Speckleless holographic display by complex modulation based on double-phase method

Yijun Qi; Chenliang Chang; Jun Xia

doi:10.1364/OE.24.030368

1. Introduction

Holographic display is regarded as the ideal technique to realize true three-dimensional (3-D) displays in the future [1]. Generally a computer-generated hologram (CGH) with coherent illumination is implemented for the holographic display [2]. However, the use of coherent light as illumination source leads to speckle noise in the reconstruction. The speckle noise in the reconstructed image comes from the uncontrolled random-phase distribution at the reconstructed plane. Here, the interference among closely adjacent pixels forms an undesired intensity pattern [3]. Although the speckle noise can be suppressed by using other less coherent light sources such as light-emitting diodes (LED), LED results to the problem of image blurring. So, Lasers are still the main light sources due to better performance in high brightness and contrast with lower power [4]. Time averaging method [5–7] is widely used to suppress speckle noises. However, this method is time consuming in calculation and requires high frame rate devices to display CGHs, which is challenging for real-time holographic display.

To date, a considerable amount of research has been dedicated to the improvement of the resultant reconstruction. Among the proposed methods, complex modulation [8, 9] using spatial light modulator (SLM) is an appealing method for high quality holographic reconstruction in recent years which is capable of modulating the amplitude and the phase of the optical field simultaneously and independently. Meanwhile, the applications of complex modulation involve not only holographic display but also pattern recognition [10], optical encryption [11], beam shaping [12], optical tweezers [13] and optical communication [14]. However, due to most commercial SLM modulating either the phase or the amplitude of the light, it remains a challenge to obtain complex modulation by using single SLM.

In order to circumvent the unavailability of complex modulation using single SLM, both encoding methods [15–18] and device methods [19, 20] have been proposed. Compared with device method, encoding method is more preferable and has been widely used nowadays because of the feasibility of experimental system.

A well-known encoding method to generate complex field is displaying an off-axis CGH on a phase-only SLM, in which the amplitude information is encoded as part of the phase information through interference with an off-axis reference plane wave [18]. Another version of this method resorts to Bessel function [21]. The essence of these methods is the first order of the expansion happens to be the desired complex object.

Another kind of encoding methods for complex modulation is double-phase hologram (DPH) [22] which decomposes a complex amplitude element into two pure phase values with constant magnitude. Arrizón et al. encoded an arbitrary complex function with a twisted nematic liquid-crystal display based on double-phase method [16, 17]. Here, it should be noted that in [23] Arrizón utilized certain binary factors to separate noise from the signal in the reconstruction plane which is different from the original DPH. Afterwards, applications based on double-phase method have been proposed to contrive more desirable complex reconstruction. Hoon Song et al. proposed an optical system for synthesizing double-phase complex CGHs using a phase-only SLM and a phase grating filter [24]. Nevertheless, the above DPH suffer loss of half of the number of pixels at the output plane of complex field. More recently, Omel Mendoza-Yero proposed a complex modulation method based on double-phase method without loss of number of pixels in the output plane of complex field [25]. In this method, the target complex object with suppressed speckle noise is obtained at the output plane of the 4-f system by isolating zero-order diffraction spatially at the frequency plane. Furthermore, this encoding method is free of pixel alignment and allows for dynamic and fast computation process. However, in this method, the reconstructed image quality is degraded by the interference of the zero-order noise brought by the phase-only SLM.

In order to improve the reconstructed image quality and further develop the DPH method to reconstruct 3-D object in the arbitrary space, we utilize a single phase-only SLM to achieve speckle-reduced holographic display based on off-axis double-phase complex modulation method. A phase-only CGH is calculated and encoded from the complex amplitude field without iteration. A blazing grating is employed to separate the zero-order information of the DPH from the zero-order noise of the phase-only SLM. Thus, the zero-order noise produced by pixel structure of the phase-only SLM is eliminated by the filtering architecture. This method also enables us to obtain high quality reconstructed 3-D object in arbitrary space with reduced speckle. The feasibility of this method is verified by both numerical simulations and optical experiments.

