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Holographic magnification system based on cylindrical lens
Accepted Manuscript Title: Holographic magnification system based on cylindrical lens Authors: Dan Xiao, Rong Wei, Su-Juan Liu, Qiong-Hua Wang PII: DOI: Reference:
S0030-4026(18)30908-2 https://doi.org/10.1016/j.ijleo.2018.06.103 IJLEO 61106
To appear in: Received date: Revised date: Accepted date:
9-10-2017 20-6-2018 21-6-2018
Please cite this article as: Xiao D, Wei R, Liu S-Juan, Wang Q-Hua, Holographic magnification system based on cylindrical lens, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.06.103 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Holographic magnification system based on cylindrical lens
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Dan Xiao, Rong Wei, Su-Juan Liu, and Qiong-Hua Wang*
(School of Electronic and Information Engineering, Sichuan University,Chengdu 610065, China)
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(*Corresponding author: [email protected])
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Abstract —
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Key words—
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In this paper, we propose a holographic magnification system based on a cylindrical lens. The cylindrical lens has the function of converge and diverge beam only in the sagittal direction. The digital cylindrical lens consists of digital orthogonal cylindrical lenses on a spatial light modulator (SLM) and a solid cylindrical lens, and the vertical zoom in the holographic magnification system can quickly be controlled. The vertical magnification can be adjusted by tuning the focal length of the two digital cylindrical lenses. In order to illustrate the capabilities of the system, some experimental results are given.
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holographic magnification system, cylindrical lens, spatial light modulator. Introduction
Holography can fully satisfy human perception of the depth-cue such as perspective, focus, binocular parallax and motion parallax. So, the holographic display is expected as the ultimate three-dimensional (3D) display [1]. In recent years,
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the rapid progresses in computer’s performance and photoelectric devices make it available using holography to display 3D images in free space, and we can obtain all of the depth cues. With the development of the spatial light modulator (SLM), some 3D display systems have already appeared by using the SLM as the key display unit [2-5]. However, one problem in computer-generated-holography (CGH), which is caused by the display performance of the SLM, is that the size of the reconstructed images is small [6]. So, at present, magnifying the reconstructed images is a common target in the holographic display. To enlarge the 3D images and the viewing zones, scientists have proposed a lot of techniques [7-15]. Some scholars [16, 17] use several SLMs and a plano convex lens with a very large diameter and long focal length in geometrical combination by a beam splitter for achieving a large image size and wide viewing zone simultaneously. In this system, the main function of the plano convex lens is the combination of the images reconstructed from different SLMs, not the enlargement of the 3D image. Other scholars [18, 19] propose the active tiling system in which the time-multiplexing and optically addressed SLM (OASLM) are employed, where an electrically addressed SLM is used to modulate the reconstructed optical field that is recorded in the OASLM by scanning block by block. However, high-speed SLMs are needed. Naturally, it is desirable for both the image size and viewing-zone angle to be large. In addition, since the demand for unidirectionally magnified images in medical filed, military field and display filed is increasing, it is necessary to study holographic magnification in a single direction. In order to increase the horizontal image size and the horizontal viewing-zone simultaneously, the number of pixels of the SLM in the horizontal direction must increase. Although it is desirable to increase the number of pixels of the SLM in the horizontal direction, the maximum number is limited for a single SLM in practice. Therefore, some scholars put some particularly fine pixel SLMs for electric view finder in a line to increase the number of the pixels in the horizontal direction [20, 21]. But in this case, the reconstructed image size and viewing zone become small in full-parallax holography since the number of SLM pixels are limited in the vertical direction. Note that an explanation for only the horizontal direction is provided since the vertical direction is exactly the same and horizontal parallax is more essential than vertical parallax for human beings. So, the vertical parallax is discarded, which makes the necessity for the number of SLM pixels to diffract the wavefront too large, and just magnify the vertical size of the reconstructed images. In recent years, some methods using optical zoomable holographic projection have also been proposed and attract more and more attention [22-25].
