Formulas of partially spatial coherent light and design algorithm for computer-generated holograms

Junyi Duan; Juan Liu; Bining Hao; Tao Zhao; Qiankun Gao; Xinhui Duan

doi:10.1364/OE.26.022284

1. Introduction

Computer generated hologram (CGH) refers to hologram generated by computer coding techniques [1]. CGH can be used in various optical systems, especially in three-dimensional (3D) dynamic holographic display. It can be loaded on the refreshable holographic display optoelectronic devices, and the image will be reconstructed dynamically. There are mainly two methods to generate CGH: one is iterative optimization method mainly used for two-dimensional display, such as GS algorithm [2, 3]; the other is a non-iterative method that is mainly used for 3D display, the diffraction propagation can be solved by point source based algorithm or polygon-based algorithm [4–17]. The iterative optimization is widely used to achieve more accurate results. However, during the process of coding, a random phase is usually added to enlarge the dynamic range of the Fourier spectrum and reduce the coding quantization error, which causes strong speckle noise in the reconstructed image. Furthermore, coherent light sources are commonly used for holography, such as lasers. This kind of perfectly coherent light source is widely used in the process of image reconstruction. When laser is diffracted in the 3D image space for display, each point in the diffraction space can be considered as a sub wave source. These sub waves are coherent with each other but with different directions of propagation and random phases. They will interfere with each other as they propagate in space and produce random bright and dark patterns, i.e., speckle noises. These noises will decay the image quality of display. Therefore, one possibility is presented to reconstruct the image under partially coherent beam illumination. G. Li proposed a general theory of coherence of the laser beam through a moving diffuser and a corresponding method of calculating speckle contrast [18]. Y. Wu applied this method to hologram reconstruction in the experimental, and the speckle noise reduction phenomenon is found [19]. And many works have been proposed so far, e.g. random-phase free computer-generated amplitude hologram [20], iterative algorithm with object-dependent quadratic phase distribution [21], single-shot with holograms with spatial jitter [22], pixel separation holographic projection [23,24], phase-only CGH based on the reconstruction of complex amplitude image [25]. All studies above are designed by the coherent light diffraction theory. However, in realistic condition, even laser cannot be perfectly coherent. So in the CGH calculation, the consideration of the coherence of the light is necessary, which is still lacking.

In addition, since the theory of partially coherent light has been proposed [26], many scholars have studied the partially coherent light, such as its Shannon entropy, its application in photovoltaic devices and its application in 3D computational fluid dynamic models [27–29]. However, in the current research, partially spatial coherent light (SPCL) has not been applied to the generation of CGH.

In this paper, combination of SPCL and CGH theory is done. Formulas of SPCL is proposed, along with a related design algorithm for CGH based on SPCL illumination. Both numerical simulation and optical experiments are performed to verify its feasibility. The evaluated parameters: the Speckle Contrast Ratio (SC) and the Peak Signal to Noise Ratio (PSNR) of the reconstructed images, are compared with the results of method base on coherent light. Speckle noises are obviously restrained without sacrificing PSNR. The simple model of SPCL propagation and the process of CGH calculation are described in section 2. The corresponding simulations and experiments are provided in section 3 and 4 respectively. Finally, section 5 gives out discussions.

2. Principle

The process of image recording and reconstruction under partially spatial coherent light illumination is shown in Figs. 1(a) and 1(b). The plane $O$ , $H$ and $O'$ are the object plane, the hologram plane and the image plane, respectively. $R$ and $R'$ , which are conjugate to each other, are partially spatial coherent light. During the recording process, $R$ is regarded as reference light illumination to the hologram. When it comes to reconstruction, $R'$ illuminates the hologram. According to the holographic principle, the image is reconstructed on the plane $O'$ . The process of this method is similar to the traditional one, except that the reference light is replaced by partially spatial coherent light. Therefore, we need to analyze the corresponding theory.

Fig. 1 The schematic of hologram recording and reconstruction under partially spatial coherent light illumination. (a) the recording process; (b) the reconstruction process.

