2. Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei 230026, China
Laser display has attracted lots of attentions for its advantages of longer lifetime, higher luminance, lower power consumption and wider color gamut^{1}. However, when an optical rough surface is illuminated by a coherent laser, speckle will appear^{2}. Speckle is a drawback of laser display for it can downgrade the image quality greatly. To solve this problem, numerous methods have been proposed, such as: using a deformable mirror^{3} or a scanning mirror^{4, 5} to realize angle diversity; using a broadarea verticalcavity surfaceemitting laser^{6} or random lasers^{7} to realize wavelength diversity; using a staircase element^{8} or a refractive steppers^{9} to realize optical path difference; rotating independent speckle patterns^{10} or a magnetooptical disk^{11} to realize polarization diversity; using a light pipe^{12} or a multimode optical fiber bundle^{13} to realize time averaging and so on.
J. I. Trisnadi firstly proposed the speckle reduction method using Hadamard phase diffuser^{14, 15}. Later, more researches about this topic^{16, 17} and its extensions^{18, 19} were presented. However, most of these methods need either a vibration system or a line scan system, which is cumbersome for laser display system. For example, in Ref.^{14}, submatrices are produced by each row of Hadamard matrix to form phase patterns through vibrating the Hadamard phase diffuser in cooperation with grating light valve (GLV). Therefore, how to reduce the speckle effect in a static way is an open research topic.
Lens array (LA) is the most commonly used device in 3D display, which can integrate elemental images into 3D images^{2022}. Inspired by this, we hope that the LA can also be integrated with the phase plate array (PPA) for speckle reduction. Thus, we propose a motionless method for speckle reduction using LA and PPA. A 4f system includes PPA, LA and main lens will be described in Section 3. A special PPA will be designed, and the relationships among each component will be discussed. The results are explained in our experiments in Section 4. Speckle reduction can be realized in a static way without any moving mechanism.
TheoryHadamard matrices H(n) are matrices of 1's and 1's whose columns are orthogonal, which satisfy:
$ \mathit{\boldsymbol{H}}\mathit{'} \times \mathit{\boldsymbol{H = }}n \times \mathit{\boldsymbol{I}}\left( n \right), $  (1) 
where, H' is the transpose of H, n is the order of Hadamard matrix, and I(n) is identity matrix. For example, H(4) has the following expression:
$ \mathit{\boldsymbol{H}}\left( 4 \right) = \left[ {\begin{array}{*{20}{c}} 1&1&1&1\\ 1&{  1}&1&{  1}\\ 1&1&{  1}&{  1}\\ 1&{  1}&{  1}&1 \end{array}} \right] $  (2) 
According to Trisnadi's theory^{14}, a diffuser based on Hadamard binary phase matrix can be used for speckle reduction. Supposing a detector resolution spot is divided into M=N_{1}×N_{2} cells, as shown in Fig. 1(a).
If the amplitude of each cell is E_{ij}, and each cell is assigned with a phase φ_{ij}, where i=1, 2, …, N_{1} and j=1, 2, …, N_{2}. Then, these cells will constitute a phase pattern, and the intensity I of the resolution spot will be:
$ I = {\left {\sum\limits_{i = 1}^{{N_1}} {\sum\limits_{j = 1}^{{N_2}} {{E_{ij}}\exp \left( {{\rm{i}}{\varphi _{ij}}} \right)} } } \right^2}. $  (3) 
When N different phase patterns are presented in one resolution spot during the integration time, which have the equal duration, the intensity I can be expressed as:
$ I = \frac{1}{N}\sum\limits_{n = 1}^N {{{\left {\sum\limits_{i = 1}^{{N_1}} {\sum\limits_{j = 1}^{{N_2}} {h_{ij}^n{E_{ij}}} } } \right}^2}, } $  (4) 
where,
If the summation of
$ \sum\limits_{n = 1}^N {{{\left( {h_{ij}^n} \right)}^ * }} h_{kl}^n = N{\delta _{ik}}{\delta _{jl}}. $  (5) 
Then, we will get the following equation:
$ I = \frac{1}{N}\sum\limits_{n = 1}^N {\sum\limits_{i = 1}^{{N_1}} {\sum\limits_{j = 1}^{{N_2}} {\sum\limits_{k = 1}^{{N_1}} {\sum\limits_{l = 1}^{{N_2}} {{{\left( {h_{ij}^n} \right)}^ * }} } } } } {\left( {{E_{ij}}} \right)^ * }h_{kl}^n{E_{kl}} = \sum\limits_{i = 1}^{{N_1}} {\sum\limits_{j = 1}^{{N_2}} {{{\left {{E_{ij}}} \right}^2}} } . $  (6) 
Comparing equation (3) and equation (6), we could find that the cross terms vanished due to the averaging. Most importantly, 'N different phase patterns' can be formed by the rows or columns of the Hadamard matrix. They are also named submatrices of Hadamard matrix (SH). For example, in Fig. 1(b), SH_{2} is one submatrix of Hadamard matrix, which is formed by the second row of H(4). If the elements of '1' and '1' in Hadamard matrix are assigned with phase '0' and 'π' respectively, each SH will represent a phase pattern. Moreover, the amplitudes of each cell E_{ij} usually have the same value.