In this paper, we propose a holographic magnification system based on a cylindrical lens. By using a combination of digital orthogonal cylindrical lenses on a SLM and a solid cylindrical lens, we can control vertical zoom in the holographic magnification system very quickly. The vertical magnification can be adjusted by tuning the focal length of the two digital orthogonal cylindrical lenses.
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Structure and operating principle In the wave optics, the optical element realizes the modulation of the wavefront by the action of the amplitude and phase of the light wave. The different delay time of the lens, because of the incident light through the change of the thickness of the lens at different positions, can realize the modulation of the incident light wave. As shown in Fig. 1, the point light source emits divergent waves and converges to a point after through the cylindrical lens L. And P1, P2 respectively represent the front and rear plane of the cylindrical lens. The height of the incident point (Q) on the P1 plane is the same as the exit point (Q’) on the P2 plane. Thus, the phase modulation of the transverse cylindrical lens can be expressed as:
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h( x, y) exp[ j ( x, y)] exp[ jkL( x, y)] ,
(1)
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where L(x, y) denotes the optical path difference between Q and Q’, which is expressed as
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L( x, y ) n( x, y ) [ 0 ( x, y)] 0 (n 1)( x, y)
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(2)
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where ∆0 represents the maximum thickness of the lens and ∆(x, y) denotes the thickness between the Q and Q’. According to Eqs. (1) and (2), h(x, y) can be shown in:
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h( x, y) exp[ jk 0 ] exp[ jk (n 1)( x, y))] ,
(3)
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Fig. 1 Light path diffidence of the cylindrical lens.
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In Fig. 1, ∆(x, y) also can be shown in:
( x, y ) 0 ( R R 2 y 2 )
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y2 0 R(1 1 2 ) R
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Under the paraxial approximation, the thickness of the lens can be transformed into:
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( x, y ) 0
y2 , 2R
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And the phase delay h(x, y) in Eq. (3), when the light passes through the lens, can be expressed as y2 h( x, y ) exp[ jk 0 ] exp[ jk ( n 1) )] 2R , 2 ky exp[ jk 0 ] exp( j ) 2 fy
(6)
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where fy represents the focal length of the horizontal cylindrical lens. The first term of Eq. (6), which contributes the phase distribution of the whole light-wave, does not affect the relative spatial distribution of the phase. Therefore, we can ignore it when analyzing. The second term, also is called a modulation term, can change the relative spatial distribution of the phase. By analyzing the phase modulation of the horizontal cylindrical lens from Eq. (6), the phase modulation characteristic of the lens can be expressed as:
h( x, y ) exp( j
ky 2 ), 2 fy
(7)
Similarly, the phase modulation characteristic of the vertical cylindrical lens can be expressed as: h( x, y ) exp( j
kx 2 ), 2 fx
(8)
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where fx represents the focal length of the vertical cylindrical lens. In order to realize a holographic magnification system, we add a solid cylindrical lens to a Fourier holographic display system in which two digital cylindrical lenses encodes lens power. Fig. 2 shows the structure of the designed holographic magnification system consisting of a blue laser (λ=473nm), a filter, a solid lens, a SLM, a computer, a solid horizontal cylindrical lens and a receiving screen. The light source with blue color illuminates the filter and then passes through the solid lens and changes into a uniform plane light from a spherical light. Then the plane light illuminates the reflective SLM loaded on the hologram coded with two digital orthogonal cylindrical lenses and passes through the solid cylindrical lens. The polarization of the laser beam is set parallel to the rubbing direction of the SLM and the solid horizontal cylindrical lens. The system includes two electrically tuning lenses: one is the digital horizontal cylindrical lens and the other is the digital vertical cylindrical lens. The Fourier transform of the hologram displayed on the SLM will be formed in the effective back focal plane due to Fraunhofer diffraction. The solid horizontal cylindrical lens reimages the pattern to the receiving screen. In the proposed system, the SLM and the solid horizontal cylindrical lens constitute the central part. They form the zoom modules in the system. We can control the vertical magnification in the holographic magnification system very quickly by tuning the focal length of the two digital orthogonal cylindrical lenses.
