which is capable of modulating both amplitude and phase of the reconstructed image simultaneously and independently [14, 15]. By controlling the complex amplitude, the destructive interference between the closely adjacent pixels of reconstructed image is suppressed and the speckle noise can be greatly reduced. In order to generate a desired complex amplitude field, the amplitude and phase information are synthesized into a phase-only CGH by different encoding scheme. For example, the amplitude information is encoded as part of the pure phase information (direct synthesizing method) [16–18] or decomposing a complex amplitude element into two pure phase values with constant magnitude (double-phase method) [19–22]. Although the complex amplitude modulation is effective in reducing the speckle noise, it has two considerable drawbacks against lensless holographic projection application. First, 4-f optical filtering system is employed to reconstruct complex amplitude, increasing the complexity of the system by the presence of bulky lenses, which requires much space due to its finite focal length and aperture. Second, the size of the reconstructed image is greatly restricted, because when the image size is larger than the CGH, the desired information of the complex amplitude image is difficult to be filtered by 4-f system. Therefore, it has troubles for reconstructing large-sized images that are required in the applications of lensless holographic projection.

In order to solve the above mentioned problems, we propose a method for lensless holographic projection with reduced speckle noise based on the reconstruction of complex amplitude image from phase-only CGH. Besides, the complex amplitude image is reconstructed with zoomable image size and the magnification can achieve the maximum diffraction bandwidth of the SLM. The feasibility of this method is verified by both monochrome and color experiments.

2. Method and simulation

Figure 1(a) shows the scheme of CGH calculation for lensless holographic projection based on double-step Fresnel diffraction. Assuming that A(x, y) represents the amplitude of a two-dimensional image. The distance from the image to the CGH plane is denoted by z. A virtual plane is established which is located between the image and the CGH plane. d1 and d2 represents the distance from the virtual plane to the CGH and image plane respectively (so we have z = d1 + d2). The image is multiplied by a spherical convergent phase factor φ(x, y). This operation is equivalent to illuminating the image with a virtual convergent light. The main and important contribution of this virtual convergent light reflects in two aspects: first, all of the information of image can widely spread to the CGH whose size is smaller than the image, without losing low-frequency information of the image [13]. Second, the light field of image is required to be focused into a small spot at the virtual plane through diffraction, which is necessary in the filtering system of the CGH reconstruction. So the focal point of the convergent light should be at the virtual plane and the focal length of the convergent light is d2. Hence the phase of the convergent light added to the image is written as φ(x,y)=π(x2+y2)/λd2, and the complex amplitude of the image is accordingly expressed by I(x,y)=A(x,y)exp[iφ(x,y)]. Here the expression of convergent phase factor is written under the paraxial approximation whose condition will be discussed in the following part of this section.

 figure: Fig. 1

Fig. 1 Principle illustration of the proposed method for lensless holographic projection. (a) Geometry model of the CGH calculation. (b) Geometry model of the CGH reconstruction.

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Supposing the sampling pitch of the image, virtual plane and CGH is dx, dv and dh respectively. The sampling number is N in both of x and y direction. Here the sampling pitch of the CGH dh is determined by the SLM which will be used in the experiment. The proposed method for CGH calculation consists of two steps. The first step is the Fresnel diffraction calculation from the image to the virtual plane, by employing the “ARSS-Fresnel diffraction” algorithm [5] which is performed by convolution algorithm involving three fast Fourier transforms (FFTs). Remarkable feature of ARSS-Fresnel diffraction algorithm is the implement of diffraction calculation with different sampling pitches on image and virtual planes. If we define the parameter s as the ratio of dx/dv, the complex amplitude of the virtual plane is given by:

VP(xv,yv)=I(x,y)exp{iπλd2[(xvsx)2+(yvsy)2]}dxdy=FFT1{FFT[I(x,y)exp(iφ1)]FFT[exp(iφ2)Rect]}.
Where Rect is a rectangular function that reduces aliasing noise, φ1 and φ2 are quadratic phase factors whose detailed expression can be found in [5].

The second step is the Fresnel diffraction calculation from the virtual plane to the CGH. The calculation is done by using the single FFT based Fresnel diffraction algorithm. The complex amplitude of the CGH is calculated by:

H(xh,yh)=VP(xv,yv)exp{iπλd1[(xhxv)2+(yhyv)2]}dxvdyv=exp[iπ(xh2+yh2)λd1]FFT[VP(xv,yv)exp[iπ(xv2+yv2)λd1]].

In Eq. (2) the Fresnel diffraction from the virtual plane to the CGH is calculated by using one FFT and two multiplication of quadratic phase factors. In this algorithm, the relations between the sampling pitch dv and dh is mutually constraint by the Nyquist theory as dv=λd1/Ndh. Here it should be mentioned that the CGH calculation process is divided into two-step Fresnel diffraction via a virtual plane instead of using one Fresnel diffraction from image to the CGH. The reason is explained as follows: If we use only one ARSS-Fresnel diffraction algorithm to directly calculate the diffraction from image to the CGH, the image information will be lost. This is because the ARSS-Fresnel diffraction algorithm is based on the theory of angular spectrum of plane waves. It is easy to find that the light is divergently diffracted from the virtual plane to the CGH, which means the plane waves component with large diffraction angle would not arrive at the region of the CGH, losing the wavefront information of the original complex image. However, this problem would not exist by using the single FFT based Fresnel diffraction algorithm. On this account double-step Fresnel diffraction is essential for the CGH calculation.