2. Method

The complex object treated here is assumed to be a two-dimensional complex image for simplicity. The two-dimensional amplitude $A (x, y)$ and phase $φ (x, y)$ of a complex field $U (x, y)$ expressed as $U (x, y)$ = $A (x, y) \exp (j φ (x, y))$ can be represented as the sum of two pure phase functions with constant amplitude as

P (u, v) = r e c t ((u - f \sin α) / ε, v / ε)

In the above expression,

B = A_{\max} / 2

is a constant and

A_{\max}

is the maximum value of

A (x, y)

. The decompositions of the given complex field,

θ_{1} (x, y)

and

θ_{2} (x, y)

can be calculated as follows:

θ_{1} (x, y) = φ (x, y) + \cos^{- 1} [A (x, y) / A_{\max}]

θ_{2} (x, y) = φ (x, y) - \cos^{- 1} [A (x, y) / A_{\max}]

We assume that $A_{\max} = 2$ , then $U (x, y)$ = $\exp (j θ_{1} (x, y)) + \exp (j θ_{2} (x, y))$ . Consequently, the desired complex field can be achieved by coherent superposition of the two pure phase elements, $θ_{1} (x, y)$ and $θ_{2} (x, y)$ .

According to the method proposed by Omel Mendoza-Yero, a phase encoding technique which utilizes complementary two-dimensional binary gratings (checkerboard patterns) and a low-pass filter is able to obtain superposition of two pure phase elements [25].

The pixelated checkerboard patterns, $M_{1} (x, y)$ and $M_{2} (x, y)$ are given as follows:

M_{1} (i Δ x, j Δ y) = {\begin{cases} 1 (i j a r e o d d o r e v e n) \\ 0 (e l s e) \end{cases}

M_{2} (i Δ x, j Δ y) = {\begin{cases} 1 (i j a r e o d d o r e v e n) \\ 0 (e l s e) \end{cases}

Here,

i, j

are the indexes of the pixel and

Δ x, Δ y

are the pixel intervals of the pattern. Apparently,

M_{1} (x, y)

and

M_{2} (x, y)

are complementary which can be expressed as

M_{1} (x, y) + M_{2} (x, y)

. Hence, the encoded CGH is presented as follows:

h (x, y) = θ_{1} (x, y) M_{1} (x, y) + θ_{2} (x, y) M_{2} (x, y)

If we use a filter $P (u, v)$ to block all diffraction orders but the zero one, the spectrum in the frequency plane in which the coordinates are $u, v$ is reduced to the expression

H (u, v) P (u, v) = \frac{1}{2} F F T {U (x, y)}

Here,

H (u, v)

is

F F T {h (x, y)}

.

F F T {•}

is denoted as fast Fourier transform(FFT). The mathematical derivation is given by Omel Mendoza-Yero [25]. The following function shows the expression of the filter

P (u, v)

:

P (u, v) = r e c t (u / ε, v / ε)

where

ε = f λ / (2 p)

. In reality,

p

is the pixel interval of the SLM and the value of

ε

is the distance from the propagation axis to the first diffraction order. In order to separate the zero-order information of the DPH from the zero-order noise of the phase-only SLM, a blazing grating is employed on the encoded CGH. Consequently, the final expression of the DPH is given by

h (x, y) = θ_{1} (x, y) M_{1} (x, y) + θ_{2} (x, y) M_{2} (x, y) + 2 π x \sin α / λ

where

λ

is the wavelength and

α

is the blaze angle. Accordingly, the filter function should be modified to be

P (u, v) = r e c t ((u - f \sin α) / ε, v / ε)

3. Results

3.1 The simulated assessment

A computer simulation is conducted to demonstrate the performance of complex modulation of the proposed method. The target complex field consists of “Lena” as amplitude image whose range is [0, 1] and “Parrots” as phase image whose range is [-π, π]. The calculation is performed under conditions when the pixel resolution of the target complex amplitude is 1024 × 1024; the wavelength is 532nm; the pixel resolution of the phase-only SLM modeled in the simulation is 1920 × 1080; the modulation range of the SLM is [0, 2π] and the number of gray levels of the SLM is 256. The blaze angle $α$ is π/4.The CGH is generated based on Eq. (9). After the calculation employing filter function, the numerical simulation result at the output plane of the 4-f system is acquired and shown in Fig. 1.

