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Holographic projection based on programmable axilens
Optics and Laser Technology 120 (2019) 105682
Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Holographic projection based on programmable axilens☆ a,b,⁎
Chuan Shen a b
a
a
a
T
a
, QingQing Hong , QinQin Zhu , Ci Zu , Sui Wei
Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei 230601, China Key Laboratory of Modern Imaging and Display Technology of Anhui Province, Auhui University, Hefei 230601, China
H I GH L IG H T S
holographic projection system based on programmable axilens extends the DOF. • The reconstruction distances can be achieved by changing the focal depth of the axilens. • Different • The reconstructed images in good focus can be simultaneously observed on a multi-plane in a range.
A R T I C LE I N FO
A B S T R A C T
Keywords: Holographic projection Axilens Liquid crystal on silicon (LCoS)
In a traditional holographic projection system, a clear reconstructed image is formed only in the focal plane of the Fourier lens. In this paper, We propose a novel method to form simultaneously holographic reconstructed images in good focus located at different planes. The large focal depth characteristic of the programmable axilens is applied to the holographic projection. First, the iterative Fourier algorithm is employed to generate a phase hologram. A new phase distribution is formed by superimposing the axilens phase on the phase hologram, and loading onto the phase-only liquid crystal on silicon (LCoS); With the programmability of the LCoS, the different focal-depth parameters of the axilens are tuned. We could obtain the desired holographic reconstructed images when the collimated laser is used to incident the LCoS. The experimental results show that compared with the holographic projection system based on Fresnel lens, the holographic reconstructed images in good focus can be observed simultaneously on a multi-plane in a range by dynamically changing the focal depth of the axilens.
2010 MSC: 00-01 99-00
1. Introduction According to the diffraction theory, holographic projection uses a spatial light modulator (SLM) to load a phase hologram, which can modulate the incident light to obtain the desired two-dimensional image [1]. Compared with traditional projection technology, holographic projection technology has the advantage of high diffraction efficiency [2]. Holographic projection can be mainly divided into Fourier holographic projection and Fresnel holographic projection based on the difference of diffraction regions. In the traditional Fourier holographic projection system, a clear reconstructed image can be formed on the back focal plane of the Fourier lens. However, a clear reconstructed image cannot be obtained on other planes that deviate from the focal plane. In order to expand the application range of holographic projection systems, researchers have carried out research work on multi-plane holographic projection [3–5], which can use Fresnel phase holograms with different propagation distances or multi-
planes with Fresnel phase lens. However, for holographic projection in different planes along the propagation direction of the light field, the above methods need to separately calculate the corresponding Fresnel phase hologram or adjust the focal length of the Fresnel phase lens, which affects the scalability of the system. Moreover, only a limited number of quantized propagation distances for holographic projection can be achieved, which leads to the limitation of the holographic projection for a number of applications such as head up display and neareye display. To overcome the shortage of limited depth of focus (DOF), the axicon produces an approximately non-diffractive Bessel beam with a small optical focus diameter and a large DOF, which has been widely used in space optical communication [6], high-precision alignment [7] and optical capture [8]. Niklas W et al. make use of the arrangements of multiple convex and concave axicons to obtain Bessel beams which applied in optical coherence tomography (OCT) to improve the DOF and achieve high resolution [9]. Notably, the axilens is an optical
☆ ⁎
Fully documented templates are available in the elsarticle package on CTAN. Corresponding author at: Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei 230601, China. E-mail addresses: [email protected] (C. Shen), [email protected] (S. Wei).
https://doi.org/10.1016/j.optlastec.2019.105682 Received 3 April 2019; Received in revised form 27 June 2019; Accepted 15 July 2019 Available online 08 August 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 120 (2019) 105682
C. Shen, et al.
Zg = 0 to Eq. (1) the phase function can be written φ (r ) = πr 2/ λf0 , thus a Fresnel lens is a particular case. Limited by the phase modulation capability of the phase-only SLM device used in subsequent experiments, the maximum phase value of the axilens is set to 2π , and the phase distribution actually applied to the SLM is φ (r ) = mod [φ (r ), 2π ], so the research work is performed using the diffraction-type axilens. The parameters f0 and Zg in Eq. (1) can be chosen according to a specific application of the axilens. Having examined the phase characteristics of an axilens, we now turn to the Fourier phase hologram. The Gerchberg-Saxton algorithm, also known as the iterative Fourier transform algorithm is a phase retrieval algorithm based on the constraint of prior knowledge or measured data [16], which iterates between the image plane and the holographic plane, and finally the retrieval phase is encoded as the required Fourier phase hologram. In a classical Fourier holographic projection system, a Fourier lens exists between the hologram field and the reconstructed field, the two fields exist in the front and back focal planes of the lens, which are related by a Fourier transform. Finally, a clear holographic reconstructed image is formed only in the back focal plane of the Fourier lens. Consider now the case of the Fourier lens is missing, as shown in Fig. 2, this time within the region of Fresnel diffraction rather than Fraunhofer diffraction. In order to form a relatively clear holographic reconstructed image within a long range along the direction of light field propagation, the phase profile of the hologram ϕ (r ) is superimposed with the phase of programmable axilens φ (r ) to obtain a new phase distribution φ1 (r ) to be uploaded onto phase-only SLM with a square geometry of pixelated structure. Suppose that the SLM is normally illuminated by a monochromatic plane wave of unit amplitude. The hologram field U0 (x 0 , y0 ) on the SLM plane and the complex amplitude field U (x , y ) on the reconstructed plane are now related by the Fresnel diffraction integral.