In order to overcome the unavailability of complex modulation using the SLM, the complex amplitude CGH H(xh, yh) need to be encoded into a phase-only CGH. We use the off-axis double-phase method [22] as the encoding procedure. Assuming that Ah(xh, yh) and φh(xh, yh) represent the amplitude and phase of H(xh, yh). The decomposition of the complex amplitude CGH into two pure phase values θ1(xh, yh) and θ2(xh, yh) can be calculated as follows:

θ1(xh,yh)=φh(xh,yh)+cos1[Ah(xh,yh)/Amax]θ2(xh,yh)=φh(xh,yh)cos1[Ah(xh,yh)/Amax].
Where Amax is the maximum value of Ah(xh, yh). Then the final expression of the phase-only CGH is given by [22]:
p(xh,yh)=θ1(xh,yh)M1(xh,yh)+θ2(xh,yh)M2(xh,yh)+2πxhsinα/λ.
Where M1 and M2 are the complementary two-dimensional binary gratings (checkerboard patterns), α is the angle of a blazing grating in xh-direction.

In off-axis double-phase method, the phase-only CGH p(xh, yh) acts as a phase grating that encodes the complex amplitude H(xh, yh) into its zero diffraction order [21]. So it is easy to see that if we use a filter to block all diffraction orders but the zero one, the reconstruction of the complex amplitude CGH can be guaranteed. Figure 1(b) shows the reconstruction process of the phase-only CGH. Illuminated by a plane wave, the zero diffraction order as well as other diffraction orders of the phase-only CGH will be back propagated to the virtual plane. On this plane, a focused spot from the zero diffraction order (namely the inverse Fresnel diffraction of the complex amplitude CGH) will be observed. The information among diffraction orders will be separated. A spatial filter is placed to select the zero order spot while block the light from other orders. Then the picked spot region will continue to back propagate to the image plane. Since the reconstruction is from the encoded complex amplitude CGH essentially, the complete complex amplitude of the image can be obtained.

A computer simulation is conducted to demonstrate the performance of the proposed method. Figure 2(a) is the original image composed of 1024 × 1024 pixels with pixel pitch 20μm. The parameters of distance are d1 = 0.1m and d2 = 0.4m respectively. The wavelength of the light is 532nm. The pixel pitch of the CGH is set as dh = 8μm in the calculation in order to keep consistent with the SLM we used in the experiment. Figure 2(b) shows the phase-only CGH calculated by the proposed method [Eq. (1)-Eq. (4)]. The amplitude of light field at the virtual plane reconstructed from the phase-only CGH is shown in Fig. 2(c). It can be seen that the focused spot from the zero diffraction order is completely separated from other diffraction orders. By selecting the spot region using a filter, the light field is further propagated to the image plane. The spot selecting is fulfilled by multiplying the wavefront at the virtual plane with a circular aperture function whose diameter is 3.3mm. Figure 2(d) and 2(e) are the reconstructed amplitude and phase of the image. The amplitude of the image can be reconstructed successfully with high image quality. The reconstructed phase is a spherical phase as a result of the initial virtual convergent light. Since the phase is smooth and slowly varied, the speckle noise can be reduced due to the nature of the obtained phase distribution [23]. It should be noted that the reconstructed amplitude is contaminated by tiny ringing artifacts that is arisen from the numerical error in Fresnel diffraction algorithm. Figure 2(f) shows another reconstruction in a different size by setting the image pixel pitch as 10μm in CGH calculation.

 figure: Fig. 2

Fig. 2 Computer simulation results. (a) The original image. (b) Phase-only CGH calculated by the proposed method. (c) Amplitude of light field at the virtual plane (filter plane). (d) Reconstructed amplitude of the image. (e) Reconstructed phase of the image. (f) Reconstructed amplitude of the image in a different size.

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The imposed initial convergent spherical phase φ(x,y)=π(x2+y2)/λd2 is written under the paraxial approximation. In our situation, the size of the diffracted light field between the image and CGH plane is restricted by the pixel pitch of the CGH. The CGH can be considered as a phase-type grating and its maximum diffraction angle is determined by θm = arcsin(λ/dh) = 532nm/8μm = 0.0665. Under this angle, the paraxial approximation condition can be satisfied when dealing with the diffraction between the image and CGH plane.

The key point of the proposed method is the setup of the virtual plane where the diffraction wavefront of the image can be focused into a spot by imposing a spherical phase distribution. The focused spot contains all of the complex amplitude information of the image which allows to be conveniently filtered from other redundant information in the reconstruction. In this way the image can be reconstructed with higher quality due to the preservation of its complex amplitude. In addition, the reconstruction of image with variable sizes is achieved by freely setting the desired sampling pitch dx during the calculation. It should be noted that there are no lenses involved in the CGH reconstruction system.

3. Experiments and results

3.1 Experiments and results in monochrome

The setup of optical experiment for lensless holographic projection using the proposed method is presented in Fig. 3(a). A phase-only SLM of Holoeye PLUTO with pixel size 8μm × 8μm is used here. The loaded phase-only CGH is illuminated by a laser source (532nm) with divergent spherical wave in order to spread the non-diffracted light of SLM on a larger area, leading to a far less noticeable of its visibility [3]. The projected beam from the SLM is transmitted through a low-pass spatial filter that consists of an iris whose diameter is 3.3mm and finally captured by a CMOS camera (Nikon D7000) at the image plane. The parameters of distances (d1 and d2) are the same with those used in the simulation. Three original images (“ABC”, “Lena” and “Cameraman”) used in the CGH calculation are 1024 × 1024 pixels. The resultant phase-only CGHs are multiplied with phase factors of positive lens in order to cancel the divergence of the illuminating wavefront.

 figure: Fig. 3

Fig. 3 (a) Optical setup for lensless holographic projection. (b) Optical reconstructions of projected images (image size is 10.24mm). (c) Optical reconstructions of projected images (image size is 20.48mm).

