Spectral-envelope modulated double-phase method for computer-generated holography

Xiaomeng Sui; Zehao He; Guofan Jin; Liangcai Cao

doi:10.1364/OE.463462

1. Introduction

Computer-generated holography (CGH) enables the reshaping of volumetric intensity patterns by modulating the coherent wavefront. This ability to control the spatial propagation of light enables its applications in fields like 3-D display [1–5], optical trapping [6], optical encryption [7], beam shaping [8] and optogenetics [9,10]. Since the digital device for complex modulation is not yet available, many current implementations of holographic wavefronts modulation are carried out by phase-type devices. The fundamental concern of generating such a phase-type hologram is basically to solve the phase-synthesis problem of a target object propagating to form a band-limited hologram with a constant intensity distribution [11,12]. Two categories of existing algorithms can solve this problem: non-convex optimization algorithms and complex converting algorithms.

Non-convex optimization algorithms solve the desired phase-type hologram from a known reconstruction and several constraints. Unlike in the general phase-retrieval situations, the band-limited uniform-intensity holographic distribution is not ensured to exist in the computational synthesis of holograms. But several phase distributions approximately satisfying the constraints can be obtained as solutions since the relation between the reconstructed wavefront and its intensity is ill-conditioned. One of suchlike approximate solutions can be searched through non-convex optimization. Corresponding algorithms include alternative projection and nonlinear minimization. The alternative projection algorithms, like Gerchberg–Saxton (GS) algorithm [13] and error reduction algorithm [14], iteratively update the solution by projecting it between a pair of wave distributions. The nonlinear minimization algorithms, including stochastic gradient descent [15], Wirtinger flow [16] and quasi-Newton gradient descent [17], search for the optimal resolution of a well-chosen loss function towards the optimizing direction of gradient descent. This class of algorithms is easy for optical implementation. However, random phases are typically introduced in the optimization, resulting in severe speckle noise in the reconstruction of the optimized holograms [18]. Although specific phase distributions can be designed to suppress the speckle noise [19,20], the hologram optimization is time-consuming.

The second category is the complex converting algorithms. The desired hologram is obtained by an error-controlled conversion from the diffracted complex wavefront to an intensity-constant distribution [21–23]. An applicable method for complex conversion is double-phase decomposition [24,25], through which the complex wavefront can be decomposed into two phase components. The phase components are encoded into a double-phase hologram (DPH) and reconstruct the complex wavefront after superposition. This method enables photorealistic reconstructions with holograms uploaded on a phase-type device. Since the complex wavefront is converted analytically, the computation of DPHs is sufficiently time-efficient to achieve a real-time reconstruction [26]. However, the spatial shifting between the two phases encoded in one hologram causes a non-perfect superposition and brings about spatial-shifting noises in reconstruction [27,28]. The optical reconstruction for DPHs thus requires a 4-f spatial filtering to block the noisy frequencies. And the reconstruction becomes a tradeoff between a higher spatial resolution and fewer spatial-shifting noises [29].

Herein, we propose a spectral-envelope modulated double-phase method (SEM DPH) to preserve the high frequency components and suppress the spatial-shifting noises simultaneously. This approach is inspired by our preceding research of signal-noise distribution analysis of a DPH [29,30] and a further derivation for the signal-noise relation of a DPH. The derivation reveals that the signal and the noise of a DPH are modulated by the overlapped envelope functions. The modulated terms of the signal and the noise are mathematically coupled. The signal-noise overlap causes the degradation of reconstructions in either spatial resolution or signal-to-noise radio. We solve this conflict by an off-axis spectral envelope modulation: a spectral transmittance function is introduced in hologram computation to reduce the signal-noise overlap; a biaxial linear phase is introduced in optical reconstruction to separate the signal from the spatial-shifting noises and the DC term on the Fourier plane.

In Section 2, we describe the transmittance function of a conventional on-axis double-phase hologram (on-axis DPH) and separate the signal from the spatial-shifting noises on the Fourier plane. In Section 3, we illustrate the computational process for the SEM DPH and optical operations for the off-axis reconstruction. In Section 4, we demonstrate that the proposed SEM DPH outperforms the conventional on-axis DPH numerically and experimentally. And the SEM DPHs are compared with phase-only holograms generated by optimization algorithms, verifying that the SEM DPH can be generated in nearly one second.