Fig. 1 Computer simulation results at the output plane of 4-f system. (a) Reconstructed amplitude image. (b) Reconstructed phase image. (c) CGH calculated by proposed method.

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As we can see in Fig. 1, both the amplitude and the phase of the optical field can be reconstructed successfully with high image quality. The numerical results demonstrate the validity of achieving simultaneous and independent modulation of amplitude and phase by encoded phase-only hologram. It should be noted that some noise can be seen in the reconstruction, which we believe may be caused by inaccurate size of filtering window in the simulation. The root-mean-square error (RMSE) is used as a metric to measure the difference of the amplitude and phase between the target complex amplitude distributions and the reconstructed complex amplitude field, which is calculated by the function $R M S E = \sqrt{\frac{1}{M N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {({X^{'}}_{i j} - X_{i j})}^{2}}$ . $X_{i j}$ is the normalized pixel’s value of either the amplitude or phase of the target complex object, while ${X^{'}}_{i j}$ is the normalized pixel’s value of either the amplitude or phase of the numerical reconstruction. In the above case, the RMSE of the reconstructed amplitude is 0.0955 and the RMSE of the reconstructed phase is 0.1304.

3.2 Experimental verification and assessment

An optical experiment is carried out to illustrate the ability to suppress speckle noise by complex modulation. The target complex object tested here is a 256-gray-level “Lena” as the amplitude image with uniform phase. The pixel resolution of the target complex object is 1024 × 1024. The optical setup in the experiment is shown in Fig. 2. We use a 532nm laser to illuminate the phase-only SLM (Holoeye Pluto, resolution:1920 × 1080, pixel interval: 8 $μ m$ , phase modulation rage: [0, 2π]). For the 4-f system, the focal length of Lens1 is 250mm and Lens2 is 300mm. Accordingly, the physical size of the iris is about 8 mm. For the on-axis double-phase method, iterative Fourier transformation algorithm (IFTA) and IFTA with time averaging, a polarizer is arranged just after the Laser before the beam splitter (BS) to ensure the polarization state is aligned with that of the SLM. We set the polarizer when the contrast of the results reaches its maximum. As for IFTA and IFTA with time averaging, a filter is used to block the zero-order beam at the frequency plane in the 4-f system. The results are recorded directly by the CMOS of the camera (Nikon D3100) which are presented in Fig. 3.

Fig. 2 Schematic of optical setup.

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Fig. 3 Optical experimental results. (a) Reconstructed amplitude image at the output plane of 4-f system by double-phase method employing the blazing grating. (b) Reconstructed amplitude image at the output plane of 4-f system by on-axis double-phase method. (c) Reconstructed amplitude image employing the blazing grating from phase-only CGH calculated by IFTA. (d) Reconstructed amplitude image employing the blazing grating by using time averaging method in which 20 phase-only CGHs calculated by IFTA are displaying in sequence at high frame rate. ((a), (b), and (c): Exposure time: 1/8 sec; ISO speed: 400; Exposure bias: + 0.3 step (d): Exposure time: 2 sec; ISO speed: 400; Exposure bias: + 0.3 step)

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As we can see in Fig. 2, the zero-order noise brought by pixel structure of SLM is eliminated by the filtering architecture automatically by virtue of adding a blazing grating. In this experiment the camera is arranged at the output plane of the 4-f system. The size and location of the iris at the frequency plane is decided by Eq. (10).