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The optical reconstructed results of projected images are exhibited in Fig. 3(b) and 3(c). The pixel pitch of the projected images is 10μm (image size is 10.24mm) in Fig. 3(b) and 20μm (image size is 20.48mm) in Fig. 3(c). It is evidently that all of the images are reconstructed successfully with acceptable visual quality for human eyes, and the speckle noises are well suppressed. Comparing with the simulation results shown in Fig. 2(d) and 2(f), the quality of optical reconstructions is a little bit degraded by inevitably introducing stray light as well as extra loss of image resolution by inaccurate control of the iris size. In addition, the ringing artifacts existed in Fig. 2(d) and 2(f) are less noticeable in the optical reconstructions, which is probably due to the superposition with other artifacts such as diffraction on structures of optical elements. Furthermore, we show two animations (Visualization 1, Visualization 2) recorded in the optical system by displaying phase-only CGHs at frame rate of 60Hz, proving that the dynamic lensless holographic projection can be easily obtained by the proposed method. Figures 4(a) and 4(b) show one frame from each reconstructed animation.

 figure: Fig. 4

Fig. 4 One frame from reconstructed animations (image size is 20.48mm). (a) Basketball (see Visualization 1). (b) Horse riding (see Visualization 2).

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3.2 Experiments and results in color

A second experiment is carried out to further prove the feasibility of projecting color images by the proposed method. The color image is divided into RGB component. The phase-only CGH of each component is calculated using the proposed double-step Fresnel diffraction algorithm described in Section 2. In the calculation of each CGH, the wavelength used is λ = 671nm for R-CGH, λ = 532nm for G-CGH and λ = 473nm for B-CGH. Other parameters are the same with the monochromatic experiment. In the experiment, we employ the time-sequential switching of illuminating lasers [24]. The phase-only CGH displayed on the SLM are matched to the wavelength of the current open laser. The experimentally capture of projected images are presented in Fig. 5. Figures 5(a)-5(c) shows the reconstruction of projected image for each used wavelength and Fig. 5(d) is the mixed color reconstruction. Figures 5(e)-5(g) are the color reconstruction of three images (“SEU”, “Baboon” and “Fruits”) with pixel pitch 20μm, while Figs. 5(h)-5(j) are the color reconstruction of “Parrots” with different pixel pitches (10μm, 20μm and 30μm). All the reconstructed results exhibit the quality that might be accepted with eliminated speckle noise.

 figure: Fig. 5

Fig. 5 Optical reconstructions of color projected images. (a) Reconstruction of red component of “Lena”. (b) Reconstruction of green component of “Lena”. (c) Reconstruction of blue component of “Lena”. (d) Reconstruction of color “Lena”. (e)-(g) Reconstruction of color “SEU”, “Baboon” and “Fruits”. (h)-(j) Reconstruction of color “Parrots” with different pixel pitches (10μm, 20μm and 30μm).

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3.3 Zoomable holographic projection

Another feature of the proposed method is the ability to project images with variable sizes in lensless system. This zoomable function is attributed to the employment of “ARSS-Fresnel diffraction” algorithm, in which we can preset pixel pitch of the image. The maximum value that the size of the projected image can reach is determined by the Nyquist theory Lmax = λz/dh (maximum diffraction bandwidth of SLM), where dh is the pixel pitch of SLM and z is the distance between the SLM and image plane. Owing to the limitation of the camera CMOS size, we performed another experiment to show the zoomable effect by capturing the image which is projected on a large size screen. The distance from the SLM to the projected image screen is increased to z = 3.3m. Meanwhile, the position of the filter (Iris) remains unchanged (d1 = 0.1m) while the distance between the filter and screen changes to 3.2m (d2 = z-d1 = 3.2m). The maximum size that the projected image can achieve is calculated by Lmax = 532nm × 3.3m/8μm≈22cm. Figure 6(a) exhibits the projected screen (a white board) used in our experiment. The screen is smooth enough to avoid extra speckles originating from the diffusive surface. The square region enclosed by solid red line in Fig. 6(a) marks the maximum reachable size of the projected image. Figures 6(b)-6(e) show the experimental results taken from the screen by the camera. The sampling pitch of the projected images in the CGH calculation is set to be 80μm, 120 μm, 160 μm and 200 μm respectively, the corresponding image size is approximately 8.2cm, 12.3cm, 16.4cm and 20.5cm. Two animations (Visualization 3, Visualization 4) show the dynamic projected results in 20.5cm. Figures 6(f)-6(h) also show three projected images in color with the image size of 20.5cm. Based on the experiment, the lensless zoomable holographic projection with high quality by the proposed method is verified.

 figure: Fig. 6

Fig. 6 Optical reconstructions of zoomable lensless holographic projection. (a) The projected screen. (b)-(e) Monochromatic reconstructions with different image sizes of 8.2cm, 12.3cm, 16.4cm and 20.5cm (Visualization 3, Visualization 4). (f)-(h) Three color reconstructions with image size of 20.5cm.