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According to the partially coherent light theory [30], the intensity of a certain point on the interference plane is:

I = I^{(1)} (Q) + I^{(2)} (Q) + 2 \sqrt{I^{(1)} (Q)} \sqrt{I^{(2)} (Q)} | γ_{12} | \cos (φ_{1} - φ_{2}),

where

I^{(1)} (Q)

and

I^{(2)} (Q)

are the intensity at point Q on the observed plane illuminated by two partially coherent point sources S₁ and S₂. Then, the quasi-monochromatic equation can be expressed as:

E = \frac{1}{2} \sqrt{1 - | γ_{12} (τ) |} \sqrt{I^{(1)} (Q) + I^{(2)} (Q)} e^{i φ_{i n}} + \sqrt{| γ_{12} (τ) |} \sqrt{I^{(1)} (Q) I^{(2)} (Q)} e^{i φ_{c o}},

where

γ_{12}

is the complex degree of coherence of two sources, and

0 \leq | γ_{12} | \leq 1

. The two sources are completely incoherent when

| γ_{12} | = 0

and completely coherent when

| γ_{12} | =1

.

φ_{c o}

is the initial phase and

φ_{i n}

is the uncertainty phase of the source.

In this way, the SPCL can be divided into two parts. The incoherent part of SPCL can be expressed as:

E_{i} = \frac{1}{2} \sqrt{1 - | γ_{12} (τ) |} \sqrt{I^{(1)} (Q) + I^{(2)} (Q)} e^{i φ_{i n}},

while the coherent part is:

E_{c} = \sqrt{| γ_{12} (τ) |} \sqrt{I^{(1)} (Q) I^{(2)} (Q)} e^{i φ_{c o}},

From Eq. (2), we can see that for the SPCL, the two parts are linearly superposed.

As we all known, in incoherent light systems, the imaging of objects can only be expressed by intensity and transmit by optical transfer function (OTF). So for diffraction limited systems, the OTF in the frequency space of a square pupil with a length of D can express as:

S (α, β) = D^{2} t r i (\frac{λ d α}{D}) (\frac{λ d β}{D}), | \frac{λ d α}{D} | \leq 1, | \frac{λ d β}{D} | \leq 1,

where d is the distance from the light source to the pupil and

(α, β)

is the coordinate of the frequency domain. So for an input intensity

I_{0}

, the output is:

I_{i} (ξ, η) = \iint_{\infty} {\tilde{I}}_{i} (α, β) \cdot S (α, β) \exp [i 2 π (ξ α + η β)] d α d β,

(ξ, η)

is the coordinate of the space domain.

{\tilde{I}}_{i} (α, β)

is the input intensity that components with

(α, β)

.

The coherent part of the image is propagated by angular spectrum method (ASM). In this theory, the complex amplitude of input wave $A_{0} (ξ, η)$ is Fourier transformed, the expression of space frequency spectrum $a_{0} (α, β)$ is:

a_{0} (α, β) = \iint_{\infty} A_{0} (ξ, η) \exp [- i 2 π (ξ α + η β)] d α d β .

a_{0} (α, β)

is the complex amplitude of input wave that components with

(α, β)

. So the output complex amplitude

A_{c} (ξ, η)

can be expressed as:

A_{c} (ξ, η) = \iint a_{0} (α, β) \exp [i k d \sqrt{1 - λ^{2} (α^{2} + β^{2})}] \exp [i 2 π (ξ α + η β)] d α d β .

In the generation of CGH with SPCL illumination, temporal coherence is not discussed, so the time factor τ is not considered. Based on the theory above, the process of generating CGH with SPCL along with its reconstruction is shown in Fig. 2. In incoherent part, according to Eq. (6), the input image $I_{0}$ is transmitted by OTF to obtain incoherent intensity $I_{i}$ at the distance d:

Fig. 2 Flow chart of CGH with partially spatial coherent light.

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I_{0} (ξ, η) \overset{O T F}{\to} I_{i} (ξ, η) .