At the same time, according to Goodman's theory^{23}, supposing we have N incoherent speckle configurations that have equal mean intensities added on an intensity basis, then speckle contrast will decrease to N^{1/2} of the previous level. Therefore, for an norder Hadamard matrix H(n), n incoherent SH can be generated, which correspond to n incoherent speckle configurations. Averaging SH in one resolution spot during the integration time of detector, the speckle contrast will reduce by the factor of n^{1/2}.
System design Optical systemA sketch diagram of the optical system for our method is shown in Fig. 2. 632 nm light is generated by a HeNe laser. A 10× beam expander (BE) (Thorlabs, BE0510A) is used to expand the incident light. A phase only spatial light modulator (SLM) is used to generate the sequentially changing PPA, and the norder PPA contains n×n subphase plates. When using the 2order PPA, the PPA, LA, main lens and screen constitute a 4f system. When using the 4order PPA, another beam expander (BE 2) is added, and the whole system contains two seamlessly connected 4f systems. More details will be described in the following paragraphs. A rough paper is used as screen and is placed on the focal plane of main lens. Speckle patterns on the screen will be captured by CCD camera.
Speckle reduction will be achieved by the averaging of the incoherent speckle patterns with the following process: paralleled laser light comes across the PPA, beams in different position will carry sequentially changing phase patterns SH; under the action of LA and main lens, incoherent speckle patterns will superpose on the screen and then be averaged.
Phase plate arraySpecially designed PPA is used for speckle reduction. One example is shown in Fig. 3(a). The PPA contains four subphase plates. These subphase plates are constituted by repeating the four phase patternsSH_{1}, SH_{2}, SH_{3} and SH_{4}, which are formed by the first, second, third and fourth row of a H(4) phase matrix respectively, as shown in Fig. 3(b).
The phase patterns in each subphase plate will change in a cycle mode to reveal all the SH formed by H(4). However, the same SH will not be revealed in more than one subphase plate in any moment. In other words, the 2order PPA is always constituted by four different SH. To understand the corresponding relationships among SH, subphase plate, PPA, lens unit of LA and LA, an example is shown in Fig. 4(a). Resolution spot A is on the focal plane of the main lens, and corresponds to four phase patterns SH (A_{1}, A_{2}, A_{3}, and A_{4}) in four subphase plates respectively, which constitute the whole PPA. To achieve speckle reduction, A_{1}A_{4} must be incoherent from one another. However, in the same subphase plate, SH could be the same. In other words, A_{1} can be the same with B_{1}, and this principle can also be applied to A_{2}A_{4} with B_{2}B_{4} equally. This is because for a standard 4f system, the same SH (A and B) will not overlap. Conversely, the position dislocation that may cause the same SH to superimpose must be avoided, which would reduce the incoherence of beams.
Therefore, not only the number and the size of the subphase plate should correspond to the lens unit of LA, but also every subphase plate should match with the lens unit of LA in position. To further understand the stated relationships, the 2×2 PPA is converted into a 1×4 PPA and its side view is given, as shown in Fig. 4(b). The following requirements need to be satisfied: The size of subphase plate should be equal to the lens unit; Along the principal axis, the position of PPA, LA, main lens and screen should satisfy a standard of 4f system; Perpendicular to the principal axis, each subphase plate should satisfy the positional correspondence with the lens unit.
Experiments and resultsIn this optical system, the unit of LA (Edmund, #63230) is 4 mm×3 mm with a focal length of 38.1 mm. The working area of SLM is 0.55 inch with a lengthwidth ratio of 16:9, which means the length and width are about 12.2 mm and 6.9 mm, respectively. Thus, to match the working area of SLM and LA, the 2×2 PPA is used. That means the sizes of subphase plate and PPA are 4 mm×3 mm and 8 mm×6 mm, respectively. The control unit of SLM is 6.4 μm×6.4 μm, and we use the 2order phase patterns SH formed by H(4) as described in Fig. 3. Therefore, the size of one phase pattern SH will be 12.8 μm×12.8 μm. A rough paper is used as screen. The focal length of the main lens is set as 40.0 mm, and then SH will be 13.4 μm×13.4 μm on the screen.