The reconstructed results are exhibited in Fig. 3. Figure 3(a) shows the reconstruction by double-phase method employing the blaze grating and Fig. 3(b) shows the reconstruction by double-phase method without employing the blaze grating. In Fig. 3(b), although a polarizer is used to improve the image contrast in the on-axis optical setup without employing the blaze grating, the image contrast is degraded because of the zero-order noise generated by the undiffracted regions of SLM. Compared Fig. 3(a) with Fig. 3(b), we can see that the contrast of the reconstructed image in Fig. 3(a) is elevated by eliminating the zero-order noise. It should be noted that some residual noises in Figs. 3(a) and 3(b) can be observed. Although the intrinsic curvature of the SLM is calibrated [26, 27] which impairs the residual noise, the size and the location of the filter, dust in the optical system, the error of calculation and imperfect calibration of SLM can still induce some residual noises. Reconstruction from phase-only CGH calculated by IFTA is shown in Fig. 3(c) [28]. The image quality is degraded due to serious speckle noise. This speckle noise can be suppressed as shown in Fig. 3(d) by time averaging method, in which 20 phase-only CGHs calculated by IFTA is displaying in sequence at high frame rate. However, due to the uncontrolled phase distribution at the image plane, some spurious interference occurs between the reconstructed pixels, which causes the observable image blur in Fig. 3(d). Compared Fig. 3(a) with Fig. 3(d), not only the speckle noise is effectively suppressed in Fig. 3(a), but also the detail of the reconstruction in Fig. 3(a) is sharper than in Fig. 3(d). Hence, the method is proved to be capable of generating high quality holographic reconstruction with reduced speckle noise by complex modulation.

In order to qualitatively evaluate the reconstructed results, multiple evaluation indexes are utilized in the following part. We firstly evaluate the quality of the reconstructed image by peak signal noise ratio (PSNR). The equation for the calculation of the PSNR for grey-level (8 bits) images is defined as

P S N R (f, g) = 10 \log_{10} (255^{2} / (\frac{1}{M N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {(f_{i j} - g_{i j})}^{2}))

M and N are the horizontal and vertical number of pixels for the original image and reconstructed image. f and g are the original image and the reconstructed image respectively [29]. The PSNR value approaches infinity as

f_{i j} - g_{i j}

approaches zero; this shows that a higher PSNR value provides a higher image quality. Another well-known objective evaluation index, structural similarity index measure (SSIM) is used to measure the similarity between the original image and the reconstructed image [29]. The SSIM is defined as:

S S I M (f, g) = l (f, g) c (f, g) s (f, g)

where

{\begin{cases} l (f, g) = \frac{2 μ_{f} μ_{g} + C_{1}}{μ_{f}^{2} + μ_{g}^{2} + C_{1}} \\ c (f, g) = \frac{2 σ_{f}^{} σ_{g}^{} + C_{2}}{σ_{f}^{2} + σ_{g}^{2} + C_{2}} \\ s (f, g) = \frac{σ_{f g} + C_{3}}{σ_{f}^{} σ_{g} + C_{3}} \end{cases}

The first term in Eq. (13) is the luminance comparison function which measures the closeness of the mean luminance ( $μ_{f}$ and $μ_{g}$ ) of the two image. This factor is maximal and equal to 1 only if $μ_{f}$ = $μ_{g}$ . The second term is the contrast comparison function which measures the closeness of the contrast of the two images. Here the contrast is measured by the standard deviation $σ_{f}^{}$ and $σ_{g}^{}$ . This term is maximal and equal to 1 only if $σ_{f}^{}$ = $σ_{g}^{}$ . The third term is the structure comparison function which measures the correlation coefficient between the two images f and g. Note that $σ_{f g}^{}$ is the covariance between f and g. The positive values of the SSIM index are in [0, 1]. A value of 0 means no correlation between images, and 1 means that f = g. The positive constants C₁, C₂ and C₃ are used to avoid a null denominator. The overall image quality MSSIM is obtained by calculating the mean of SSIM values over all windows as in Eq. (14):

M S S I M (f, g) = \frac{1}{M N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} S S I M_{i j}

The third evaluation index, speckle contrast, is introduced to quantitatively assess the degradation of image quality due to speckle noise [30]. The value of speckle contrast, C is given by:

C (g) = \frac{σ_{g}}{μ_{g}}

where

σ_{g}

is the standard deviation of reconstructed image and

μ_{g}

is the mean value of the reconstructed image. Equation (15) indicates that speckle contrast is the standard deviation of the pixel intensity divided by the mean value of the intensity. Smaller C indicates better speckle reduction and higher image quality. The assessment of reconstruction shown in Fig. 3 by the above objective evaluation indexes is shown Table 1.