While in coherent part, $I_{0}$ is transformed by the complex amplitude mode:

A_{0} (ξ, η) = \sqrt{I_{0} (ξ, η)} \cdot \exp (i φ_{r}),

where

φ_{r}

is the initial random phase added on the image. Then

A_{0} (ξ, η)

is transmitted by ASM, so the complex amplitude

A_{c}

at the distance d can be obtained:

A_{0} (ξ, η) \overset{A S M}{\to} A_{c} (ξ, η) .

So the corresponding intensity and phase can be expresses as:

I_{c} (ξ, η) = {| A_{c} (ξ, η) |}^{2},

φ_{c} (ξ, η) = a n g l e [A_{c} (ξ, η)] .

According to the theory mentioned above and the principle of phase hologram generation, the intensity of incoherent and coherent light propagation is linearly superimposed in the phase of the hologram A along with

φ_{c}

:

A (ξ, η) = \exp {i [(1 - γ) I_{i} (ξ, η) + γ I_{c} (ξ, η)]} \cdot \exp [i φ_{c} (ξ, η)],

so the influence of both parts can be considered.

In reconstruction, A is transmitted back by OTF⁻¹ and ASM separately. Finally, the reconstructed image is composed by incoherent and coherent intensity $I_{i}^{'}$ and ${| E_{c} |}^{2}$ , which is obtained from OTF⁻¹ and ASM respectively:

{| A (ξ, η) |}^{2} \overset{O T F^{- 1}}{\to} I_{i}^{'} (ξ, η),

A (ξ, η) \overset{A S M}{\to} E_{c} (ξ, η) .

So the final reconstructed image can be expressed as:

I = I_{i}^{'} + {| E_{c} |}^{2},

and can be used as the input image for the further iteration algorithm.

SC [31] and PSNR are used to evaluate the image quality. The expressions of the two parameters are shown as:

S C = \frac{\sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(p_{i} - \bar{I})}^{2}}}{\bar{I}},

P S N R = 10 \lg (\frac{{(2^{n} - 1)}^{2}}{M S E}),

where N is the number of pixels, p_i is the intensity of each pixel in the image,

\bar{I}

is the average intensity of all pixels, and MSE is the mean square error between the original image and the processed image.

3. Numerical simulation

3.1 The result without iteration

In order to verify the correctness of the above theory, we carry out the computational simulation. Binary and gray images with 512 × 512 are used in numerical simulation. The wavelength is 532 nm, and the propagation distance d is set to 50mm. Theoretically speaking, the introduction of incoherent light in the calculation will introduce the corresponding background noise, so the quality of the image will be severely affected when degree of coherence γ is too low. This is verified in the actual simulation. In this case, the purpose of improving the image quality is not achieved. Therefore, we try to find the appropriate γ value among 0.9~1. The numerical results of the simulation are shown in Fig. 3 with non-iteration.

Fig. 3 The numerical simulation results. (a) and (b) are reconstructed images when γ = 1; (c) and (d) are reconstructed images when γ = 0.98; (e) and (f) are PSNR and SC of binary and gray images as γ increases, respectively.

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The reconstruction results of the binary image and the gray image when γ = 1 and 0.98 are shown in Figs. 3(a)-3(d), and the details are enlarged. As can be seen from the details, speckle noises changes and the edge is slightly blurred which is more obvious for gray image.

The evaluation results of the image quality are shown in Figs. 3(e) and 3(f). Since the initial phase of the images is random, the calculation results will fluctuate within a certain range, taking the average of five calculation results as the final result. As shown in Fig. 3, The PSNR of the binary and grayscale images fluctuates in the range of 25.49-25.65 and 33.7-34.8, and the SC fluctuates between 0.221 and 0.229 and 0.367-0.380, respectively. From Figs. 3(e) and 3(f), we note that when γ = 0.98, the SC reach a small value for both binary and gray image as the PSNR of both image keeps relatively high. So we pick 0.98 as the degree of coherence that can optimize the image on speckle reduction. This is the basis for further research.