2. On-axis DPH

Starting from the on-axis DPH, a free-space propagated complex wavefront can be described by amplitude A and phase φ, given as:

(1)$$u({x,y} )= A({x,y} ){e^{i\varphi ({x,y} )}}. $$

This complex wavefront is converted to a pixelated transmittance function of a DPH encoded on a phase-type element, written as:

(2)$$h({x,y} )= \textrm{rect}\left( {\frac{x}{{{L_x}}},\frac{y}{{{L_y}}}} \right)[{{h_a}({x,y} )+ {h_d}({x,y} )} ], $$

where L_x and L_y are the length and width of the phase-type element, respectively. All the symbols representing for parameters of the phase-type element are annotated in Fig. 1(a). The rectangle function limits the spatial extension of the transmittance function according to the size of the panel. ${h_a}({x,y} )$ and ${h_d}({x,y} )$ are transmittance functions for the pixel-active window and the dead area of the element, respectively, given by:

(3)$${h_a}({x,y} )= {\omega _{ab}}({x,y} )\ast \left[ {\sum\limits_{n ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {{e^{i{\theta_{n,m}}}} \cdot \delta ({x - n\alpha ,y - m\beta } )} } } \right], $$

(4)$${h_d}({x,y} )= \left\{ {[{{\omega_{\alpha \beta }}({x,y} )- {\omega_{ab}}({x,y} )} ]\ast \sum\limits_{n ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {\delta ({x - n\alpha ,y - m\beta } )} } } \right\}{A_0}({x,y} ){e^{i{\varphi _0}({x,y} )}}, $$

where ${A_0}({x,y} ){e^{i{\varphi _0}({x,y} )}}$ is the modulated incident wave of the dead area. ${\omega _{ab}}({x,y} )$ represents for the pixel-active window of dimensions $({a,b} )$ and ${\omega _{\alpha \beta }}({x,y} )$ represents for the rectangle structure determined by the pixel intervals $({\alpha ,\beta } )$, written as:

(5)$${\omega _{ab}}({x,y} )= \textrm{rect}\left( {\frac{x}{a}} \right)\textrm{rect}\left( {\frac{y}{b}} \right), $$

(6)$${\omega _{\alpha \beta }}({x,y} )\textrm{ = rect}\left( {\frac{x}{\alpha }} \right)\textrm{rect}\left( {\frac{y}{\beta }} \right). $$

Fig. 1. The phase-type element panel to encode the double-phase hologram. (a) The element panel with annotations of parameters. (b) The cell structure of a single-pixel DPH. (c) The cell structure of a macro-pixel DPH.

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The relation between $({a,b} )$ and $({\alpha ,\beta } )$ is determined by the filling factor of the phase-type element. ${h_a}({x,y} )$ contributes an effective modulation on the incident wavefront and ${h_d}({x,y} )$ provides a DC term in diffraction.

The phase ${e^{i{\theta _{n,m}}}}$ in Eq. (3) can be encoded by the single-pixel double-phase cell structure as is shown in Fig. 1(b), given by [25]:

(7)$${e^{i{\theta _{n,m}}}} = {\cos ^2}\left[ {\frac{{\pi ({n + m} )}}{2}} \right]{e^{i\theta _{n,m}^{(1 )}}} + {\cos ^2}\left[ {\frac{{\pi ({n + m + 1} )}}{2}} \right]{e^{i\theta _{n,m}^{(2 )}}}, $$

where $\cos [{\cdot} ]$ operator generates a checkboard-type binary grating for complementary sampling, with periods of 2α and 2β along the axes x and y. The decomposed phase pairs are described by,

(8)$$\theta _{n,m}^{(1 )}\textrm{ = }{\varphi _{n,m}} - {\cos ^{ - 1}}({{A_{n,m}}} ), $$

(9)$$\theta _{n,m}^{(2 )} = {\varphi _{n,m}} + {\cos ^{ - 1}}({{A_{n,m}}} ). $$

According to our preceding research, the reconstructed accuracy has a slight improvement if the single-pixel structure is converted to a macro-pixel cell structure (Fig. 1(c)) through upsampling, with the same size and pixel number of the element [28,30]. We therefore implement a macro-pixel cell structure in this work.