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3.4 Comparison with conventional methods

The advantage of the proposed method is analyzed and compared with two existing methods of lensless holographic projection based on one phase-only CGH. One is the iterative CGH method and the other is the random phase-free (RPF) method [12]. In iterative method, the phase-only CGH is generated by iterative calculation based on ARSS-Fresnel diffraction algorithm. The iteration number is 20. The optical setup is the same with that in Fig. 6. The comparison results are presented in Fig. 7. In the iterative method [Fig. 7(a)] and the proposed method [Fig. 7(c)], the pixel pitch of the projected image is 214μm, the image size is approximately 22cm which fill up the reachable maximum size on the screen (the region enclosed by red line in Fig. 6(a)). It can be seen that the quality of reconstruction using the proposed method is higher than that using the iterative method, with lower speckle noise and a sharper image. When comparing the reconstruction of the proposed method with the RPF method [Fig. 7(b)], there is no big difference in the image quality. However, due to the existing of the zero-order and conjugated component, the maximum size of the projected image by RPF method is limited. The pixel pitch of the projected image in Fig. 7(b) is 100μm (image size is 10.2cm or so) which is less than half of the image size using the proposed method. Moreover, the projected image cannot be easily separated from the zero-order and conjugated component via filtering operation in lensless optical system because of their specific dimensions. The image contrast is also decreased due to the existence of zero-order component. Hence, it can be concluded that the proposed method is capable of generating high quality holographic image projection with reduced speckle noise as well as large image size up to the maximum diffraction bandwidth of SLM.

 figure: Fig. 7

Fig. 7 Comparison of the results by different methods. (a) The iterative method. (b) The random phase-free (RPF) method. (c) The proposed method. (C = speckle contrast.)

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The quantitative quality of the projected images is analyzed by using the concept of speckle contrast. The speckle contrast in the chosen region (sub-region enclosed by dotted yellow lines in each projected image in Fig. 7) is calculated. The value of speckle contrast C is given by C = σ/μ, where σ is the standard deviation of the region and μ is the mean value of the region. The resulting values are presented in the top of each subfigure. Smaller C indicates better speckle reduction and higher image quality. The speckle contrast by the proposed method is 0.0936 while by the iterative method is up to 0.1989, which demonstrates the speckle reduction effect by the proposed method.

4. Discussions

4.1 The position of the virtual plane

The position of the virtual plane (filtering plane) is pivotal to the successful generation of CGH in double-step Fresnel diffraction algorithm. In this section we discuss the requirement for determining the position of the virtual plane (namely the parameter d1 and d2) in the calculation. We assume that the distance between the image and the CGH is given as z. The image size L takes value L = Νdx where N is the total pixel number and dx is the pixel pitch of the image. The size of the CGH is calculated by Lc = Ndh where dh is the pixel pitch of the CGH. The transmitted wavefront from the image toward to the CGH will be convergent and focused at the virtual plane because of the imposed initial virtual convergent light. Three geometry models of optical path in Fig. 8 present different focusing behavior of the image wavefront. The convergent angle θ denotes the convergence degree in each geometry model.

 figure: Fig. 8

Fig. 8 Geometry models of different positions of the virtual plane. (a) d1 = ds. (b) d1>ds. (c) d1<ds. (Parameters in the simulation: λ = 532nm, N = 1024, z = 0.5m, dh = 8μm).

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In the geometry of Fig. 8(a), the convergent angle θ is determined by connecting the across corners of the image and the CGH. In this way, all of the image wavefront can arrive at the region of the CGH exactly and fitly. So the value of d1 and d2 can be deduced according to the relations of similar triangles as d1 = zLc/(Lc + L), d2 = zL/(Lc + L). Here we define the value of zLc/(Lc + L) as the standard distance denoted by ds = zLc/(Lc + L). Figure 8(b) shows the geometry model in which d1>ds (the virtual plane is closer to the image). We can see that, due to the enlarge of θ, part of the image wavefront will propagate to the area which exceeds the region of CGH [marked as blue color in Fig. 8(b)]. However, the exceeded wavefront will still be calculated to the CGH according to the feature of Fresnel diffraction algorithm we used, introducing extra aliasing noise in the reconstructed image. Figure 8(c) shows the geometry model in which d1<ds (the virtual plane is closer to the CGH). In this model, all the image wavefront will propagate to central zone of the CGH whereas the rest region of the CGH is unused. Although some part of the CGH is not utilized, it can still reconstruct the image because it records all of the information of image wavefront. The simulation results of the reconstructed images from CGH in each geometry model are presented below in Fig. 8. Two groups of simulation are performed with image pixel pitch dx = 10μm and 20μm respectively. In each group, the standard distance ds is calculated by zLc/(Lc + L) and listed in Fig. 8. The value of d1 in the numerical calculation is marked at upper left of each reconstructed image. It can be seen that in Fig. 8(b) the aliasing noise is observable while in other two situations the complete images can be reconstructed successfully. Based on the above analysis, the criterion of determining the position of virtual plane in the CGH calculation be concluded as: the distance d1 should be no greater than the standard distance ds.

According to the expression of the standard distance ds, we can easily deduce that the value of ds is inversely proportional with the image size L. While the image has a maximum reachable size determined by the Nyquist limit of Lmax = λz/dh, the minimum value of ds is accordingly to be dsmin = zLc/(Lc + Lmax). This means that for more convenient design of the lensless optical system, we can just preset the distance of d1 by d1 = dsmin = zLc/(Lc + Lmax). Followed by the above criterion, the system can work well to guarantee the correct reconstruction of images in arbitrary sizes without considering the relations between d1 and ds (because d1 is already the minimum of all possible values of ds). Hence, in the lensless holographic projection based on the proposed method, the parameters of distances among the image, filter (Iris) and CGH can be chosen freely followed by the criterion, making the design of the system more flexible and compact.