3.2 The results after iterations

In order to get better results, iterative algorithm is introduced and the results are shown in Fig. 4. Apparently the properties of the reconstructed images is better as the iteration n increases. Since one could tell from Figs. 4(e) and 4(f) that with the increase of the iterations, the quality of the reconstructed binary and gray images increased obviously. The value of SC goes down gently when the iteration reaches to 10. Compared with the results by non-iteration process, SC decreases by 44.0% for both binary and gray image. Also PSNR becomes better in the same condition, but a little bit fluctuation still happens. It becomes to 26.3587 and 35.0800 when iteration reaches 10 for binary and gray images respectively.

Fig. 4 The numerical simulation results with iteration algorithm when γ = 0.98. (a) and (b) are reconstructed images when iteration n = 1; (c) and (d) are reconstructed images when iteration n = 10; (e) and (f) are PSNR and SC of binary and gray images as iterations increase, respectively.

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Also, comparison between proposed method and ASM, which is the basis of the method, is done. From Figs. 5(a) and 5(c) (or same as Figs. 5(b) and 5(d)), it can be concluded that the results generated from proposed method is smoother. And we can see from Table 1 that SC decreases by 45.1% and 39.6% for binary and gray images respectively. And also slight improvement is observed in PSNR evaluation.

Fig. 5 The numerical simulation results. (a) and (b) are the reconstructed images using ASM; (c) and (d) are the reconstructed images using proposed method.

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Table 1. SC and PSNR of the simulated reconstructed images

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4. Optical experiments and discussions

We perform the corresponding experiments to verify the proposed CGH generation method. The setup is shown in Fig. 6. A 532nm diode laser is used as the light source. The beam is filtered and collimated by a spatial filter. Then it is irradiated through an aperture whose diameter is variable, along with a rotating ground glass before the beam is expanded, after which partially spatial coherent light can be generated. After that, a phase only spatial light modulator (SLM) (HOLOEYE PLUTO) is illuminated, as well as a 4-f system and a high-pass filter are adopted to remove the zero-order noise that is reflected by the SLM. The reconstructed image will be recorded by a CCD.

Fig. 6 Schematic view of the optical experimental setup.

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On the basis of Van Cittert-Zernike theorem [24], the degree of coherence γ is defined by the time average of the vibrations of two points, and can be measured by the intensity of interference with monochromatic light form an extended source. We can control the transverse coherence width l_t of the source illuminating on the SLM by adjusting the diameter of the SPCL source, and make sure the part which is greater than a given γ in the distribution of degree of coherence can cover the hologram. The smaller the aperture is, the greater l_t is, and the γ on the edge of cover area on hologram can be increased. And it is a coherent light source when there is no aperture and vibrating ground glass.

According to Eq. (12), when γ = 1, the incoherent part in the phase decreases to zero. That means only coherent part, which is obtained by ASM, is left in the hologram. The ASM (γ = 1) result which is based on coherent beam theory are shown in Fig. 7(b). The results using proposed method are shown in Figs. 7(c) and 7(d) with γ = 0.98, γ = 0.92 respectively. It can be seen from Fig. 7 that the speckle noise of the reconstructed image obtained by proposed method is more suppressed. And same experiment results with gray image is shown in Fig. 8.

Fig. 7 Reconstructed results with coherent light. (a) is the original binary image; (b) is result by ASM (γ = 1); (c) and (d) are results by proposed method with γ = 0.98, γ = 0.92 respectively.

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Fig. 8 Reconstructed results with coherent light. (a) the original gray image; (b) image by ASM (γ = 1); (c), (d) are results by proposed method with γ = 0.98, γ = 0.92 respectively.

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Also SC and PSNR evaluation of both experiment result is given in Tables 2 and 3. We can see from Table 2 that for binary image, the SC is reduced by about 56.7% and 59.0% respectively. But as γ decreases, the reconstructed image gets more blur and the PSNR gets lower. The main reason is that the low coherence of the source, which means larger diameter of the PSCL source, can bring in random dynamic wave front to the reconstructed image. So the speckle noises in the images can be averaged in some extent. But when the coherence is more decreased, there will be more disorder in the wave front of reconstructing light. So incoherence part is brought in that cannot be modulated. And same conclusion can be obtain from Table 3 in which SC is reduced by about 46.8% and 57.4%. But PSNR goes down a little bit with γ increases. Considering both SC and PSNR, we might take reconstruction when γ = 0.98 for relatively superior result.