Lens array  Main lens  CCD  Phase plate array  
Focal length  The size of lens unit  Focal length  Integration time  Focal length  Pupil size  The size of phase unit  The size of subphase plate 
38.1 mm  4 mm×3 mm  40 mm  30 ms  60 mm  3 mm  6.4 µm×6.4 µm  4 mm×3 mm 
A CCD (Thorlabs, DCC1545M) with pixel size of 8.5 μm×8.5 μm is used to capture the speckle patterns. The integration time, focal length, pupil size and Fnumber (F/#) of CCD are set as 30 ms, 60 mm, 3 mm and 20, respectively. The speckle size on CCD will be 2.44λ(F/#)= 31 μm, which is greater than the CCD unit cell's size of 8.5 μm×8.5 μm. Hence, the CCD can digitize the speckle pattern without any significant spatial averaging effects making the measurement accurate^{16}. The size of CCD resolution spot on the screen is 2.44λ(F/#)=30.8 μm, which means that on the screen the 2×2 phase cells belonging to one picture pixel fall in one resolution area of the camera lens. Thus, a phase pattern SH can be contained in one resolution spot^{17}.
Our experimental results using 2order PPA are shown in Fig. 5. The speckle contrast ratio C is defined as the ratio of standard deviation to mean value of a speckle image. Before loading the designed 2order PPA, the speckle contrast is C_{1}=0.57. After loading the designed 2order PPA, the speckle contrast reduces to C_{2}=0.30. Here, we define the relative ratio of speckle contrast as γ=C_{2}/C_{1}. Thus γ= 52.63%, which is close to the theoretical value 50%. Further, as we have mentioned above, if a larger number of different SH are used, speckle will be reduced further. Therefore, we study a more complicated PPA, which consists of 16 subphase plates. Obviously, this 4order PPA contains 16 4order SH, which are formed by the 16order Hadamard matrix H(16). Theoretically, the speckle contrast will reduce to 1/4 of the previous level. Considering the limited size of SLM, the subphase plates are set as 2 mm×1.5 mm, making sure that the 4×4 PPA can be contained in the working area. However, the unite size of LA is still 4 mm×3 mm. Therefore, the PPA formed by SLM need to be magnified twice. In such way, the subphase plate can match with the size of lens unit.
As shown in Fig. 2, another beam expander (BE 2) is added on the basis of original optical system, so that subphase plates and lens units are corresponding one by one. The PPA, lens 'a' and lens 'b' constitute a 4f system, as well as the LA, main lens and screen. Moreover, these two 4f systems are seamlessly connected. The focal lengths of lens 'a' and 'b' are set as 3 cm and 6 cm, respectively, which enlarge the phase patterns with a 2× magnification. A 4order phase pattern on SLM is 25.6 μm×25.6 μm. After the first 4f system, the size changes to 51.2 μm×51.2 μm. After the second 4f system, its size changes to 53.8 μm×53.8 μm. The camera's parameters are reset as: the integration time 30 ms, focal length 120 mm, pupil size 3 mm and (F/#)=40. Therefore, the size of CCD resolution spot on the screen is 2.44λ(F/#)=61.8 μm, which is larger than a phase pattern. Thus, one SH can be contained in one resolution spot on the screen, and the 4×4 phase cells belonging to one picture pixel fall in one resolution area of the camera lens.
The experimental results using 4order PPA are shown in Fig. 6. Before loading the PPA, the speckle contrast is C_{1}=0.45. After loading the PPA, the speckle contrast reduces to C_{2}= 0.12. Thus, the relative ratio of speckle contrast γ= 26.67%, which is close to the theoretical value 25%.
Therefore, as shown in Fig. 7, the higher order the PPA, the lower the speckle contrast. When using an norder PPA, speckle contrast will decrease to 1/n of its original value, which means the relative speckle contrast is inversely proportional to the order of PPA. The experimental results coincide with the theoretical values. Although speckle can be further eliminated by using a higherorder PPA, the restricted size of SLM and LA and the requirements for precise alignment of optical elements are two main obstacles.
ConclusionsIn this paper, a solution for speckle reduction using PPA and LA in a motionless way is proposed. The PPA, LA, main lens and screen constitute a 4f system. For the 2order PPA, subphase plates are constituted by four phase patterns formed by a 4order Hadamard matrix. Incident light will be incoherent after passing through PPA and superpose on the screen under the action of LA and main lens. Speckle reduction will be achieved by the averaging of incoherent speckle patterns. We also analyze the size and positional relationships among each component. Experiments show that a relative ratio of 52.63% is achieved, which is close to the theoretical value 50%. For the 4order PPA, subphase plates are constituted by sixteen different phase patterns, which are formed by a 16order Hadamard matrix. In order to match the working area of SLM with the size of lens unit, another beam expander is added, which also satisfies a 4f system standard. Thus, the whole system contains two seamlessly connected 4f systems. Experiments show that a relative ratio of 26.67% is achieved, which is close to the theoretical value 25%. The results prove that using an norder PPA will decrease the speckle contrast to 1/n of its original value. Speckle can be further suppressed by using a higherorder PPA. However, the light loss is inevitable due to the existence of diffraction. How to improve the light efficiency is our next work.