4.2 Iris size in filtering

The employing of iris in the filtering process is a key point to the reconstruction of both amplitude and phase of the image. However, this complex amplitude modulation method is realized with the reduction of the space-bandwidth product (SBP). Because while the size of the SLM does not change, the bandwidth of the spectrum domain (referring to the virtual plane) is lessened by filtering, which predicates that the effective resolution decreases according to Whittaker-Shannon sampling theorem. The quality of the reconstructed image will be degraded by the loss of the image resolution. In order to evaluate this factor, we calculated the peak signal noise ratio (PSNR) between the reconstructed images and the original image under different iris sizes. The result is plotted in Fig. 9. The diameter of the iris is set from 0mm to 6.6mm (bandwidth at virtual plane) in the numerical reconstruction of the CGH. Three sub-images in Fig. 9 show the details of the reconstructed images with tagged PSNR values. As the iris size decreases, the image is blurred due to more serious loss of the resolution. Meanwhile, the image contrast reconstructed via large iris size would be descend by introducing more stray light. Therefore, a moderate physical size (around 3.3mm) can be selected based on a trade-off between the influence of resolution loss and stray light. Despite the fact that the reconstructed complex amplitude of image is facing the restriction of finite SBP, the proposed method avoids the speckle noise as well as optimizing the optical system in lensless holographic projection.

 figure: Fig. 9

Fig. 9 Measurement of reconstructed image quality by PSNR under different iris diameters. (Parameters in the simulation: λ = 532nm, N = 1024, z = 0.5m, dh = 8μm, dx = 20μm, d1 = 0.1m, d2 = 0.4m).

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4.3 Current problem of the proposed method

The proposed method is effective and useful in lensless holographic image projection. The projected image has high quality by reconstructing both of its amplitude and phase. The phase distribution is a spherical phase factor that is equal to the phase of initial assigned virtual convergent light. Based on the feature of the phase distribution, angular variance of the light rays forming the image is quite limited, leading to an enlarged depth of focus (DoF) [25]. The extended DoF is highly suitable for the holographic 2D projection. But unfortunately, it brings limitation in terms of displaying 3D objects. The reasons are listed as follows: Firstly, owing to the fact that the wavefront of reconstructed image is accompanied by a spherical phase distribution, the shape of the image wavefront will be deformed upon further propagates along optical axis, resulting in a serious distortion problem when reconstructing 3D object with depth space information. Secondly, the phenomenon of extended DoF means that the defocusing of the projected image is rather inconspicuous when the screen is moved back and forth in a large range. In this way the depth information of large sized 3D object cannot be clearly observed. So in this paper we just use 2D images instead of 3D objects in simulation and optical experiments. Even considering this defect, we believe that the proposed work still shows potential in the application of lensless holographic projection.

5. Conclusion

In this paper, a method for speckle reduced lensless holographic projection from phase-only CGH is proposed. The phase-only CGH is generated by double-step Fresnel diffraction calculation from a 2D image which has an initial spherical convergent phase. The phase-only CGH can project high quality zoomable image in both of monochromatic and color in lensless optical system. The speckle noise in the projected image is reduced attributed to the reconstruction of complex amplitude of the image. The size of the projected image can achieve the maximum diffraction bandwidth of the SLM, which permits for the large screen projection. The proposed method is assumed to be more suitable for the application of dynamic lensless holographic projection and display.

Funding

Program 863 (2015AA016301); Program 973 (2013CB328803); National Natural Science Foundation of China (NSFC) (61605080, 61306140); Natural Science Foundation of Jiangsu Province (BK20130618).

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11. Y. Mori, T. Fukuoka, and T. Nomura, “Speckle reduction in holographic projection by random pixel separation with time multiplexing,” Appl. Opt. 53(35), 8182–8188 (2014). [CrossRef]   [PubMed]  

12. T. Shimobaba and T. Ito, “Random phase-free computer-generated hologram,” Opt. Express 23(7), 9549–9554 (2015). [CrossRef]   [PubMed]  

13. T. Shimobaba, T. Kakue, Y. Endo, R. Hirayama, D. Hiyama, S. Hasegawa, Y. Nagahama, M. Sano, M. Oikawa, T. Sugie, and T. Ito, “Random phase-free kinoform for large objects,” Opt. Express 23(13), 17269–17274 (2015). [CrossRef]   [PubMed]  

14. R. D. Juday and J. M. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991). [CrossRef]  

15. A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Near-perfect hologram reconstruction with a spatial light modulator,” Opt. Express 16(4), 2597–2603 (2008). [CrossRef]   [PubMed]  

16. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38(23), 5004–5013 (1999). [CrossRef]   [PubMed]  

17. X. Li, J. Liu, J. Jia, Y. Pan, and Y. Wang, “3D dynamic holographic display by modulating complex amplitude experimentally,” Opt. Express 21(18), 20577–20587 (2013). [CrossRef]   [PubMed]  

18. G. Xue, J. Liu, X. Li, J. Jia, Z. Zhang, B. Hu, and Y. Wang, “Multiplexing encoding method for full-color dynamic 3D holographic display,” Opt. Express 22(15), 18473–18482 (2014). [CrossRef]   [PubMed]  

19. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef]   [PubMed]  