Fig. 2 Structure of the holographic magnification system (side view).
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The analysis of the zoom modules in our holographic magnification system is shown in Fig. 3. The image planes for the horizontal and vertical axes must be coincident. For the vertical axis, the system can be represented as a two-lens system. However, for the horizontal axis, the system is a single-lens system since the solid horizontal cylindrical lens has no effect. P is the light source, P1 is the horizontal image location after the digital orthogonal cylindrical lenses, P2 is the vertical image location after the digital orthogonal cylindrical lenses, the receiving screen is the reimage location after the solid cylindrical lens, d1 is the distance between the light source position P and the digital orthogonal cylindrical lenses, d2 is the distance between the digital orthogonal cylindrical lenses and the solid cylindrical lens, d3 is the distance between the solid cylindrical lens and the receiving screen, f1 is the focal length of the digital horizontal cylindrical lens in the orthogonal cylindrical lenses, f2 is the focal length of the digital vertical cylindrical lens in the orthogonal cylindrical lenses and f3 is the focal length of the solid cylindrical lens. In our system, the light source is collimated, so d1 equals infinity.
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Fig. 3 Zoom modules of the holographic magnification system with different magnifications in the (a) horizontal and (b) vertical directions.
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A cylindrical lens has the effect of phase transformation, and its phase modulation can be described as: (9) ( x, y, , f ) ( x s i n y c o s ) 2 , f where λ is the wavelength, f is the focal length of the cylindrical lens and is the angle between the axle of the cylindrical lens and the horizontal direction. Then we
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can express the phase factor introduced by a set of orthogonal cylindrical lenses: ( x, y, , f1 , f 2 )
( x s i n y c o s ) 2 ( x s i n y c o s ) 2 , f1 f 2
(10)
where θ is the angle between the axle of the arbitrary cylindrical lens in the orthogonal cylindrical lenses and the horizontal direction, and f1 and f2 are the different focal lengths of the cylindrical lenses in the orthogonal cylindrical lenses. In order to achieve the function of the digital orthogonal cylindrical lenses using the SLM, the
phase distribution of the phase modulation diagram on the SLM satisfies the following equation: ( x, y, , f1 , f 2 ) mod 2 [
where
( x sin y cos ) 2 ( x sin y cos ) 2 ], f1 f 2
(11)
means the modulo operation. We calculate the phases of the object and
mod 2
y 2 x 2 ], f1 f 2
(12)
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0 [
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the digital orthogonal cylindrical lenses respectively and add them together to get a new phase hologram. When θ equals to 0° and 90°, respectively, and the new phase can be expressed as follows:
where 0 is the phase of the object on the hologram, f1 is the focal length of the
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eikf1 i iku 2 i 2 exp[ ( x 2 y 2 )] {U (u, v) exp( ) exp[ ( xu yv )]}dudv, if1 f1 2 f1 f
(13)
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horizontal cylindrical lens, and f2 is the focal length of the vertical cylindrical lens. According to the diffraction theory, the complex amplitude distribution in the horizontal direction behind the SLM is
The complex amplitude distribution in the vertical direction behind the SLM is eikf 2 i 2 ikv2 i 2 exp[ ( x y 2 )] {U (u, v) exp( ) exp[ ( xu yv )]}dudv. (14) if 2 f 2 2 f2 f
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In Fig. 2, the distance between the digital orthogonal cylindrical lenses and the horizontal image P1 is f1, the distance between the digital orthogonal cylindrical lenses and the vertical image P2 is f2, the size of the reconstructed image in horizontal is hx f1 / p (p is the pixel pitch of the SLM), and the size of the reconstructed image
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in vertical is hy f 2 / p . In this case, the object position of the solid horizontal cylindrical lens is s2 y d 2 f 2 , and the image position of the solid cylindrical lens
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satisfies the following equation:
1 1 1 . d3 d 2 f 2 f3
(15)
From the above formula, we can see that when the SLM and receiving screen remain stationary, the value of f 2 is determined by f 3 , and distinctly f1 d 2 d 3 . The
lateral magnification of the solid horizontal cylindrical lens is M y d3 /( d 2 f 2 ) , so the size of the reconstructed image in the vertical direction on the receiving screen is hy' M y hy
f 2 d3 p( f 2 d 2 )
.