component that benefits from both Fresnel zone plate and axicon. Many researchers have also carried out the corresponding work in the optical imaging applications for the purpose of increasing the DOF of the system. Chen et al. studied the effect of the focusing characteristics of the axicon on the DOF of the imaging system [10]. Savoia R et al. applied the ability of an axicon lens to traditional optical microscopy systems, which can abtain an extended focused image located at different depths [11]. In 2015, Heuke S et al. applied axilens to microscopic imaging systems to improve the resolution of the imaging system [12]. Grzegorz M et al. applied an axilens with line focus characteristics to the imaging system to increase its system depth [13]. Its optical behavior can be tuned by changing the design parameters of the axilens, and the programmable axilens may be realized by SLM which enable to rapidly changing phase modulation in a reliable and potentially miniature device. In 2015, Romero et al. explored programmable diffractive optical elements (PDOEs) to extand the DOF in ophthalmic optics, and these DOEs were written onto a reflective LCoS to compare the performances of the PDOEs with those of multifocal lenses [14]. For the best of our knowledge, the programmable axilens has never been applied in holographic projection. Therefore, we combine the large DOF properties of the axilens and the phase hologram onto a single phase-only SLM. Different with the previous methods, we would like to reconstruct the holographic images with continuous depths rather than quantized depths in the same field of view, which increases the flexibility and compactness of the holographic projection system. In this paper, a holographic projection system based on programmable axilens is proposed to enhance the DOF. The main idea is to superimpose the phase function of the axilens on the phase hologram, and then the new phase distribution is loaded on the SLM. With the help of large focal depth characteristics of the axilens, we will show that the application of axilens to holographic projection can realize simultaneously the reconstruction of holographic images in good focus located at different planes. The proposed method is also compared with a common approach based on Fresnel phase lens.
U (x , y ) =
∞
2π
.
where U0 (x 0 , y0 ) = exp{i [φ1 (r )]}, (x 0 , y0 ) represents the coordinates of the plane in which the SLM is located, and (x , y ) represent the coordinates of the reconstructed plane, and the distance between the SLM plane and the reconstructed plane is z. Benefits from the long focal depth characteristics of the axilens, the approximate clear holographic reconstructed images at different observation distance z are formed simultaneously. When Zg = 0 , φ (r ) = πr 2/ λf0 in Eq. (1), if the observation distances satisfy z = f , then the quadratic phase factors in Eq. (2) are exactly offset, equivalent to the phase transformation function of the lens. It is observed that the complex amplitude distribution on the reconstructed plane is the Fraunhofer diffraction pattern of exp{i [ϕ (r )]} . Due to the presence of the quadratic phase factor that precedes the integral (2), the Fourier transform relation between the input complex amplitude transmittance and the focal-plane complex amplitude distribution is not a exact one, but it is of interest to observe
The principle of focusing the input plane wave by the axilens is shown in Fig. 1. The phase function of an axilens is given by [15]
R
k
(2)
2.1. Design of a programmable axilens
π r2 λ f + Zg r 2 2 0
(x 2 + y 2 )]
× ∬−∞ {U0 (x 0 , y0 )exp[j 2z (x 02 + y02 )]}exp[−j λz (xx 0 + yy0 )] dx 0 dy0
2. Optical design and characteristics
φ (r ) =
1 k exp(jkz )exp[j 2z jλz
(1)
where λ is the wavelength of the incident light, R is the radius of the element, f0 is the starting focal length, Zg is the non-diffracting region, i.e. the focal depth of the axilens, and r is the radial coordinate (r = x 02 + y02 ). The axilens is located in the (x 0 , y0 ) plane. Applying the
Fig. 2. Schematic diagram of the holographic projection system without the use of a Fourier lens.
Fig. 1. Geometric diagram of focusing of an axilens. 2
Optics and Laser Technology 120 (2019) 105682
C. Shen, et al.
Fig. 3. Images of Fresnel diffraction along the axis direction for an axilens with focal length f0 = 900 mm (a) focal depth Zg = 100 mm (b) focal depth Zg = 300 mm (c) focal depth Zg = 500 mm (d) focal depth Zg = 700 mm.
Fig. 4. Images of Fresnel diffraction along the axis direction for an axilens with focal depth Zg = 100 mm (a) focal length f0 = 1000 mm (b) focal length f0 = 1100 mm (c) focal length f0 = 1200 mm (d) focal length f0 = 1300 mm.
the light intensity distribution on the image plane, namely, I (x , y ) = |U (x , y )|2 , so the quadratic phase factor is of no consequence. 3. Results 3.1. Simulation result of linear focus characteristics of axilens In order to verify the characteristics of the axilens, two sets of experiments are designed to observe the relative intensity distribution corresponding to the Fresnel diffraction along optical axis when the axilens is illuminated by a unit-amplitude, normally incident, monochromatic plane wave. Firstly, let us consider the following design parameters: the radius of axilens R = 10 mm, the initial focal length
Fig. 5. (a) Image of the object (b) Image of the phase hologram.
