Formulas of partially spatial coherent light are derived and its corresponding design algorithm is proposed for generating computer-generated holograms based on partially spatial coherent light. The partially coherent light is divided into two terms: spatially absolute coherent part and incoherent part. The former is propagated by angular spectrum method, while the latter is based on the optical transfer function. The related formulas are derived where the coherent function (degree of coherence) is related. A modified iterative algorithm is further developed for optimizing the phase distributions. Numerical simulations and optical experiments are both performed to verify the proposed algorithm. The results obtained by the proposed method and the traditional method are compared, and it is clear that the speckle contrasts in optical experiments are improved more than 46%, and the image quality is obviously improved. This method could also provide new applications for three-dimensional holographic display, beam shaping, and other wave-front modulation techniques.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Computer generated hologram (CGH) refers to hologram generated by computer coding techniques . 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 . Y. Wu applied this method to hologram reconstruction in the experimental, and the speckle noise reduction phenomenon is found . And many works have been proposed so far, e.g. random-phase free computer-generated amplitude hologram , iterative algorithm with object-dependent quadratic phase distribution , single-shot with holograms with spatial jitter , pixel separation holographic projection [23,24], phase-only CGH based on the reconstruction of complex amplitude image . 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 , 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.
The process of image recording and reconstruction under partially spatial coherent light illumination is shown in Figs. 1(a) and 1(b). The plane, and are the object plane, the hologram plane and the image plane, respectively. and , which are conjugate to each other, are partially spatial coherent light. During the recording process, is regarded as reference light illumination to the hologram. When it comes to reconstruction, illuminates the hologram. According to the holographic principle, the image is reconstructed on the plane. 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.
According to the partially coherent light theory , the intensity of a certain point on the interference plane is:
In this way, the SPCL can be divided into two parts. The incoherent part of SPCL can be expressed as: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:
The coherent part of the image is propagated by angular spectrum method (ASM). In this theory, the complex amplitude of input wave is Fourier transformed, the expression of space frequency spectrum is:
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 is transmitted by OTF to obtain incoherent intensity at the distance d:
While in coherent part, is transformed by the complex amplitude mode:
In reconstruction, A is transmitted back by OTF−1 and ASM separately. Finally, the reconstructed image is composed by incoherent and coherent intensity and , which is obtained from OTF−1 and ASM respectively:
SC  and PSNR are used to evaluate the image quality. The expressions of the two parameters are shown as:
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.
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.
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.
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.
On the basis of Van Cittert-Zernike theorem , 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 lt 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 lt 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.
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.
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.
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.
National Natural Science Founding of China (NSFC) (61575024, 61420106014) and UK-CIAPP\369.
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