(16)
hx' hx
f1 p
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The solid horizontal cylindrical lens expands the size of the image only in the vertical direction, and it is served as a plate glass in the horizontal direction. So, in the horizontal direction, the image size on the receiving screen is (17)
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In our system, we can adjust the focal length of the digital orthogonal cylindrical lenses encoded to the hologram loaded on the SLM. According to the above equations, we can adjust the vertical magnification of the reconstructed image.
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Experiment and discussion We have a basic experiment to confirm that a digital cylindrical lens can just expand the size of the image in only one direction. With the purpose of illustrating the capabilities of the digital cylindrical lens we performed some experiences just using a blue laser, a filter, a SLM, two solid lenses and a receiving screen. The blue laser, the filter and the solid lens are used to generate a collimated light source. Then the light
0 0 and 90 0 are shown in Fig. 4. It is easy to see that if
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cylindrical lens with
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source illuminates the SLM. The digital cylindrical lens loaded on the SLM modulates the light by effecting its phase. The light passes through the solid lens sitten behind the SLM to focus on the receiving screen. The holograms of the digital
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00 , we recover the expression for a horizontal cylindrical lens. If 900 , we recover the expression for a vertical cylindrical lens. Fig. 4(a) is the hologram of the
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horizontal cylindrical lens with focal length 50cm, and Fig. 4(b) is the hologram of the vertical cylindrical lens with focal length 50cm.
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Fig. 4 Holograms of the (a) horizontal and (b) vertical cylindrical lenses.
The digital horizontal and vertical cylindrical lenses are loaded on the SLM. Three different results are shown in Fig. 4. Fig. 5(a) is the reconstructed image
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without any digital lens loaded on the SLM and Fig. 5(b) is the reconstructed image with the digital horizontal cylindrical lens. Fig. 5(c) is the reconstructed image with the digital vertical digital cylindrical lens.
Fig. 5 Experimental images shown on the receiving screen by means of the digital cylindrical lens: (a) No digital lens loaded on the SLM; (b) digital horizontal cylindrical lens; (c) digital vertical cylindrical lens.
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The vertical stretching is shown in Fig. 5(b) and the horizontal stretching is shown in Fig. 5(c). So, the digital horizontal cylindrical lens can make images stretching vertically. The digital vertical cylindrical lens can make images stretching horizontally. To demonstrate a holographic magnification system based on a cylindrical lens, we adopt the reflective SLM with the pixel number 1920×1080 and with pixel pitch 8 μm. The solid horizontal cylindrical lens is placed behind the SLM. The distance between the SLM and the solid cylindrical lens is 10cm. The focal length of the solid cylindrical lens is 40cm, because only one solid cylindrical lens is available. We first calculate the phase-only hologram of the image by Gaschberg-Saxton (GS) algorithm. Fig. 6 (a) is the phase-only hologram of the original image, and Fig. 6(b) is the hologram of the digital orthogonal cylindrical lenses which have a positive focal length f1 in the x direction and a different positive focal length f2 in the y direction, so a new phase hologram can be generated by combining the hologram of the origin image with that of the digital orthogonal cylindrical lenses, as shown in Fig. 6(c). Then the new phase hologram is loaded on the SLM.
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(c) Fig. 6 Process of the holograms: (a) Hologram of the original image, (b) hologram of the
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digital orthogonal cylindrical lenses, (c) combined hologram.