Fig. 6. In the case of an image size of 512 ∗ 512 pixels, simulated reconstruction images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens), the image size is 512 ∗ 512 pixels. 3
Optics and Laser Technology 120 (2019) 105682
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Fig. 7. In the case of an image size of 2048 ∗ 2048 pixels, simulated reconstruction images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens).
concentration in the beam within the focal range (f0 , f0 + Zg ) . In the case of the same DOF, the focal length increases, the line focal length shifts along the axial direction, which leads to a focal depth does not change. As is evident, the DOF, over which the intensity along the axis is constant, is approximately several times greater for this axilens than for the Fresnel lens. From the simulation results of Figs. 3 and 4, the large DOF characteristics of the axilens can be observed clearly. Therefore, by setting the corresponding parameter value of the axilens, the phase distribution obtained by the Eq. (1) is written on a phase-only SLM. The ability of modern SLMs to provide a freedom that allows the calculation of diffraction axilens with different parameters of focal length and focal depth, thus the programmable axilens can be
value was kept constant at f0 = 900 mm, and the DOF parameter was changed. The focal depth Zg was taken as 100 mm, 300 mm, 500 mm, and 700 mm, respectively. The Fresnel diffraction patterns of an axilens along the optical axis are shown in Fig. 3. In order to observe the simulation results clearly, the colormap (pink) function in MATLAB is used. And the full width at half maximum (FWHM)[17] approach has been employed to describe the different focal depths and the initial focal lengths. Secondly, the radius of axilens is also R = 10 mm, since the focal depth is remains unchanged, Zg = 100 mm, the different initial focal length values f0 are set to 1100 mm, 1200 mm, 1300 mm, 1400 mm, respectively. As shown in Fig. 4, It can be seen that reasonable energy 4
Optics and Laser Technology 120 (2019) 105682
C. Shen, et al.
Fig. 8. In the case of grayscale image, simulated reconstruction images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens).
Fig. 9. RMSE value as a function of the axial distance around the focal region for the axilens and a Fresnel lens: (a) the image size of 512 ∗ 512 pixels (b) the image size of 2048 ∗ 2048 pixels (c) grayscale image.
image can be observed only at the focal plane. Secondly, set R = 10 mm and keep the same focal length as the Fresnel lens. The phase of the axilens and the phase hologram are combined, consider different focusdepth parameters and observe the holographic reconstruction images at different propagation distances. Set the focal depth Zg = 100 mm, the propagation distance is 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, respectively. The simulated holographic reconstruction images are shown in Fig. 6(b). The clearest image can be observed at the reconstruction distance of 950 mm, and a relatively clear reconstruction image can be obtained at the reconstruction distance of 900 mm, 950 mm and 1000 mm. Then, set the focal depth Zg = 300 mm, the reconstruction distance is 500 mm, 700 mm, 900 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, 1800 mm, and the simulated holographic reconstruction images are as shown in Fig. 6(c). The clearest image is observed at the reconstruction distance of 1000 mm, and the relatively clear reconstruction images are obtained
realized. 3.2. Simulation results The Fourier phase hologram is calculated by the iterative Fourier algorithm. The original image (a Chinese character) and the generated phase distribution are shown in Fig. 5(a) and (b), respectively. The image size is 512 ∗ 512 pixels. We assume that the sampling pitch is 12 μm , which is equivalent to the pixel pitch of the LCoS used in the experiments. The wavelength of the incident light is λ = 532 nm. We consider next several examples of holographic projection. Firstly, of ultimate interest, is that the phase distribution of a Fresnel lens and the phase hologram are combined, and the focal length is f0 = 900 mm. Different propagation distances of 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1100 mm, 1200 mm and 1300 mm are used to observe the holographic images. As illustrated in Fig. 6(a), a clear holographic 5
Optics and Laser Technology 120 (2019) 105682
C. Shen, et al.
Fig. 10. The schematic of holographic projection.
Fig. 11. The experimental system of holographic projection.
Fig. 12. Holographic images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens).
contrast than those images with 512 ∗ 512 pixels. Thus, it can be inferred that increasing the number of pixels will also increase the contrast of the reconstructed image. The second case, a grayscale image (Cameraman) is utilized, which keeps the same simulation parameters as well as the above binary image (512 ∗ 512 pixels). The results shown in Fig. 8 prove that our experiment can be effectively applied to the case of grayscale images. In order to further verify the accuracy of the holographic reconstructed image, the error function RMSE is defined as
on the planes of the reconstruction distances of 900 mm, 1000 mm, and 1200 mm, and note that the holographic images obtained on the plane exceeding the focus range became blurred. It is preliminarily verified that the validity of the axilens which is applied to the holographic projection. And a set of relatively clearer images can be observed on multiple reconstruction planes along the focal depth of the axilens. As is evident, the contrast of the holographic image in the focal plane for the case of axilens is lower than for the case of Fresnel lens, and there is a trade-off between the depth of field and the image sharpness. In this section, two more cases are presented that provide more insight about our proposed method. The first case, the image size is set to 2048 ∗ 2048 pixels. The reconstructed images along the direction of light field propagation are shown in Fig. 7. It is clear to observe that these holographic reconstruction images have significantly higher
RMSE =
∑x , y [|I1 (x , y )| − |I2 (x , y )|]2 M×N
(3)
where |I2 (x , y )| is the intensity distribution of the reconstructed image, |I1 (x , y )| is the intensity distribution of the original image, and the 6
Optics and Laser Technology 120 (2019) 105682
C. Shen, et al.
has also been presented to verify the proposed method. The performances of different combined hologram with superimposing the diffractive axilens and Fresnel lens are compared. Finally, using our method, the approximate clear holographic images can be observed simultaneously on multiple planes in a long range area in comparison with conventional Fourier holographic projection.