20. H. Song, G. Sung, S. Choi, K. Won, H.-S. Lee, and H. Kim, “Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter,” Opt. Express 20(28), 29844–29853 (2012). [CrossRef]   [PubMed]  

21. O. Mendoza-Yero, G. Mínguez-Vega, and J. Lancis, “Encoding complex fields by using a phase-only optical element,” Opt. Lett. 39(7), 1740–1743 (2014). [CrossRef]   [PubMed]  

22. Y. Qi, C. Chang, and J. Xia, “Speckleless holographic display by complex modulation based on double-phase method,” Opt. Express 24(26), 30368–30378 (2016). [CrossRef]   [PubMed]  

23. C. Chang, J. Xia, L. Yang, W. Lei, Z. Yang, and J. Chen, “Speckle-suppressed phase-only holographic three-dimensional display based on double-constraint Gerchberg-Saxton algorithm,” Appl. Opt. 54(23), 6994–7001 (2015). [CrossRef]   [PubMed]  

24. M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shiraki, N. Masuda, and T. Ito, “Time-division color electroholography using one-chip RGB LED and synchronizing controller,” Opt. Express 19(13), 12008–12013 (2011). [CrossRef]   [PubMed]  

25. M. Makowski, T. Shimobaba, and T. Ito, “Increased depth of focus in random-phase-free holographic projection,” Chin. Opt. Lett. 14, 120901 (2016).

References

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  1. E. Buckley, “Holographic projector using one lens,” Opt. Lett. 35(20), 3399–3401 (2010).
    [Crossref] [PubMed]
  2. E. Buckley, “Holographic Laser Projection,” J. Disp. Technol. 7(3), 135–140 (2011).
    [Crossref]
  3. M. Makowski, I. Ducin, K. Kakarenko, J. Suszek, M. Sypek, and A. Kolodziejczyk, “Simple holographic projection in color,” Opt. Express 20(22), 25130–25136 (2012).
    [Crossref] [PubMed]
  4. T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Lensless zoomable holographic projection using scaled Fresnel diffraction,” Opt. Express 21(21), 25285–25290 (2013).
    [Crossref] [PubMed]
  5. T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15(7), 536–544 (2013).
    [Crossref]
  6. M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005).
    [Crossref]
  7. L. Golan and S. Shoham, “Speckle elimination using shift-averaging in high-rate holographic projection,” Opt. Express 17(3), 1330–1339 (2009).
    [Crossref] [PubMed]
  8. W. Qu, H. Gu, and Q. Tan, “Holographic projection with higher image quality,” Opt. Express 24(17), 19179–19184 (2016).
    [Crossref] [PubMed]
  9. Y. Takaki and M. Yokouchi, “Speckle-free and grayscale hologram reconstruction using time-multiplexing technique,” Opt. Express 19(8), 7567–7579 (2011).
    [Crossref] [PubMed]
  10. M. Makowski, “Minimized speckle noise in lens-less holographic projection by pixel separation,” Opt. Express 21(24), 29205–29216 (2013).
    [Crossref] [PubMed]
  11. Y. Mori, T. Fukuoka, and T. Nomura, “Speckle reduction in holographic projection by random pixel separation with time multiplexing,” Appl. Opt. 53(35), 8182–8188 (2014).
    [Crossref] [PubMed]
  12. T. Shimobaba and T. Ito, “Random phase-free computer-generated hologram,” Opt. Express 23(7), 9549–9554 (2015).
    [Crossref] [PubMed]
  13. T. Shimobaba, T. Kakue, Y. Endo, R. Hirayama, D. Hiyama, S. Hasegawa, Y. Nagahama, M. Sano, M. Oikawa, T. Sugie, and T. Ito, “Random phase-free kinoform for large objects,” Opt. Express 23(13), 17269–17274 (2015).
    [Crossref] [PubMed]
  14. R. D. Juday and J. M. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
    [Crossref]
  15. A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Near-perfect hologram reconstruction with a spatial light modulator,” Opt. Express 16(4), 2597–2603 (2008).
    [Crossref] [PubMed]
  16. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38(23), 5004–5013 (1999).
    [Crossref] [PubMed]
  17. X. Li, J. Liu, J. Jia, Y. Pan, and Y. Wang, “3D dynamic holographic display by modulating complex amplitude experimentally,” Opt. Express 21(18), 20577–20587 (2013).
    [Crossref] [PubMed]
  18. G. Xue, J. Liu, X. Li, J. Jia, Z. Zhang, B. Hu, and Y. Wang, “Multiplexing encoding method for full-color dynamic 3D holographic display,” Opt. Express 22(15), 18473–18482 (2014).
    [Crossref] [PubMed]
  19. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007).
    [Crossref] [PubMed]
  20. H. Song, G. Sung, S. Choi, K. Won, H.-S. Lee, and H. Kim, “Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter,” Opt. Express 20(28), 29844–29853 (2012).
    [Crossref] [PubMed]
  21. O. Mendoza-Yero, G. Mínguez-Vega, and J. Lancis, “Encoding complex fields by using a phase-only optical element,” Opt. Lett. 39(7), 1740–1743 (2014).
    [Crossref] [PubMed]
  22. Y. Qi, C. Chang, and J. Xia, “Speckleless holographic display by complex modulation based on double-phase method,” Opt. Express 24(26), 30368–30378 (2016).
    [Crossref] [PubMed]
  23. C. Chang, J. Xia, L. Yang, W. Lei, Z. Yang, and J. Chen, “Speckle-suppressed phase-only holographic three-dimensional display based on double-constraint Gerchberg-Saxton algorithm,” Appl. Opt. 54(23), 6994–7001 (2015).
    [Crossref] [PubMed]
  24. M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shiraki, N. Masuda, and T. Ito, “Time-division color electroholography using one-chip RGB LED and synchronizing controller,” Opt. Express 19(13), 12008–12013 (2011).
    [Crossref] [PubMed]
  25. M. Makowski, T. Shimobaba, and T. Ito, “Increased depth of focus in random-phase-free holographic projection,” Chin. Opt. Lett. 14, 120901 (2016).