We put corresponding parameters into Eq. (7). We get d2=10cm and d3=f1-d2. According to Eqs. (8)-(9), we can get the following equation: d 2 d 3 (d 2 d 3 ) f 3 40( f1 10) 10 . d3 f 3 f1 50
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When f1 is given, we can get f2 from Eq. (10) and f1= d2+d3. When the focal length of the digital horizontal cylindrical lens changes, the focal length of the digital vertical cylindrical lens can be calculated correspondingly according to Eq. (18). The focal length of the digital horizontal cylindrical lens can range from 20cm to 80cm, and we can change it by programming. Fig. 6 is the reconstructed result when the digital orthogonal cylindrical lenses with different focal lengths are loaded on the SLM. From the results we can see that the images can be changed. We can see the full images from Figs. 7(a)-(f).
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Fig. 7 Results of the holographic display system using a solid lens with (a) f1=35cm, M’x=1, and results of the proposed holographic magnification system with, (b) f1=40cm, M’x=3, (c) f1=20cm, M’x=4, (d) f2=80cm, M’y=4, (e) f2=70cm, M’y=4.5, (f) f1=60cm, M’y=5.
From the results we can see that by changing the focal lens of the digital orthogonal cylindrical lenses loaded on the SLM, the magnification of the reconstructed image changes. By measuring the change of the width of the image in the x direction or in the y direction, we can measure the magnification M x' and M y' .
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We define unit magnification of the observed image when the focal length of the digital horizontal cylindrical lens is 90cm and the focal length of the digital vertical cylindrical lens is 50cm, respectively. The zoom ratio is defined as the ratio of the magnification to be the unit magnification. The proposed system still has some unsolved issues. Since rings cannot be encoded when the ring spacing is less than the pixel spacing [26], the minimum focal
length is f N . Where N is the smaller number of pixels in the x or y direction, 2
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=8μm and λ=473nm, the minimum focal length is f=14.2cm.
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pixels with
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is the pixel spacing, and is the wavelength. For an SLM that has 1920 1080
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Programming focal lengths shorter than this Nyquist limit directs energy into replica images that are formed because of diffraction by the SLM. Finally, the number of
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pixels on the SLM is limited, which affects the resolution of the optical element. In the other hand, in order to adjust the vertical focus plane to the horizontal image field we use a solid cylindrical lens, so the quality of the reconstructed image decreases. And we discard the vertical parallax. So, in the next work, we will use lenticular sheet to expand the vertical viewing zone. We can revise the aspect ratio of the data to get the desired reconstructed holographic image. Then we will try to set up multiple SLMs continuously in the horizontal direction to expand the size and the viewing zone of the reconstructed image in the horizontal direction and use the proposed holographic magnification system combined with SLMs loaded with digital orthogonal cylindrical lenses and solid cylindrical lens to expand the size of
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reconstructed image in the vertical direction. We will try to solve the problems mentioned above, which can make our system better. Conclusion We propose a holographic magnification system based on a cylindrical lens. In the proposed system, the SLM and the solid horizontal cylindrical lens constitute the zoom modules. We can control the vertical magnification in the holographic
magnification system very quickly by tuning the focal length of the two digital orthogonal cylindrical lenses loaded on the SLM. The magnification in the vertical direction can increase from 1 to 5 times. In short, the proposed system is simple and flexible, which has wide applications in holographic displays. Acknowledgements
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This work is supported by National Key R&D Program of China under Grant No. 2017YFB1002900 and the Equipment Research Program in Advance of China under Grant No. JZX2016-0293/Y031-1.