image is M × N pixels. Different holographic images are calculated with step change of 50 mm at propagation distance from 500 mm to 2000 mm. The RMSE values with respect to the different propagation distance are shown in Fig. 9, which the programmable axilens of the focal depth Zg = 100 mm and Zg = 300 mm. For comparison, we also present the corresponding RMSE for a Fresnel lens that has the same aperture and a focal length of 900 mm. Fig. 9(a) and (b) show that increasing the number of pixels reduces the error value. As is evident, the DOF, over which the RMSE is lower for axilens than for the Fresnel lens at a long distance from the optical elements. Unfortunately, due to the characteristics of the axilens, it concentrates only a small fraction of the energy into the focused beam, which results in an extremely larger RMSE value than the Fresnel lens at the constant focal plane f0 = 900 mm. As shown in Fig. 9(c), in the case of grayscale image, the difference in error value is even more pronounced.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61605002, 61501001, 61301296), Natural Science Foundation of Anhui Province (1608085QF161), Natural Science Project of Anhui Higher Education Institutions of China (KJ2016A029). References
4. Optical setup and experimental results [1] A. Georgiou, J. Christmas, J. Moore, a.A.D. A Jeziorska-Chapman, W.A.C.N. Collings, Liquid crystal over silicon device characteristics for holographic projection of high-definition television images, Appl. Opt. 47 (26) (2008) 4793–4803, https://doi.org/10.1364/AO.47.004793. [2] E. Buckley, Holographic laser projection, J. Display Technol. 7 (3) (2011) 135–140, https://doi.org/10.1109/JDT.2010.2048302. [3] M. Makowski, M. Sypek, A. Kolodziejczyk, G. Mikula, J. Suszek, Iterative design of multiplane holograms: experiments and applications, Opt. Eng. 46 (4) (2007) 045802, https://doi.org/10.1117/1.2727379. [4] G.A. Alkan, J.B. Keith, Cascaded diffractive optical elements for improved multiplane image reconstruction, Appl. Opt. 52 (15) (2013) 3608–3616, https://doi.org/ 10.1364/AO.52.003608. [5] Chao-fu Ying, New method for the design of a phase-only computer hologram for multiplane reconstruction, Opt. Eng. 50 (5) (2011) 055802, https://doi.org/10. 1364/AO.52.003608. [6] G. Gibson, J. Courtial, M. Vasnetsov, M.J. Padgett, S. Frankearnold, S.M. Barnett, V. Pas’Ko, Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express 12 (22) (2004) 5448–5456, https://doi.org/10. 1364/OPEX.12.005448. [7] P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A.T. Friberg, Electromagnetic analysis of nonparaxial bessel beams generated by diffractive axicons, J. Opt. Soc. Am. A Opt. Image Sci. 14 (14) (1997) 1817–1824, https://doi.org/10.1364/JOSAA. 14.001817. [8] K.D. Leake, A.R. Hawkins, H. Schmidt, All-optical particle trap using orthogonally intersecting beams [invited], Photon. Res. 1 (1) (2013) 47–51, https://doi.org/10. 1364/PRJ.1.000047. [9] W. Niklas, S. Dominik, S. Andreas, Z. Hans, Highly compact imaging using bessel beams generated by ultraminiaturized multi-micro-axicon systems, J. Opt. Soc. Am. A Opt. Image Sci. Vision 29 (5) (2012) 808, https://doi.org/10.1364/JOSAA.29. 000808. [10] C.Z.W. FengTie, Effect of axicon line focusing characteristics on imaging system’s focal depth, J. Optoelectron. Laser. doi:https://doi.org/10.16136/j.joel.2017.05. 0851. [11] R. Savoia, M. Matrecano, P. Memmolo, M. Paturzo, P. Ferraro, Investigation on axicon transformation in digital holography for extending the depth of focus in biomicrofluidics applications, J. Display Technol. 11 (10) (2017) 861–866, https://doi. org/10.1109/JDT.2015.2395074. [12] D. Akimov, F.B. Legesse, J. Dellith, J. Popp, M. Schmitt, S. Heuke, U. Hübner, Bessel beam coherent anti-stokes raman scattering microscopy, J. Opt. Soc. Am. B 32 (9) (2015) 1773, https://doi.org/10.1364/JOSAB.32.001773. [13] G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, Diffractive elements for imaging with extended depth of focus, Opt. Eng. 44 (5) (2005) 058001–058007, https://doi.org/10.1117/1.1905481. [14] L.A. Romero, M.S. Millán, Z. Jaroszewicz, A. Kolodziejczyk, Programmable diffractive optical elements for extending the depth of focus in ophthalmic optics, in: International Symposium on Medical Information Processing & Analysis, vol. 9287. doi:https://doi.org/10.1117/12.2073866. [15] N. Davidson, A.A. Friesem, E. Hasman, Holographic axilens: high resolution and long focal depth, Opt. Lett. 16 (7) (1991) 523, https://doi.org/10.1364/OL.16. 000523. [16] R.W. Gerchberg, A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik 35 (1972) 237. [17] E.O. Brinchmann-Hansen O, Microphotometry of the blood column and the light streak on retinal vessels in fundus photographs, Acta Ophthalmol. (1986) 9–19, https://doi.org/10.1111/j.1755-3768.1986.tb00698.x.