AcknowledgementsWe thank Z. Wang and Y. Zhou for the helpful discussion.
Competing interestsThe authors declare no competing financial interests.
1. 
Chellappan K V, Erden E, Urey H. Laserbased displays: a review. Appl Opt
49, F79F98 (2010) [Crossref] 
2. 
Goodman J W. Some fundamental properties of speckle. J Opt Soc Am
66, 11451150 (1976) [Crossref] 
3. 
Tran T K T, Chen X Y, Svensen Ø, Akram M N. Speckle reduction in laser projection using a dynamic deformable mirror. Opt Express
22, 1115211166 (2014) [Crossref] 
4. 
Akram M N, Tong Z M, Ouyang G M, Chen X Y, Kartashov V. Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror. Appl Opt
49, 32973304 (2010) [Crossref] 
5. 
Tong Z M, Chen X Y, Akram M N, Aksnes A. Compound speckle characterization method and reduction by optical design. J Dis Technol
8, 132137 (2012) [Crossref] 
6. 
Riechert F, Craggs G, Meuret Y, van Giel B, Thienpont H et al. Lowspeckle laser projection with a broadarea verticalcavity surfaceemitting laser in the nonmodal emission regime. Appl Opt
48, 792798 (2009) [Crossref] 
7. 
Redding B, Choma M A, Cao H. Specklefree laser imaging using random laser illumination. Nat Photon
6, 355359 (2012) [Crossref] 
8. 
Mizuyama Y, Leto R. Laser speckle reduction element
(USA: US8902506B2, 2011).

9. 
Lutz Y, Poyet J M. Methodology for the conception of speckle reduction elements in the case of short pulse illumination. Proc SPIE
9649, 96490L (2015) [Crossref] 
10. 
Tong Z M, Chen X Y. Speckle contrast for superposed speckle patterns created by rotating the orientation of laser polarization. J Opt Soc Am A
29, 20742079 (2012) [Crossref] 
11. 
Liu C S, Chang Y C, Lin K W, Lin P H. Speckle reduction in laser imaging applications using rotating magnetooptical disk. J Opt Soc Am A
31, 1620 (2014) [Crossref] 
12. 
Deng Q L, Lin B S, Wu P J, Chiu K Y, Fan P L et al. A hybrid temporal and spatial specklesuppression method for laser displays. Opt Express
21, 3106231071 (2013) [Crossref] 
13. 
Mehta D S, Naik D N, Singh R K, Takeda M. Laser speckle reduction by multimode optical fiber bundle with combined temporal, spatial, and angular diversity. Appl Opt
51, 18941904 (2012) [Crossref] 
14. 
Trisnadi J I. Hadamard speckle contrast reduction. Opt Lett
29, 1113 (2004) [Crossref] 
15. 
Trisnadi J I. Speckle contrast reduction in laser projection displays. Proc SPIE
4657, 131137 (2002) [Crossref] 
16. 
Gao W H, Tong Z M, Kartashov V, Akram M N, Chen X Y. Replacing twodimensional binary phase matrix by a pair of onedimensional dynamic phase matrices for laser speckle reduction. J Dis Technol
8, 291295 (2012) [Crossref] 
17. 
Koers G, Ocket I, Feng Q, Tavakol V, Jäger I et al. Study of active millimeterwave image speckle reduction by Hadamard phase pattern illumination. J Opt Soc Am A
25, 312317 (2008) [Crossref] 
18. 
Akram M N, Kartashov V, Tong Z M. Speckle reduction in linescan laser projectors using binary phase codes. Opt Lett
35, 444446 (2010) [Crossref] 
19. 
Lapchuk A, Kryuchyn A, Petrov V, Yurlov V, Klymenko V. Full speckle suppression in laser projectors using two Barker codetype diffractive optical elements. J Opt Soc Am A
30, 2231 (2013) [Crossref] 
20. 
Wang Z, Wang A T, Wang S L, Ma X H, Ming H. Resolutionenhanced integral imaging using two microlens arrays with different focal lengths for capturing and display. Opt Express
23, 2897028977 (2015) [Crossref] 
21. 
Ma X H, Wang S L, Liu W W, Ma F H, Wang A T et al. Optimized stereo matching algorithm for integral imaging microscopy and its potential use in precise 3D optical manipulation. Opt Commun
430, 374379 (2019) [Crossref] 
22. 
Wang Z, Wang A T, Ma X H, Liu B Y, Ma F H et al. Integral floating 3D display using two retroreflector arrays. IEEE Photon J
9, 7000108 (2017) [Crossref] 
23. 
Goodman J W. Speckle Phenomena in Optics: Theory and Applications
(Roberts and Company Publishers: Englewood, 2007).
[Crossref]