2016 (3)

2015 (3)

2014 (3)

2013 (4)

2012 (2)

2011 (3)

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2005 (1)

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005).
[Crossref]

1999 (1)

1991 (1)

R. D. Juday and J. M. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Arrizón, V.

Bernet, S.

Buckley, E.

E. Buckley, “Holographic Laser Projection,” J. Disp. Technol. 7(3), 135–140 (2011).
[Crossref]

E. Buckley, “Holographic projector using one lens,” Opt. Lett. 35(20), 3399–3401 (2010).
[Crossref] [PubMed]

Campos, J.

Carrada, R.

Chang, C.

Chen, J.

Choi, S.

Cottrell, D. M.

Davis, J. A.

Ducin, I.

Endo, Y.

Florence, J. M.

R. D. Juday and J. M. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Fukuoka, T.

Golan, L.

González, L. A.

Gu, H.

Hasegawa, S.

Hirayama, R.

Hiyama, D.

Hu, B.

Ito, T.

Jesacher, A.

Jia, J.

Juday, R. D.

R. D. Juday and J. M. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Kakarenko, K.

Kakue, T.

Kim, H.

Kolodziejczyk, A.

M. Makowski, I. Ducin, K. Kakarenko, J. Suszek, M. Sypek, and A. Kolodziejczyk, “Simple holographic projection in color,” Opt. Express 20(22), 25130–25136 (2012).
[Crossref] [PubMed]

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005).
[Crossref]

Lancis, J.

Lee, H.-S.

Lei, W.

Li, X.

Liu, J.

Makowski, M.

Masuda, N.

Maurer, C.

Mendoza-Yero, O.

Mikula, G.

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005).
[Crossref]

Mínguez-Vega, G.

Moreno, I.

Mori, Y.

Nagahama, Y.

Nakayama, H.

Nomura, T.

Oikawa, M.

Okada, N.

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Lensless zoomable holographic projection using scaled Fresnel diffraction,” Opt. Express 21(21), 25285–25290 (2013).
[Crossref] [PubMed]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15(7), 536–544 (2013).
[Crossref]

Pan, Y.

Qi, Y.

Qu, W.

Ritsch-Marte, M.

Ruiz, U.

Sano, M.

Schwaighofer, A.

Shimobaba, T.

Shiraki, A.

Shoham, S.

Song, H.

Sugie, T.

Sung, G.

Suszek, J.

Sypek, M.

M. Makowski, I. Ducin, K. Kakarenko, J. Suszek, M. Sypek, and A. Kolodziejczyk, “Simple holographic projection in color,” Opt. Express 20(22), 25130–25136 (2012).
[Crossref] [PubMed]

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005).
[Crossref]

Takaki, Y.

Tan, Q.

Wang, Y.

Won, K.

Xia, J.

Xue, G.

Yamaguchi, Y.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15(7), 536–544 (2013).
[Crossref]

Yang, L.

Yang, Z.

Yoda, T.

Yokouchi, M.

Yzuel, M. J.

Zhang, Z.

Appl. Opt. (3)

Chin. Opt. Lett. (1)

J. Disp. Technol. (1)

E. Buckley, “Holographic Laser Projection,” J. Disp. Technol. 7(3), 135–140 (2011).
[Crossref]

J. Opt. (1)

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15(7), 536–544 (2013).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005).
[Crossref]

Opt. Express (14)

L. Golan and S. Shoham, “Speckle elimination using shift-averaging in high-rate holographic projection,” Opt. Express 17(3), 1330–1339 (2009).
[Crossref] [PubMed]

W. Qu, H. Gu, and Q. Tan, “Holographic projection with higher image quality,” Opt. Express 24(17), 19179–19184 (2016).
[Crossref] [PubMed]

Y. Takaki and M. Yokouchi, “Speckle-free and grayscale hologram reconstruction using time-multiplexing technique,” Opt. Express 19(8), 7567–7579 (2011).
[Crossref] [PubMed]

M. Makowski, “Minimized speckle noise in lens-less holographic projection by pixel separation,” Opt. Express 21(24), 29205–29216 (2013).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “Random phase-free computer-generated hologram,” Opt. Express 23(7), 9549–9554 (2015).
[Crossref] [PubMed]

T. Shimobaba, T. Kakue, Y. Endo, R. Hirayama, D. Hiyama, S. Hasegawa, Y. Nagahama, M. Sano, M. Oikawa, T. Sugie, and T. Ito, “Random phase-free kinoform for large objects,” Opt. Express 23(13), 17269–17274 (2015).
[Crossref] [PubMed]

H. Song, G. Sung, S. Choi, K. Won, H.-S. Lee, and H. Kim, “Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter,” Opt. Express 20(28), 29844–29853 (2012).
[Crossref] [PubMed]

X. Li, J. Liu, J. Jia, Y. Pan, and Y. Wang, “3D dynamic holographic display by modulating complex amplitude experimentally,” Opt. Express 21(18), 20577–20587 (2013).
[Crossref] [PubMed]