A holographic projection system was constructed to verify the proposed method, the schematic diagram of the system is shown in Fig. 10. The LCoS (MD1280, Three-Five Systems Company) was used as a phase-only modulation SLM, which resolution is 1280 ∗ 1024 pixels, the pixel pitch is 12 μm . A green laser with a wavelength of 532 nm is used, after the laser is beam-expanded and collimated, a uniform plane light wave is obtained to illuminate the LCoS. A picture of the experimental setup is shown in Fig. 11. The phase-only hologram generated by the iterative Fourier algorithm is loaded onto the LCoS after superimposing the phase of the axilens or Fresnel lens. Along the direction of propagation of the light field, the holographic reconstructed images at different reconstructed planes from the LCoS are taken. We compared the performances of different combined hologram with superimposing the phase of the axilens and Fresnel lens. The experiment results are shown in Fig. 12. Fig. 12(a) shows reconstructed images which are taken at the distances of 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1100 mm, 1200 mm, and 1300 mm from the LCoS plane when the focal length of Fresnel lens is f0 = 900 mm; Fig. 12(b) gives the reconstructed images which are taken at 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, respectively, when the focal length of axilens is f0 = 900 mm, the focal depth is Zg = 100 mm. In the other case of the focal depth Zg = 300 mm, Fig. 12(c) shows the reconstructed images taken at the distances of 500 mm, 700 mm, 900 mm, 1000 mm, 1200 mm, 1400 mm, and 1600 mm, respectively. As is evident, the results of experimental holographic reconstructions basically confirm the results of the numerical simulations. In a case of Fresnel lens, a clear holographic image is formed only in the focal plane. In contrast, the approximate clear holographic images are formed in a long range area along the propagation direction of the light field when the axilens are applied to the holographic projection. Considering the phase-only LCoS with a limited fill factor, its single pixel structure is square, which causes some artifacts on the holographic images. In order to separate the diffraction orders, a linear phase term was added to the combined hologram. The elimination of zero-order light and multi-level reproduction is not the focus of this paper, only the +1 order image is observed in the experiment. 5. Conclusions In this paper, we discussed the implementation programmable axilens for holographic projection system. The large focal depth characteristics and the flexibility of this approach in different configurations of axilens are analyzed. Holographic projection system based on LCoS
7
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Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Holographic projection based on programmable axilens☆ a,b,⁎
Chuan Shen a b
a
a
a
T
a
, QingQing Hong , QinQin Zhu , Ci Zu , Sui Wei
Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei 230601, China Key Laboratory of Modern Imaging and Display Technology of Anhui Province, Auhui University, Hefei 230601, China
H I GH L IG H T S
holographic projection system based on programmable axilens extends the DOF. • The reconstruction distances can be achieved by changing the focal depth of the axilens. • Different • The reconstructed images in good focus can be simultaneously observed on a multi-plane in a range.
A R T I C LE I N FO
A B S T R A C T
Keywords: Holographic projection Axilens Liquid crystal on silicon (LCoS)
In a traditional holographic projection system, a clear reconstructed image is formed only in the focal plane of the Fourier lens. In this paper, We propose a novel method to form simultaneously holographic reconstructed images in good focus located at different planes. The large focal depth characteristic of the programmable axilens is applied to the holographic projection. First, the iterative Fourier algorithm is employed to generate a phase hologram. A new phase distribution is formed by superimposing the axilens phase on the phase hologram, and loading onto the phase-only liquid crystal on silicon (LCoS); With the programmability of the LCoS, the different focal-depth parameters of the axilens are tuned. We could obtain the desired holographic reconstructed images when the collimated laser is used to incident the LCoS. The experimental results show that compared with the holographic projection system based on Fresnel lens, the holographic reconstructed images in good focus can be observed simultaneously on a multi-plane in a range by dynamically changing the focal depth of the axilens.
2010 MSC: 00-01 99-00
1. Introduction According to the diffraction theory, holographic projection uses a spatial light modulator (SLM) to load a phase hologram, which can modulate the incident light to obtain the desired two-dimensional image [1]. Compared with traditional projection technology, holographic projection technology has the advantage of high diffraction efficiency [2]. Holographic projection can be mainly divided into Fourier holographic projection and Fresnel holographic projection based on the difference of diffraction regions. In the traditional Fourier holographic projection system, a clear reconstructed image can be formed on the back focal plane of the Fourier lens. However, a clear reconstructed image cannot be obtained on other planes that deviate from the focal plane. In order to expand the application range of holographic projection systems, researchers have carried out research work on multi-plane holographic projection [3–5], which can use Fresnel phase holograms with different propagation distances or multi-
planes with Fresnel phase lens. However, for holographic projection in different planes along the propagation direction of the light field, the above methods need to separately calculate the corresponding Fresnel phase hologram or adjust the focal length of the Fresnel phase lens, which affects the scalability of the system. Moreover, only a limited number of quantized propagation distances for holographic projection can be achieved, which leads to the limitation of the holographic projection for a number of applications such as head up display and neareye display. To overcome the shortage of limited depth of focus (DOF), the axicon produces an approximately non-diffractive Bessel beam with a small optical focus diameter and a large DOF, which has been widely used in space optical communication [6], high-precision alignment [7] and optical capture [8]. Niklas W et al. make use of the arrangements of multiple convex and concave axicons to obtain Bessel beams which applied in optical coherence tomography (OCT) to improve the DOF and achieve high resolution [9]. Notably, the axilens is an optical
☆ ⁎
Fully documented templates are available in the elsarticle package on CTAN. Corresponding author at: Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei 230601, China. E-mail addresses: [email protected] (C. Shen), [email protected] (S. Wei).