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S0030-4026(18)30908-2 https://doi.org/10.1016/j.ijleo.2018.06.103 IJLEO 61106
To appear in: Received date: Revised date: Accepted date:
9-10-2017 20-6-2018 21-6-2018
Please cite this article as: Xiao D, Wei R, Liu S-Juan, Wang Q-Hua, Holographic magnification system based on cylindrical lens, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.06.103 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Holographic magnification system based on cylindrical lens
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Dan Xiao, Rong Wei, Su-Juan Liu, and Qiong-Hua Wang*
(School of Electronic and Information Engineering, Sichuan University,Chengdu 610065, China)
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(*Corresponding author: [email protected])
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Abstract —
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Key words—
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In this paper, we propose a holographic magnification system based on a cylindrical lens. The cylindrical lens has the function of converge and diverge beam only in the sagittal direction. The digital cylindrical lens consists of digital orthogonal cylindrical lenses on a spatial light modulator (SLM) and a solid cylindrical lens, and the vertical zoom in the holographic magnification system can quickly be controlled. The vertical magnification can be adjusted by tuning the focal length of the two digital cylindrical lenses. In order to illustrate the capabilities of the system, some experimental results are given.
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holographic magnification system, cylindrical lens, spatial light modulator. Introduction
Holography can fully satisfy human perception of the depth-cue such as perspective, focus, binocular parallax and motion parallax. So, the holographic display is expected as the ultimate three-dimensional (3D) display [1]. In recent years,
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the rapid progresses in computer’s performance and photoelectric devices make it available using holography to display 3D images in free space, and we can obtain all of the depth cues. With the development of the spatial light modulator (SLM), some 3D display systems have already appeared by using the SLM as the key display unit [2-5]. However, one problem in computer-generated-holography (CGH), which is caused by the display performance of the SLM, is that the size of the reconstructed images is small [6]. So, at present, magnifying the reconstructed images is a common target in the holographic display. To enlarge the 3D images and the viewing zones, scientists have proposed a lot of techniques [7-15]. Some scholars [16, 17] use several SLMs and a plano convex lens with a very large diameter and long focal length in geometrical combination by a beam splitter for achieving a large image size and wide viewing zone simultaneously. In this system, the main function of the plano convex lens is the combination of the images reconstructed from different SLMs, not the enlargement of the 3D image. Other scholars [18, 19] propose the active tiling system in which the time-multiplexing and optically addressed SLM (OASLM) are employed, where an electrically addressed SLM is used to modulate the reconstructed optical field that is recorded in the OASLM by scanning block by block. However, high-speed SLMs are needed. Naturally, it is desirable for both the image size and viewing-zone angle to be large. In addition, since the demand for unidirectionally magnified images in medical filed, military field and display filed is increasing, it is necessary to study holographic magnification in a single direction. In order to increase the horizontal image size and the horizontal viewing-zone simultaneously, the number of pixels of the SLM in the horizontal direction must increase. Although it is desirable to increase the number of pixels of the SLM in the horizontal direction, the maximum number is limited for a single SLM in practice. Therefore, some scholars put some particularly fine pixel SLMs for electric view finder in a line to increase the number of the pixels in the horizontal direction [20, 21]. But in this case, the reconstructed image size and viewing zone become small in full-parallax holography since the number of SLM pixels are limited in the vertical direction. Note that an explanation for only the horizontal direction is provided since the vertical direction is exactly the same and horizontal parallax is more essential than vertical parallax for human beings. So, the vertical parallax is discarded, which makes the necessity for the number of SLM pixels to diffract the wavefront too large, and just magnify the vertical size of the reconstructed images. In recent years, some methods using optical zoomable holographic projection have also been proposed and attract more and more attention [22-25].
In this paper, we propose a holographic magnification system based on a cylindrical lens. By using a combination of digital orthogonal cylindrical lenses on a SLM and a solid cylindrical lens, we can control vertical zoom in the holographic magnification system very quickly. The vertical magnification can be adjusted by tuning the focal length of the two digital orthogonal cylindrical lenses.