We transfer the transmittance function $h({x,y} )$ to its Fourier spectrum $H({u,v} )$, written as:

(10)$$H({u,v} )= {L_x}{L_y}\textrm{sinc}({{L_x}u} )\textrm{sinc}({{L_y}v} )\ast [{{H_a}({u,v} )+ {H_d}({u,v} )} ]. $$

${H_a}({u,v} )$ and ${H_d}({u,v} )$ are the Fourier spectra of the pixel-active window ${h_a}({x,y} )$ and the dead area ${h_d}({x,y} )$ respectively, given by:

(11)$${H_a}({u,v} )= {E_{ab}}({u,v} )\sum\limits_{n ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {{e^{i{\theta _{n,m}}}} \cdot {e^{ - i2\pi ({n\alpha u + m\beta v} )}}} } , $$

(12)$${H_d}({u,v} )= \left\{ {[{{E_{\alpha \beta }}({x,y} )- {E_{ab}}({x,y} )} ]\cdot \sum\limits_{n ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {{e^{ - i2\pi ({n\alpha u + m\beta v} )}}} } } \right\} \cdot \mathrm{{\cal F}}\{{{A_0}({x,y} ){e^{i{\varphi_0}({x,y} )}}} \}, $$

where $\mathrm{{\cal F}}\{{\cdot} \}$ is the Fourier transform operator. In Eqs. (11) and (12), the modulating envelopes ${E_{ab}}({u,v} )$ and ${E_{\alpha \beta }}({x,y} )$ are

(13)$${E_{ab}}({u,v} )= ab\textrm{sinc}({au} )\textrm{sinc}({bv} ), $$

(14)$${E_{\alpha \beta }}({u,v} )= \alpha \beta \textrm{sinc}({\alpha u} )\textrm{sinc}({\beta v} ). $$

We expand the inner phase term of Eq. (11) based on the macro-pixel cell structure. Then Eq. (11) can be rewritten in the form of:

(15)$${H_a}({u,v} )\textrm{ = }{E_{ab}}({u,v} )\sum\limits_{n ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {\left\{ \begin{array}{l} {e^{i\theta_{n,m}^{(1 )}}} \cdot [{{e^{i\pi ({\alpha u + \beta v} )}} + {e^{i\pi ({ - \alpha u - \beta v} )}}} ]\\ + {e^{i\theta_{n,m}^{(2 )}}} \cdot [{{e^{i\pi ({ - \alpha u + \beta v} )}} + {e^{i\pi ({\alpha u - \beta v} )}}} ]\end{array} \right\} \cdot {e^{ - i2\pi ({2n\alpha u + 2m\beta v} )}}} }. $$

Considering Eqs. (8,9), ${H_a}({u,v} )$ can be expanded through Euler’s formula and rewritten into the superposition of a signal term and a noise term:

(16)$${H_a}({u,v} )= {H_S}({u,v} )+ {H_N}({u,v} ). $$

${H_S}({u,v} )$ is the signal term detailed as:

(17)$${H_S}({u,v} )\textrm{ = }{E_{ab}}({u,v} ){P_S}({u,v} ){U_S}({u,v} ), $$

where ${P_S}({u,v} )$ is an envelope function modulating the signal term and given as:

(18)$${P_S}({u,v} )= 4\cos ({\pi \alpha u} )\cos ({\pi \beta v} ). $$

And ${U_S}({u,v} )$ is the Fourier transform of the desired complex wavefront, whose first nonzero diffraction order is centered at the spatial-frequency coordinates ranging in $[{ - 1/4\alpha ,1/4\alpha } ]$ and $[{ - 1/4\beta ,1/4\beta } ]$, written as:

(19)$${U_S}({u,v} )= \mathrm{{\cal F}}\{{A({x,y} ){e^{i\varphi ({x,y} )}}} \}. $$

In Eq. (16), ${H_N}({u,v} )$ is the noise term detailed as:

(20)$${H_N}({u,v} )\textrm{ = }{E_{ab}}({u,v} ){P_N}({u,v} ){U_N}({u,v} ), $$

where ${P_N}({u,v} )$ is an envelope function modulating the noise term and given as:

(21)$${P_N}({u,v} )= 4\sin ({\pi \alpha u} )\sin ({\pi \beta v} ). $$

And ${U_N}({u,v} )$ is the inner complex term of the noise term and we also convert it into a form of Fourier transform:

(22)$${U_N}({u,v} )= \mathrm{{\cal F}}\left\{ {\sqrt {1 - {A^2}({x,y} )} \cdot {e^{i\;\left[ {\frac{\pi }{2} - \varphi ({x,y} )} \right]}}} \right\}. $$

${U_S}({u,v} )$ and ${U_N}({u,v} )$ share a similar frequency distribution but are different in the intensity. Since $A({x,y} )$ is normalized to $[{0,\; 1} ]$, Eq. (22) indicates that the spatial-shifting noise is associated with the low-intensity components in the complex wavefront. This provides an explanation for the experimental fact that DPH reconstructs severe spatial-shifting noises when large areas of low-intensity components are contained in the complex wavefront.

3. SEM method for DPH

3.1 Computational modulation

The computational modulation is carried out by a spectral transmittance function. This constraint on the Fourier plane could control the spatial frequency distribution effectively and reduce the bandwidth occupied by the noise term. As is shown in Fig. 2(a), the complex wavefront generated from the diffraction of an object is filtered by the spectral transmittance function through convolution and produces a modulated complex wavefront. The modulated wavefront is decomposed according to Eqs. (8,9) and then encoded into a SEM DPH.

Fig. 2. (a) The computation process of the SEM DPH. The on-axis Fourier spectra of (b) the DCE DPH, (c) the SEM DPH modulated by P_M₁, and (d) the SEM DPH modulated by P_M₂.

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One basic form of the spectral transmittance function is converted from the envelope of signal term, given by:

(23)$${P_{M1}}({u,v} )= \textrm{norm}|{{E_{ab}}({u,v} )\cdot {P_S}({u,v} )} |, $$

where $\textrm{norm}[{\cdot} ]$ is the operation for the normalization. This distribution of P_M₁ is designed to keep the shape of spectral distributions approximately the same before and after the modulation.

In another common case, the complex wavefront with fewer low-intensity areas can be filled into a rectangle panel. The corresponding Fourier spectrum processes more frequency components along the axes u and v. We therefore define a spectral transmittance function P_M₂ to modulate the complex wavefronts cut off by the rectangle window of the element, given by:

(24)$${P_{M2}}({u,v} )= 1 - \textrm{norm}|{{E_{ab}}({u,v} )\cdot {P_N}({u,v} )} |. $$

The spectral transmittance function P_M₂ could provide a better preservation of the low-noise spatial frequencies, especially for the frequencies along the axes. It should be noted that the frequency ranges of P_M₁ and P_M₂ are matched with the frequency ranges of the complex wavefronts. For a single-pixel DPH, the spatial frequencies range in $[{ - 1/2\alpha ,1/2\alpha } ]$ and $[{ - 1/2\beta ,1/2\beta } ]$, while for a macro-pixel DPH the spatial frequencies range in $[{ - 1/4\alpha ,1/4\alpha } ]$ and $[{ - 1/4\beta ,1/4\beta } ]$.

The effectiveness of computational spectral-envelope modulation with P_M₁ and P_M₂ is also illustrated in Fig. 2. The DC-eliminated DPH (DCE DPH), in which the DC term is eliminated by numerically setting the filling factor as 1, and the SEM DPHs (modulated by P_M₁ and P_M₂) are utilized to encode 30 complex wavefront matrixes. These complex wavefront matrixes are constructed by randomly distributed amplitudes ranging in $[{0,\; 1} ]$ and randomly distributed phases ranging in $[{0,\; 2\pi } ]$. The Fourier spectra for each type of DPHs are numerically computed, and their averaged distributions are presented in Figs. 2(b)-(d). The signal and the noise of the DCE DPH have a severe overlap as is shown in Fig. 2(b), while this overlap is sufficiently reduced in the spectrum of SEM DPH with the modulation of P_M₁ as is shown in Fig. 2(c). This separation of the desired signal is achieved at a sacrifice of the bandwidth. P_M₂ is thus utilized to make up for the bandwidth loss. As is shown in Fig. 2(d), the spectrum presents a better preservation of the spatial frequencies along the axes on the basis of signal-noise separation.