https://doi.org/10.1016/j.optlastec.2019.105682 Received 3 April 2019; Received in revised form 27 June 2019; Accepted 15 July 2019 Available online 08 August 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 120 (2019) 105682
C. Shen, et al.
Zg = 0 to Eq. (1) the phase function can be written φ (r ) = πr 2/ λf0 , thus a Fresnel lens is a particular case. Limited by the phase modulation capability of the phase-only SLM device used in subsequent experiments, the maximum phase value of the axilens is set to 2π , and the phase distribution actually applied to the SLM is φ (r ) = mod [φ (r ), 2π ], so the research work is performed using the diffraction-type axilens. The parameters f0 and Zg in Eq. (1) can be chosen according to a specific application of the axilens. Having examined the phase characteristics of an axilens, we now turn to the Fourier phase hologram. The Gerchberg-Saxton algorithm, also known as the iterative Fourier transform algorithm is a phase retrieval algorithm based on the constraint of prior knowledge or measured data [16], which iterates between the image plane and the holographic plane, and finally the retrieval phase is encoded as the required Fourier phase hologram. In a classical Fourier holographic projection system, a Fourier lens exists between the hologram field and the reconstructed field, the two fields exist in the front and back focal planes of the lens, which are related by a Fourier transform. Finally, a clear holographic reconstructed image is formed only in the back focal plane of the Fourier lens. Consider now the case of the Fourier lens is missing, as shown in Fig. 2, this time within the region of Fresnel diffraction rather than Fraunhofer diffraction. In order to form a relatively clear holographic reconstructed image within a long range along the direction of light field propagation, the phase profile of the hologram ϕ (r ) is superimposed with the phase of programmable axilens φ (r ) to obtain a new phase distribution φ1 (r ) to be uploaded onto phase-only SLM with a square geometry of pixelated structure. Suppose that the SLM is normally illuminated by a monochromatic plane wave of unit amplitude. The hologram field U0 (x 0 , y0 ) on the SLM plane and the complex amplitude field U (x , y ) on the reconstructed plane are now related by the Fresnel diffraction integral.
component that benefits from both Fresnel zone plate and axicon. Many researchers have also carried out the corresponding work in the optical imaging applications for the purpose of increasing the DOF of the system. Chen et al. studied the effect of the focusing characteristics of the axicon on the DOF of the imaging system [10]. Savoia R et al. applied the ability of an axicon lens to traditional optical microscopy systems, which can abtain an extended focused image located at different depths [11]. In 2015, Heuke S et al. applied axilens to microscopic imaging systems to improve the resolution of the imaging system [12]. Grzegorz M et al. applied an axilens with line focus characteristics to the imaging system to increase its system depth [13]. Its optical behavior can be tuned by changing the design parameters of the axilens, and the programmable axilens may be realized by SLM which enable to rapidly changing phase modulation in a reliable and potentially miniature device. In 2015, Romero et al. explored programmable diffractive optical elements (PDOEs) to extand the DOF in ophthalmic optics, and these DOEs were written onto a reflective LCoS to compare the performances of the PDOEs with those of multifocal lenses [14]. For the best of our knowledge, the programmable axilens has never been applied in holographic projection. Therefore, we combine the large DOF properties of the axilens and the phase hologram onto a single phase-only SLM. Different with the previous methods, we would like to reconstruct the holographic images with continuous depths rather than quantized depths in the same field of view, which increases the flexibility and compactness of the holographic projection system. In this paper, a holographic projection system based on programmable axilens is proposed to enhance the DOF. The main idea is to superimpose the phase function of the axilens on the phase hologram, and then the new phase distribution is loaded on the SLM. With the help of large focal depth characteristics of the axilens, we will show that the application of axilens to holographic projection can realize simultaneously the reconstruction of holographic images in good focus located at different planes. The proposed method is also compared with a common approach based on Fresnel phase lens.
U (x , y ) =
∞
2π
.
where U0 (x 0 , y0 ) = exp{i [φ1 (r )]}, (x 0 , y0 ) represents the coordinates of the plane in which the SLM is located, and (x , y ) represent the coordinates of the reconstructed plane, and the distance between the SLM plane and the reconstructed plane is z. Benefits from the long focal depth characteristics of the axilens, the approximate clear holographic reconstructed images at different observation distance z are formed simultaneously. When Zg = 0 , φ (r ) = πr 2/ λf0 in Eq. (1), if the observation distances satisfy z = f , then the quadratic phase factors in Eq. (2) are exactly offset, equivalent to the phase transformation function of the lens. It is observed that the complex amplitude distribution on the reconstructed plane is the Fraunhofer diffraction pattern of exp{i [ϕ (r )]} . Due to the presence of the quadratic phase factor that precedes the integral (2), the Fourier transform relation between the input complex amplitude transmittance and the focal-plane complex amplitude distribution is not a exact one, but it is of interest to observe
The principle of focusing the input plane wave by the axilens is shown in Fig. 1. The phase function of an axilens is given by [15]
R
k
(2)
2.1. Design of a programmable axilens
π r2 λ f + Zg r 2 2 0
(x 2 + y 2 )]
× ∬−∞ {U0 (x 0 , y0 )exp[j 2z (x 02 + y02 )]}exp[−j λz (xx 0 + yy0 )] dx 0 dy0
2. Optical design and characteristics
φ (r ) =
1 k exp(jkz )exp[j 2z jλz
(1)
where λ is the wavelength of the incident light, R is the radius of the element, f0 is the starting focal length, Zg is the non-diffracting region, i.e. the focal depth of the axilens, and r is the radial coordinate (r = x 02 + y02 ). The axilens is located in the (x 0 , y0 ) plane. Applying the
Fig. 2. Schematic diagram of the holographic projection system without the use of a Fourier lens.