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Structure and operating principle In the wave optics, the optical element realizes the modulation of the wavefront by the action of the amplitude and phase of the light wave. The different delay time of the lens, because of the incident light through the change of the thickness of the lens at different positions, can realize the modulation of the incident light wave. As shown in Fig. 1, the point light source emits divergent waves and converges to a point after through the cylindrical lens L. And P1, P2 respectively represent the front and rear plane of the cylindrical lens. The height of the incident point (Q) on the P1 plane is the same as the exit point (Q’) on the P2 plane. Thus, the phase modulation of the transverse cylindrical lens can be expressed as:
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h( x, y) exp[ j ( x, y)] exp[ jkL( x, y)] ,
(1)
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where L(x, y) denotes the optical path difference between Q and Q’, which is expressed as
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L( x, y ) n( x, y ) [ 0 ( x, y)] 0 (n 1)( x, y)
,
(2)
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where ∆0 represents the maximum thickness of the lens and ∆(x, y) denotes the thickness between the Q and Q’. According to Eqs. (1) and (2), h(x, y) can be shown in:
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h( x, y) exp[ jk 0 ] exp[ jk (n 1)( x, y))] ,
(3)
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Fig. 1 Light path diffidence of the cylindrical lens.
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In Fig. 1, ∆(x, y) also can be shown in:
( x, y ) 0 ( R R 2 y 2 )
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y2 0 R(1 1 2 ) R
,
(4)
Under the paraxial approximation, the thickness of the lens can be transformed into:
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( x, y ) 0
y2 , 2R
(5)
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And the phase delay h(x, y) in Eq. (3), when the light passes through the lens, can be expressed as y2 h( x, y ) exp[ jk 0 ] exp[ jk ( n 1) )] 2R , 2 ky exp[ jk 0 ] exp( j ) 2 fy
(6)
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where fy represents the focal length of the horizontal cylindrical lens. The first term of Eq. (6), which contributes the phase distribution of the whole light-wave, does not affect the relative spatial distribution of the phase. Therefore, we can ignore it when analyzing. The second term, also is called a modulation term, can change the relative spatial distribution of the phase. By analyzing the phase modulation of the horizontal cylindrical lens from Eq. (6), the phase modulation characteristic of the lens can be expressed as:
h( x, y ) exp( j
ky 2 ), 2 fy
(7)
Similarly, the phase modulation characteristic of the vertical cylindrical lens can be expressed as: h( x, y ) exp( j
kx 2 ), 2 fx
(8)
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where fx represents the focal length of the vertical cylindrical lens. In order to realize a holographic magnification system, we add a solid cylindrical lens to a Fourier holographic display system in which two digital cylindrical lenses encodes lens power. Fig. 2 shows the structure of the designed holographic magnification system consisting of a blue laser (λ=473nm), a filter, a solid lens, a SLM, a computer, a solid horizontal cylindrical lens and a receiving screen. The light source with blue color illuminates the filter and then passes through the solid lens and changes into a uniform plane light from a spherical light. Then the plane light illuminates the reflective SLM loaded on the hologram coded with two digital orthogonal cylindrical lenses and passes through the solid cylindrical lens. The polarization of the laser beam is set parallel to the rubbing direction of the SLM and the solid horizontal cylindrical lens. The system includes two electrically tuning lenses: one is the digital horizontal cylindrical lens and the other is the digital vertical cylindrical lens. The Fourier transform of the hologram displayed on the SLM will be formed in the effective back focal plane due to Fraunhofer diffraction. The solid horizontal cylindrical lens reimages the pattern to the receiving screen. In the proposed system, the SLM and the solid horizontal cylindrical lens constitute the central part. They form the zoom modules in the system. We can control the vertical magnification in the holographic magnification system very quickly by tuning the focal length of the two digital orthogonal cylindrical lenses.
Fig. 2 Structure of the holographic magnification system (side view).
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The analysis of the zoom modules in our holographic magnification system is shown in Fig. 3. The image planes for the horizontal and vertical axes must be coincident. For the vertical axis, the system can be represented as a two-lens system. However, for the horizontal axis, the system is a single-lens system since the solid horizontal cylindrical lens has no effect. P is the light source, P1 is the horizontal image location after the digital orthogonal cylindrical lenses, P2 is the vertical image location after the digital orthogonal cylindrical lenses, the receiving screen is the reimage location after the solid cylindrical lens, d1 is the distance between the light source position P and the digital orthogonal cylindrical lenses, d2 is the distance between the digital orthogonal cylindrical lenses and the solid cylindrical lens, d3 is the distance between the solid cylindrical lens and the receiving screen, f1 is the focal length of the digital horizontal cylindrical lens in the orthogonal cylindrical lenses, f2 is the focal length of the digital vertical cylindrical lens in the orthogonal cylindrical lenses and f3 is the focal length of the solid cylindrical lens. In our system, the light source is collimated, so d1 equals infinity.