We numerically reconstructed test images from DCE DPHs and SEM DPHs (P_M₁ and P_M₂). In the reconstruction, a numerical circular apertured is utilized for spatial filtering. According to Shannon’s sampling theorem, the Fourier spectrum of the DPH is sampled at the intervals of

(25)$${\Delta _u} = \frac{{\lambda f}}{{2\alpha N}},\;{\Delta _v} = \frac{{\lambda f}}{{2\beta M}}. $$

The computation and numerical reconstruction are carried out under a condition of wavelength $\lambda = 532\;\textrm{nm}$, the focal length of the lens $f = 100\;\textrm{mm}$, the pixel pitch $\alpha = \beta = 3.74\;\mathrm{\mu m}$ and the pixel number $M = N = 1024$. The sampling intervals in numerical reconstruction are therefore calculated to be ${\Delta _u} = {\Delta _v} = 6.95\;\mathrm{\mu m}$. Since a zero-padding operation is introduced to avoid the circular convolution error, the sampling number of the spectrum of DPH is expanded to $2048 \times 2048$. And accordingly, the scale of the spectrum is calculated to be ${L_u} = {L_v} = 14.23\;\textrm{mm}$. The diameter of the circular aperture used in numerical reconstruction is thus adjusted within the scale of the spectrum and referred to as “aperture size” in the following numerical reconstructions.

The structural similarity index measurements (SSIMs) for reconstructions are plotted along the axes. As is presented in Fig. 3, these test images can be classified into two categories: relatively smooth images (Figs. 3(a)-(d)) and relatively fluctuant images (Figs. 3(e)-(h)). The Fourier spectra of the test images are present after an operation of $\ln ({1 + |U |} )$, given the existence of extreme peak values on the centers of the spectra. The spectra for the relatively smooth images present concentrated energy distributions on axes and the original point. This spectral characteristic enables the reconstructions of these images to attain their highest SSIMs of over 0.95 with spectral-envelope modulation by P_M₂ and indicate a reduction of SSIM within 0.1 when the aperture size is doubled. However, the reconstructions of the relatively fluctuant images present limited performance in accuracy and are more sensitive to the aperture size. But for all the images tested, the SEM DPHs perform well in keeping improved SSIMs with the variation of the aperture size. It verifies the capability of the SEM DPH to achieve the reconstruction with both high spatial resolution and few spatial-shifting noises. The SEM DPH modulated by P_M₂ outperforms that modulated by P_M₁ for most tested images, except for Fig. 3(g) where the spectrum shows a more isotropic distribution with relatively fewer frequencies distributed along the axes.

Fig. 3. Numerical reconstructing comparisons of SSIMs for test images computed by the DCE DPH and the SEM DPHs (P_M₁ and P_M₂) with different reconstructing aperture sizes. (a)-(h) are sorted according to the spectral frequency components.

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3.2 Off-axis optical reconstruction

The optical strategy for the off-axis reconstruction of SEM DPHs is utilized to further achieve a well separation of the desired order of signal term from other terms. The existence of spatial-shifting noise complicates the diffraction pattern on the Fourier plane for on-axis reconstruction, which contains a DC term, a multi-order signal term and a multi-order noise term. To shift the spatial position of the desired signal to an area of the least overlap with other terms, a biaxial linear phase $\gamma ({x,y} )$ is defined as [31]:

(26)$$\gamma ({x,y} )= \frac{{2\pi }}{\lambda }({x\cos {\vartheta_x} + y\cos {\vartheta_y}} ). $$

As is shown in Fig. 4(a), the center point of ${H_a}({u,v} )$ is labeled as A and the center point of ${H_d}({u,v} )$ is labeled as O’. Coordinate xOy is the plane for the element with a DPH and coordinate uO’v represents for the Fourier plane of a lens. The two-dimensional spatial shifting ${\Delta _x}$ and ${\Delta _y}$ are associated with the deflection angles ${\vartheta _x}$ and ${\vartheta _y}$, described by:

(27)$${\vartheta _x} = \textrm{arctan}\left( {\frac{{{\Delta_x}}}{f}} \right),\;{\vartheta _y} = \textrm{arctan}\left( {\frac{{{\Delta_y}}}{f}} \right). $$

Fig. 4. (a) The off-axis reconstruction carried out by the biaxial linear phase. (b) The Fourier spectrum of an off-axis DPH captured in optical reconstruction.