G. Xue, J. Liu, X. Li, J. Jia, Z. Zhang, B. Hu, and Y. Wang, “Multiplexing encoding method for full-color dynamic 3D holographic display,” Opt. Express 22(15), 18473–18482 (2014).
[Crossref] [PubMed]

M. Makowski, I. Ducin, K. Kakarenko, J. Suszek, M. Sypek, and A. Kolodziejczyk, “Simple holographic projection in color,” Opt. Express 20(22), 25130–25136 (2012).
[Crossref] [PubMed]

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Lensless zoomable holographic projection using scaled Fresnel diffraction,” Opt. Express 21(21), 25285–25290 (2013).
[Crossref] [PubMed]

A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Near-perfect hologram reconstruction with a spatial light modulator,” Opt. Express 16(4), 2597–2603 (2008).
[Crossref] [PubMed]

Y. Qi, C. Chang, and J. Xia, “Speckleless holographic display by complex modulation based on double-phase method,” Opt. Express 24(26), 30368–30378 (2016).
[Crossref] [PubMed]

M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shiraki, N. Masuda, and T. Ito, “Time-division color electroholography using one-chip RGB LED and synchronizing controller,” Opt. Express 19(13), 12008–12013 (2011).
[Crossref] [PubMed]

Opt. Lett. (2)

Proc. SPIE (1)

R. D. Juday and J. M. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Supplementary Material (4)

NameDescription
» Visualization 1: MOV (1101 KB)      projected animation of "basketball" in lensless holographic projection system
» Visualization 2: MOV (1574 KB)      projected animation of "horse-riding" in lensless holographic projection system
» Visualization 3: MOV (1136 KB)      projected animation of large-sized "basketball" in lensless holographic projection system
» Visualization 4: MOV (1703 KB)      projected animation of large-sized "horse-riding" in lensless holographic projection system

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Figures (9)

Fig. 1
Fig. 1 Principle illustration of the proposed method for lensless holographic projection. (a) Geometry model of the CGH calculation. (b) Geometry model of the CGH reconstruction.
Fig. 2
Fig. 2 Computer simulation results. (a) The original image. (b) Phase-only CGH calculated by the proposed method. (c) Amplitude of light field at the virtual plane (filter plane). (d) Reconstructed amplitude of the image. (e) Reconstructed phase of the image. (f) Reconstructed amplitude of the image in a different size.
Fig. 3
Fig. 3 (a) Optical setup for lensless holographic projection. (b) Optical reconstructions of projected images (image size is 10.24mm). (c) Optical reconstructions of projected images (image size is 20.48mm).
Fig. 4
Fig. 4 One frame from reconstructed animations (image size is 20.48mm). (a) Basketball (see Visualization 1). (b) Horse riding (see Visualization 2).
Fig. 5
Fig. 5 Optical reconstructions of color projected images. (a) Reconstruction of red component of “Lena”. (b) Reconstruction of green component of “Lena”. (c) Reconstruction of blue component of “Lena”. (d) Reconstruction of color “Lena”. (e)-(g) Reconstruction of color “SEU”, “Baboon” and “Fruits”. (h)-(j) Reconstruction of color “Parrots” with different pixel pitches (10μm, 20μm and 30μm).
Fig. 6
Fig. 6 Optical reconstructions of zoomable lensless holographic projection. (a) The projected screen. (b)-(e) Monochromatic reconstructions with different image sizes of 8.2cm, 12.3cm, 16.4cm and 20.5cm (Visualization 3, Visualization 4). (f)-(h) Three color reconstructions with image size of 20.5cm.
Fig. 7
Fig. 7 Comparison of the results by different methods. (a) The iterative method. (b) The random phase-free (RPF) method. (c) The proposed method. (C = speckle contrast.)
Fig. 8
Fig. 8 Geometry models of different positions of the virtual plane. (a) d1 = ds. (b) d1>ds. (c) d1<ds. (Parameters in the simulation: λ = 532nm, N = 1024, z = 0.5m, dh = 8μm).
Fig. 9
Fig. 9 Measurement of reconstructed image quality by PSNR under different iris diameters. (Parameters in the simulation: λ = 532nm, N = 1024, z = 0.5m, dh = 8μm, dx = 20μm, d1 = 0.1m, d2 = 0.4m).

Equations (4)

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V P ( x v , y v ) = I ( x , y ) exp { i π λ d 2 [ ( x v s x ) 2 + ( y v s y ) 2 ] } d x d y = F F T 1 { F F T [ I ( x , y ) exp ( i φ 1 ) ] F F T [ exp ( i φ 2 ) R e c t ] } .
H ( x h , y h ) = V P ( x v , y v ) exp { i π λ d 1 [ ( x h x v ) 2 + ( y h y v ) 2 ] } d x v d y v = exp [ i π ( x h 2 + y h 2 ) λ d 1 ] F F T [ V P ( x v , y v ) exp [ i π ( x v 2 + y v 2 ) λ d 1 ] ] .
θ 1 ( x h , y h ) = φ h ( x h , y h ) + cos 1 [ A h ( x h , y h ) / A max ] θ 2 ( x h , y h ) = φ h ( x h , y h ) cos 1 [ A h ( x h , y h ) / A max ] .
p ( x h , y h ) = θ 1 ( x h , y h ) M 1 ( x h , y h ) + θ 2 ( x h , y h ) M 2 ( x h , y h ) + 2 π x h sin α / λ .

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