Table 1. Image quality measurement of results in Fig. 3 by different evaluation indexes.

View Table

According to data in Table 1, the performance of the proposed method is better than the on-axis double-phase method, IFTA and IFTA with time averaging in terms of PSNR, MSSIM and speckle contrast. To be specific, the PSNRs of the proposed method and IFTA with time averaging are higher than IFTA and on-axis double-phase method. Due to degradation caused by zero-order noise, the PSNR of on-axis double-phase method is lower than the other three methods. In comparison of MSSIM, the proposed method still performs best and the MSSIM of on-axis double-phase method is higher than IFTA with time averaging and IFTA. The results of speckle contrast quantitatively verify the effectiveness of speckle reducing by complex modulation. In Table 1, we can see that the speckle contrast of proposed method is lower than the other three methods. The speckle contrast of IFTA with time averaging is higher than that of on-axis double-phase method which reveals that the time averaging method can suppress speckle noise but is not as effective as complex modulation method. It is clear that the speckle contrast of IFTA without time averaging is much higher than the other three methods which can be observed in Fig. 3 as well. Note that, the area we used to calculate PSNR and MSSIM is selected with a red dashed box and the area we used to calculate speckle contrast is selected with a yellow dashed box shown in Fig. 3. If we don’t take the diffraction efficiency of SLM into consideration, the diffraction efficiency of IFTA and IFTA with time averaging by numerical calculation is 100%. Since the intensity of the usable image is divided by the integrated intensity departing from the SLM in both proposed method and on-axis double-phase method, the efficiency of energy utilization should be considered. The efficiency of energy is calculated numerically by the function

η_{e} = [(\sum_{i = 1}^{M} \sum_{j = 1}^{N} I_{i j}^{2}) / (\sum_{i = 1}^{M} \sum_{j = 1}^{N} O_{i j}^{2})] \times 100 %

where M and N are the horizontal and vertical number of pixels for the CGH and the reconstructed image.

I

is the amplitude value of CGH and

O

is the amplitude value of the reconstruction. The diffraction efficiency of the proposed method and the on-axis double-phase method is identical, about 30%.

3.3 3-D holographic display by complex amplitude modulation

In order to evaluate the performance to reconstruct 3-D object, an optical experiment for multi-plane holographic display is conducted with unchanged conditions. The complex object tr and $d_{4}$ from the output plane of the 4-f system. The schematic of optical setup for multi-plane holographic display is shown in Fig. 4. As shown in Fig. 4, the complex hologram is reconstructed by DPH at the output plane of the 4-f system. After the output plane of the 4-f system, target characters are reconstructed at four different planes from the complex CGH. The results are recorded by the CMOS of the camera respectively. The simulation and optical multi-plane resultant reconstructions at various distances are shown in Figs. 5(a)-5(h). We also record this reconstruction by moving the camera back and forth between the first focused plane and the fourth focused plane. The whole process is vividly shown in Visualization 1.

Fig. 4 Schematic of optical setup for multi-plane holographic display.

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Fig. 5 Simulation results and Optical results of multi-plane holographic display. (Exposure time: 1/8 sec; ISO speed: 800; Exposure bias: + 0.3 step) (a) and (e) Simulation and optical reconstruction at 0.1m after the output plane of 4-f system. (b) and (f) Simulation and optical reconstruction at 0.15m after the output plane of 4-f system. (c) and (g) Simulation and optical reconstruction at 0.2m after the output plane of 4-f system. (d) and (h) Simulation and optical reconstruction at 0.25m after the output plane of 4-f system (Visualization 1).