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The values of ${\vartheta _x}$ and ${\vartheta _y}$ are adjusted to furthest avoid the overlap of the signal and other terms. As is shown in Fig. 4(b), the point A together with ${H_a}({u,v} )$ have both horizontal and vertical spatial shifts over the point O’, which is separated on the Fourier plane. With the combination of computational modulation and off-axis optical reconstruction, a high-bandwidth noise-suppressed reconstruction of a DPH could be obtained.

4. Reconstruction assessment

4.1 Numerical reconstruction

The DPHs with 1024×1024 pixels are computed on a PC with an Intel Xeon 2.4 GHz CPU and 32 GB of RAM, and an NVIDIA GeForce GTX 1060 GPU. Objects for hologram computation are 2D test images. Numerical diffraction of the objects is conducted by the band-limited angular spectrum theory [32,33] for a near-field propagation at a distance of 50 mm. The reconstructions are evaluated by peak signal-to-noise ratio (PSNR), SSIM and normalized error.

We numerically reconstruct conventional on-axis DPHs, conventional DPHs with off-axis reconstructing strategy (off-axis DPHs) and SEM DPHs. The reconstructions of on-axis DPHs (Figs. 5(b) and (f)) are simulated according to the transmittance function Eqs. (3,4), assuming a 90% filling factor of the phase-type element. The reconstructions of off-axis DPHs (Figs. 5(c) and (g)) are simulated through superposing the biaxial linear phase given by Eq. (26) on to the on-axis DPHs before a numerical propagation. The reconstructions of SEM DPHs (Figs. 5(d) and (h)) include both the superposition of a biaxial linear phase and the computational modulation. Figures 5(d) and (h) are modulated by P_M₁ and P_M_2, respectively. As is indicated, the introduction of the biaxial linear phase in numerical reconstruction has a significant effect on removing DC term and brings about 10.99% and 13.46% improvements in PSNRs for Figs. 5(c) and (g), respectively. And the computational modulation enables the suppression of the spatial-shifting noise and results in 17.21% and 30.00% further improved PSNRs in Figs. 5(d) and (h), respectively.

Fig. 5. Numerical reconstructions demonstrating the effectiveness of SEM DPHs. (a),(e) The 2D test images used as objects. (b),(f) Numerical reconstructions for the on-axis DPHs. (c),(g) Numerical reconstructions for the off-axis DPHs. (d),(h) Numerical reconstructions for the SEM DPHs.

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A wide comparison is carried out between SEM DPHs and the phase-only holograms generated by the non-convex optimization algorithms introduced in Section 1, including alternative projection algorithms and nonlinear minimization algorithms. The alternative projection algorithm used in the comparison is GS algorithm, which is one of the most universally used algorithms to optimize phase-only holograms. The implementation of nonlinear minimization algorithm here is based on quasi-Newton gradient descent algorithm. As is presented in Figs. 6(b) and (f), the GS algorithm takes ∼18 s for 50 iterations to reach the convergence. The numerical reconstructions quite suffer from excessive speckle noises. The quasi-Newton algorithm produces higher PSNRs and SSIMs in numerical reconstruction, as is shown in Figs. 6(c) and (g). But this algorithm is much more time-consuming, taking more than 525 s for 50 iterations to reach the convergence. The SEM DPHs provide more accurate reconstructions compared with holograms generated by the quasi-Newton algorithm, and the computation is finished in nearly one second.

Fig. 6. Numerical reconstructions comparing SEM DPHs with the phase-only holograms generated by non-convex optimization algorithms. (a),(e) The 2D test images used as objects. (b),(f) Numerical reconstructions for the phase-only holograms generated by GS algorithm . (c),(g) Numerical reconstructions for the phase-only holograms generated by quasi-Newton algorithm. (d),(h) Numerical reconstructions for the SEM DPHs.