Fig. 1. Geometric diagram of focusing of an axilens. 2
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Fig. 3. Images of Fresnel diffraction along the axis direction for an axilens with focal length f0 = 900 mm (a) focal depth Zg = 100 mm (b) focal depth Zg = 300 mm (c) focal depth Zg = 500 mm (d) focal depth Zg = 700 mm.
Fig. 4. Images of Fresnel diffraction along the axis direction for an axilens with focal depth Zg = 100 mm (a) focal length f0 = 1000 mm (b) focal length f0 = 1100 mm (c) focal length f0 = 1200 mm (d) focal length f0 = 1300 mm.
the light intensity distribution on the image plane, namely, I (x , y ) = |U (x , y )|2 , so the quadratic phase factor is of no consequence. 3. Results 3.1. Simulation result of linear focus characteristics of axilens In order to verify the characteristics of the axilens, two sets of experiments are designed to observe the relative intensity distribution corresponding to the Fresnel diffraction along optical axis when the axilens is illuminated by a unit-amplitude, normally incident, monochromatic plane wave. Firstly, let us consider the following design parameters: the radius of axilens R = 10 mm, the initial focal length
Fig. 5. (a) Image of the object (b) Image of the phase hologram.
Fig. 6. In the case of an image size of 512 ∗ 512 pixels, simulated reconstruction images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens), the image size is 512 ∗ 512 pixels. 3
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Fig. 7. In the case of an image size of 2048 ∗ 2048 pixels, simulated reconstruction images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens).
concentration in the beam within the focal range (f0 , f0 + Zg ) . In the case of the same DOF, the focal length increases, the line focal length shifts along the axial direction, which leads to a focal depth does not change. As is evident, the DOF, over which the intensity along the axis is constant, is approximately several times greater for this axilens than for the Fresnel lens. From the simulation results of Figs. 3 and 4, the large DOF characteristics of the axilens can be observed clearly. Therefore, by setting the corresponding parameter value of the axilens, the phase distribution obtained by the Eq. (1) is written on a phase-only SLM. The ability of modern SLMs to provide a freedom that allows the calculation of diffraction axilens with different parameters of focal length and focal depth, thus the programmable axilens can be
value was kept constant at f0 = 900 mm, and the DOF parameter was changed. The focal depth Zg was taken as 100 mm, 300 mm, 500 mm, and 700 mm, respectively. The Fresnel diffraction patterns of an axilens along the optical axis are shown in Fig. 3. In order to observe the simulation results clearly, the colormap (pink) function in MATLAB is used. And the full width at half maximum (FWHM)[17] approach has been employed to describe the different focal depths and the initial focal lengths. Secondly, the radius of axilens is also R = 10 mm, since the focal depth is remains unchanged, Zg = 100 mm, the different initial focal length values f0 are set to 1100 mm, 1200 mm, 1300 mm, 1400 mm, respectively. As shown in Fig. 4, It can be seen that reasonable energy 4
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Fig. 8. In the case of grayscale image, simulated reconstruction images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens).
Fig. 9. RMSE value as a function of the axial distance around the focal region for the axilens and a Fresnel lens: (a) the image size of 512 ∗ 512 pixels (b) the image size of 2048 ∗ 2048 pixels (c) grayscale image.
image can be observed only at the focal plane. Secondly, set R = 10 mm and keep the same focal length as the Fresnel lens. The phase of the axilens and the phase hologram are combined, consider different focusdepth parameters and observe the holographic reconstruction images at different propagation distances. Set the focal depth Zg = 100 mm, the propagation distance is 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, respectively. The simulated holographic reconstruction images are shown in Fig. 6(b). The clearest image can be observed at the reconstruction distance of 950 mm, and a relatively clear reconstruction image can be obtained at the reconstruction distance of 900 mm, 950 mm and 1000 mm. Then, set the focal depth Zg = 300 mm, the reconstruction distance is 500 mm, 700 mm, 900 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, 1800 mm, and the simulated holographic reconstruction images are as shown in Fig. 6(c). The clearest image is observed at the reconstruction distance of 1000 mm, and the relatively clear reconstruction images are obtained
realized. 3.2. Simulation results The Fourier phase hologram is calculated by the iterative Fourier algorithm. The original image (a Chinese character) and the generated phase distribution are shown in Fig. 5(a) and (b), respectively. The image size is 512 ∗ 512 pixels. We assume that the sampling pitch is 12 μm , which is equivalent to the pixel pitch of the LCoS used in the experiments. The wavelength of the incident light is λ = 532 nm. We consider next several examples of holographic projection. Firstly, of ultimate interest, is that the phase distribution of a Fresnel lens and the phase hologram are combined, and the focal length is f0 = 900 mm. Different propagation distances of 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1100 mm, 1200 mm and 1300 mm are used to observe the holographic images. As illustrated in Fig. 6(a), a clear holographic 5
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Fig. 10. The schematic of holographic projection.
Fig. 11. The experimental system of holographic projection.