Table 2. SC and PSNR of reconstructed binary image

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Table 3. SC and PSNR of reconstructed gray image

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Additionally, considering the possibility that speckle reduction effect originates from just a low coherence of a light source. To clarify this point, experimental results for conventional CGH and proposed method CGH with partially coherent illumination of same degree of coherent (Fig. 9) is also provided. And the SC and PSNR evaluation is given in Table. 4. It is obvious (from Figs. 8(b) and 9(a)) that low coherence of illumination indeed has contribution on speckle suppression to some extent. But in the results (Fig. 9 and Table 4), it can be seen that the details in the red square that the SNR in the reconstructed picture is indeed increased with proposed method. Conventional CGHs are designed with the assumption that the reference light and the object light is totally coherent, which means coherent length and coherent area are infinite. But in real experimental hologram reconstruction process, even laser cannot reach this ideal condition. The proposed method is aiming at designing CGHs under a more realistic illumination. So it works better for partially coherent illumination reconstruction in experiment.

Fig. 9 Reconstructed results for (a) conventional (ASM) CGH and (b) proposed method CGH with both partially coherent illumination of same degree of coherent (γ = 0.98).

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Table 4. SC and PSNR of reconstructed images for conventional and proposed CGH with same partially coherent illumination.

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5. Conclusion

In brief, formulas of SPCL and a corresponding design method of CGH are proposed. This method regards SPCL as a linear superposition of coherent part and incoherent part. Both the numerical simulation and optical experiments verify the feasibility of this method. Compared with traditional method based on coherent light, the SC of image with SPCL decreased more than 46% without sacrificing PSNR. In this experiment, this method is currently only used as a preliminary calculation and discussion of flat CGH. Other than necessary laser requirements in traditional methods, it provides a potential way for new holographic display using partially special illumination, e.g. LED, CCFL etc. which can increase the portability and decrease the cost. Theoretically, the proposed method could provide a useful information and approaches for reducing the speckle noise in future 3D holographic display and other laser applications. But the perceived depth of the scene may be affected by the incoherence part of the light, this issue of perceived depth will be studied further. Moreover, this method also could be applied in beam shaping, microscopy, laser projection and other wave-front modulation techniques.

Funding

National Natural Science Founding of China (NSFC) (61575024, 61420106014) and UK-CIAPP\369.

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Method	SC	PSNR
ASM (γ = 1)	0.2918	31.5423
γ = 0.98	0.1264	31.5435
γ = 0.92	0.1194	31.4364

Method	SC	PSNR
ASM (γ = 1)	0.3274	25.5510
γ = 0.98	0.1743	25.1558
γ = 0.92	0.1396	24.5090

Method	SC	PSNR
ASM	0.2210	23.3548
proposed method	0.1912	25.2631

Method	SC	PSNR
ASM (γ = 1)	0.2918	31.5423
γ = 0.98	0.1264	31.5435
γ = 0.92	0.1194	31.4364

Method	SC	PSNR
ASM (γ = 1)	0.3274	25.5510
γ = 0.98	0.1743	25.1558
γ = 0.92	0.1396	24.5090

Formulas of partially spatial coherent light and design algorithm for computer-generated holograms

Abstract

1. Introduction

2. Principle

3. Numerical simulation

3.1 The result without iteration

3.2 The results after iterations

4. Optical experiments and discussions

5. Conclusion

Funding

References and links

Cited By

Figures (9)

Tables (4)

Equations (19)

Optics Express

Image	Method	SC	PSNR
Binary	ASM	0.2283	26.0021
Binary	proposed method	0.1228	26.3587
Gray	ASM	0.4323	34.8532
Gray	proposed method	0.2612	35.0800