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4.2 Optical reconstructions

The experimental validation for the proposed SEM DPH is implemented with the optical setup presented in Fig. 7. A divergent laser beam at a wavelength of 532 nm acts as the source for illumination. The laser beam is collimated before passing through the polarizer. And it is directed by a beam splitter to a reflective phase-type spatial light modulator (SLM). The phase-type SLM has a pixel number of 4160 × 2464, pixel pitch of 3.74 µm, filling factor of 90%, and frame rate of 60 Hz. The 2D object is adjusted to fit the parameters of SLM with a sampling number of 2048 × 2048 padded to 4160 × 2464 and a sampling interval of 3.74 µm. The beam splitter allows both the plane wave illumination on the SLM and the reflection towards a 4-f system. The signal term on the Fourier plane is spatially shifted away from the DC term and the noises with the biaxial linear phase and separated by a filtering iris. The complex wavefront is reconstructed on the back focal plane of Lens₃ and propagates for a distance to form the desired intensity pattern. The aperture size of the iris can be adjusted to attain the best reconstruction and it is set at 8.5 mm in this research. The optically reconstructed intensity patterns are captured by a complementary metal oxide semiconductor (CMOS) detector.

Fig. 7. Optical setup for the experimental validation of the SEM DPH.

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The improvement brought by the SEM DPH is demonstrated by the comparison of optical reconstructions of on-axis DPHs, off-axis DPHs and SEM DPHs. The optical reconstructions of off-axis DPHs indicate that the DC term and part of the spatial-shifting noises can be reduced as is shown in Fig. 8(c) and (f), presenting 51.50% and 157.48% higher SSIMs compared with the reconstructions of on-axis DPHs shown in Figs. 8(b) and (f). And the optical reconstructions of SEM DPHs enable the further suppression of the spatial-shifting noises, presenting 78.37% and 15.30% higher SSIMs compared with the reconstructions of off-axis DPHs. It is consistent with the numerical reconstruction that the SEM DPH indicates greater effectiveness while encoding the hologram generated from a relatively smooth test image like Fig. 8(e). Concluded from Figs. 8(f)-(h), SEM DPHs bring about total improvements of 9.54% in PSNR from 14.99 dB to 16.42 dB, and 196.86% in SSIM from 0.2072 to 0.6151.

Fig. 8. Optical experiments validating the improved reconstructing performance with SEM DPHs. (a),(e)The 2D test images used as objects. (b),(f) Optical reconstructions for the on-axis DPHs. (c),(g) Optical reconstructions for the off-axis DPHs. (d),(h) Optical reconstructions for the SEM DPHs.

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The comparison between SEM DPHs and the phase-only holograms generated by the non-convex optimization algorithms are demonstrated, as is shown in Fig. 9. The holograms generated by GS algorithm reconstruct speckled intensity patterns, as is shown in Figs. 9(b) and (f). The holograms generated by quasi-Newton algorithm perform relatively better with slightly higher PSNRs and SSIMs, but the speckle noise cannot be avoided, as is shown in Figs. 9(c) and (g). The proposed SEM DPH enables the reconstruction with relatively higher spatial resolution and minimal noises, as is shown in Figs. 9(d) and (h).

Fig. 9. Optical reconstructions comparing SEM DPHs with the phase-only holograms generated by non-convex optimization algorithms. (a),(e) 2D test images used as objects. (b),(f) Optical reconstructions for the phase-only holograms generated by GS algorithm . (c),(g) Optical reconstructions for the phase-only holograms generated by quasi-Newton algorithm. (d),(h) Optical reconstructions for the SEM DPHs.

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5. Conclusion

An off-axis spectral-envelop modulated double-phase method is proposed to preserve the high-frequency components and suppress the spatial-shifting noises in the encoding of complex holograms. The SEM DPH improves the reconstructing accuracy through the combination of computational modulation and optical off-axis reconstruction. We demonstrate the SEM DPH outperforming the conventional on-axis DPH in reconstructing accuracy, presenting 9.54% and 196.86% improvements for PSNR and SSIM in optical reconstructions, respectively. The high reconstruction accuracy of SEM DPHs enables the precise light field control applied in optical trapping and bio-optical activation. The SEM DPH also possesses an advantage in computing efficiency. This enables the further applications in real-time holography and interactive holography.

Funding

National Natural Science Foundation of China (62035003, 61775117).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Spectral-envelope modulated double-phase method for computer-generated holography

Abstract

1. Introduction

2. On-axis DPH

3. SEM method for DPH

3.1 Computational modulation

3.2 Off-axis optical reconstruction

4. Reconstruction assessment

4.1 Numerical reconstruction

4.2 Optical reconstructions

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (27)

Optics Express