Fig. 12. Holographic images of superimposed the phase of axilens and Fresnel lens at different distances with (a) focal length f0 = 900 mm (Fresnel lens) (b) focal length f0 = 900 mm, focal depth Zg = 100 mm (axilens) (c) focal length f0 = 900 mm, focal depth Zg = 300 mm (axilens).
contrast than those images with 512 ∗ 512 pixels. Thus, it can be inferred that increasing the number of pixels will also increase the contrast of the reconstructed image. The second case, a grayscale image (Cameraman) is utilized, which keeps the same simulation parameters as well as the above binary image (512 ∗ 512 pixels). The results shown in Fig. 8 prove that our experiment can be effectively applied to the case of grayscale images. In order to further verify the accuracy of the holographic reconstructed image, the error function RMSE is defined as
on the planes of the reconstruction distances of 900 mm, 1000 mm, and 1200 mm, and note that the holographic images obtained on the plane exceeding the focus range became blurred. It is preliminarily verified that the validity of the axilens which is applied to the holographic projection. And a set of relatively clearer images can be observed on multiple reconstruction planes along the focal depth of the axilens. As is evident, the contrast of the holographic image in the focal plane for the case of axilens is lower than for the case of Fresnel lens, and there is a trade-off between the depth of field and the image sharpness. In this section, two more cases are presented that provide more insight about our proposed method. The first case, the image size is set to 2048 ∗ 2048 pixels. The reconstructed images along the direction of light field propagation are shown in Fig. 7. It is clear to observe that these holographic reconstruction images have significantly higher
RMSE =
∑x , y [|I1 (x , y )| − |I2 (x , y )|]2 M×N
(3)
where |I2 (x , y )| is the intensity distribution of the reconstructed image, |I1 (x , y )| is the intensity distribution of the original image, and the 6
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has also been presented to verify the proposed method. The performances of different combined hologram with superimposing the diffractive axilens and Fresnel lens are compared. Finally, using our method, the approximate clear holographic images can be observed simultaneously on multiple planes in a long range area in comparison with conventional Fourier holographic projection.
image is M × N pixels. Different holographic images are calculated with step change of 50 mm at propagation distance from 500 mm to 2000 mm. The RMSE values with respect to the different propagation distance are shown in Fig. 9, which the programmable axilens of the focal depth Zg = 100 mm and Zg = 300 mm. For comparison, we also present the corresponding RMSE for a Fresnel lens that has the same aperture and a focal length of 900 mm. Fig. 9(a) and (b) show that increasing the number of pixels reduces the error value. As is evident, the DOF, over which the RMSE is lower for axilens than for the Fresnel lens at a long distance from the optical elements. Unfortunately, due to the characteristics of the axilens, it concentrates only a small fraction of the energy into the focused beam, which results in an extremely larger RMSE value than the Fresnel lens at the constant focal plane f0 = 900 mm. As shown in Fig. 9(c), in the case of grayscale image, the difference in error value is even more pronounced.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61605002, 61501001, 61301296), Natural Science Foundation of Anhui Province (1608085QF161), Natural Science Project of Anhui Higher Education Institutions of China (KJ2016A029). References
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A holographic projection system was constructed to verify the proposed method, the schematic diagram of the system is shown in Fig. 10. The LCoS (MD1280, Three-Five Systems Company) was used as a phase-only modulation SLM, which resolution is 1280 ∗ 1024 pixels, the pixel pitch is 12 μm . A green laser with a wavelength of 532 nm is used, after the laser is beam-expanded and collimated, a uniform plane light wave is obtained to illuminate the LCoS. A picture of the experimental setup is shown in Fig. 11. The phase-only hologram generated by the iterative Fourier algorithm is loaded onto the LCoS after superimposing the phase of the axilens or Fresnel lens. Along the direction of propagation of the light field, the holographic reconstructed images at different reconstructed planes from the LCoS are taken. We compared the performances of different combined hologram with superimposing the phase of the axilens and Fresnel lens. The experiment results are shown in Fig. 12. Fig. 12(a) shows reconstructed images which are taken at the distances of 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1100 mm, 1200 mm, and 1300 mm from the LCoS plane when the focal length of Fresnel lens is f0 = 900 mm; Fig. 12(b) gives the reconstructed images which are taken at 500 mm, 700 mm, 900 mm, 950 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, respectively, when the focal length of axilens is f0 = 900 mm, the focal depth is Zg = 100 mm. In the other case of the focal depth Zg = 300 mm, Fig. 12(c) shows the reconstructed images taken at the distances of 500 mm, 700 mm, 900 mm, 1000 mm, 1200 mm, 1400 mm, and 1600 mm, respectively. As is evident, the results of experimental holographic reconstructions basically confirm the results of the numerical simulations. In a case of Fresnel lens, a clear holographic image is formed only in the focal plane. In contrast, the approximate clear holographic images are formed in a long range area along the propagation direction of the light field when the axilens are applied to the holographic projection. Considering the phase-only LCoS with a limited fill factor, its single pixel structure is square, which causes some artifacts on the holographic images. In order to separate the diffraction orders, a linear phase term was added to the combined hologram. The elimination of zero-order light and multi-level reproduction is not the focus of this paper, only the +1 order image is observed in the experiment. 5. Conclusions In this paper, we discussed the implementation programmable axilens for holographic projection system. The large focal depth characteristics and the flexibility of this approach in different configurations of axilens are analyzed. Holographic projection system based on LCoS
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