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The optical results are consistent well with the corresponding simulation results. As the optical results shown, in each figure of Figs. 5(e)-5(h), when one character is clear, the other three are blur. By using complex modulation method, the phase distribution is controlled, so that the characters can be successfully reconstructed at different planes with reduced speckle noise. It is clear that 3-D object can be reconstructed by complex modulation based on our method with high image quality.

Another optical experiment is carried out to realize the 3-D object reconstruction based on CGH calculated by a nonuniform sampled wavefront recording plane method [31]. Apart from the distance between the camera and the output plane of 4-f system, the optical setup and conditions are identical with the above optical experiment for multi-plane holographic display. The results of further verification are shown in Fig. 6, where Figs. 6(a) and 6(b) are the intensity map of model car and depth map of model car; Figs. 6(c) and 6(d) are the simulation results at two selected focused planes using the nonuniform sampled wavefront recording plane method; Figs. 6(e) and 6(f) are the simulation results at two selected focused planes after the proposed method; Figs. 6(g) and 6(h) are the optical reconstruction at two selected focused planes of the proposed method.

Fig. 6 Simulation results and Optical results of 3-D holographic display. (a) Intensity map of model car. (b) Depth map of model car. (c) Simulation results at 0.4m after the output plane of 4-f system using the above 3-D object reconstructing method. (d) Simulation results at 0.42m after the output plane of 4-f system using the above 3-D object reconstructing method. (e) Simulation results at 0.4m after the output plane of 4-f system by double-phase complex encoding. (f) Simulation results at 0.42m after the output plane of 4-f system by double-phase complex encoding. (g) Optical reconstruction at 0.4m after the output plane of 4-f system by the proposed method. (h) Optical reconstruction at 0.42m after the output plane of 4-f system by the proposed method. (Exposure time: 1/8 sec; ISO speed: 800; Exposure bias: + 0.3 step)

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Compared the simulation results of Figs. 6(e) and 6(f) with Figs. 6(c) and 6(d), error due to encoding and inaccurate size of filtering window can give rise to mild speckle noise on the simulation results. The RMSEs of Figs. 6(e) and 6(f) are 0.0826 and 0.0819 respectively. The optical results are consistent well with the simulation results. It can be observed that when the focused plane is at 0.4m, the front part of the car in Figs. 6(c), 6(e) and 6(g) (closer to the hologram) are clearer than the rest. The clear content moves to the back in Figs. 6(d), 6(f) and 6(h), when the focused distance is changed to 0.42m. The above observations will become even more apparent when the details are amplified. The optical experiments for reconstructing multi-plane 3-D object and 3-D object with continuous depth map verify that the proposed method is effective to reconstruct 3-D object with reduced speckle noise.

4. Conclusion

In this paper, we utilize a single phase-only SLM to achieve speckle-reduced 3-D holographic display based on off-axis double-phase complex modulation method without iteration. The target image can be reconstructed successfully with reduced noise at arbitrary distance after the 4-f system. We expect that this method can be employed in various optical systems, in which both amplitude and phase information need to be controlled, such as performing special tasks in optical information processing or to synthesize light beams. Given that the encoding function allows for a single-pixel mathematical operation, instead of using an iterative algorithm or an array of subpixels to codify each pixel of the input plane, this method is assumed to be more suitable to be applied to 3-D dynamic display for improving the image quality.

Funding

Program 973 (2013CB328803); Program 863 (2015AA016301); National Natural Science Foundation of China (NSFC) (61605080).

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	*PSNR*	*MSSIM*	C
Proposed method	17.40	0.4874	0.0827
On-axis double-phase method	13.21	0.3556	0.0975
IFTA	15.64	0.1449	0.2667
IFTA with time averaging	17.35	0.2909	0.1014

Speckleless holographic display by complex modulation based on double-phase method

Abstract

1. Introduction

2. Method

3. Results

3.1 The simulated assessment

3.2 Experimental verification and assessment

3.3 3-D holographic display by complex amplitude modulation

4. Conclusion

Funding

References and links

Supplementary Material (1)

Cited By

Figures (6)

Tables (1)

Equations (16)

Optics Express