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Fig. 3 Zoom modules of the holographic magnification system with different magnifications in the (a) horizontal and (b) vertical directions.
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A cylindrical lens has the effect of phase transformation, and its phase modulation can be described as: (9) ( x, y, , f ) ( x s i n y c o s ) 2 , f where λ is the wavelength, f is the focal length of the cylindrical lens and is the angle between the axle of the cylindrical lens and the horizontal direction. Then we
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can express the phase factor introduced by a set of orthogonal cylindrical lenses: ( x, y, , f1 , f 2 )
( x s i n y c o s ) 2 ( x s i n y c o s ) 2 , f1 f 2
(10)
where θ is the angle between the axle of the arbitrary cylindrical lens in the orthogonal cylindrical lenses and the horizontal direction, and f1 and f2 are the different focal lengths of the cylindrical lenses in the orthogonal cylindrical lenses. In order to achieve the function of the digital orthogonal cylindrical lenses using the SLM, the
phase distribution of the phase modulation diagram on the SLM satisfies the following equation: ( x, y, , f1 , f 2 ) mod 2 [
where
( x sin y cos ) 2 ( x sin y cos ) 2 ], f1 f 2
(11)
means the modulo operation. We calculate the phases of the object and
mod 2
y 2 x 2 ], f1 f 2
(12)
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0 [
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the digital orthogonal cylindrical lenses respectively and add them together to get a new phase hologram. When θ equals to 0° and 90°, respectively, and the new phase can be expressed as follows:
where 0 is the phase of the object on the hologram, f1 is the focal length of the
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eikf1 i iku 2 i 2 exp[ ( x 2 y 2 )] {U (u, v) exp( ) exp[ ( xu yv )]}dudv, if1 f1 2 f1 f
(13)
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U x ( x, y)
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horizontal cylindrical lens, and f2 is the focal length of the vertical cylindrical lens. According to the diffraction theory, the complex amplitude distribution in the horizontal direction behind the SLM is
The complex amplitude distribution in the vertical direction behind the SLM is eikf 2 i 2 ikv2 i 2 exp[ ( x y 2 )] {U (u, v) exp( ) exp[ ( xu yv )]}dudv. (14) if 2 f 2 2 f2 f
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U y ( x, y )
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In Fig. 2, the distance between the digital orthogonal cylindrical lenses and the horizontal image P1 is f1, the distance between the digital orthogonal cylindrical lenses and the vertical image P2 is f2, the size of the reconstructed image in horizontal is hx f1 / p (p is the pixel pitch of the SLM), and the size of the reconstructed image
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in vertical is hy f 2 / p . In this case, the object position of the solid horizontal cylindrical lens is s2 y d 2 f 2 , and the image position of the solid cylindrical lens
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satisfies the following equation:
1 1 1 . d3 d 2 f 2 f3
(15)
From the above formula, we can see that when the SLM and receiving screen remain stationary, the value of f 2 is determined by f 3 , and distinctly f1 d 2 d 3 . The
lateral magnification of the solid horizontal cylindrical lens is M y d3 /( d 2 f 2 ) , so the size of the reconstructed image in the vertical direction on the receiving screen is hy' M y hy
f 2 d3 p( f 2 d 2 )
.
(16)
hx' hx
f1 p
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The solid horizontal cylindrical lens expands the size of the image only in the vertical direction, and it is served as a plate glass in the horizontal direction. So, in the horizontal direction, the image size on the receiving screen is (17)
.
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In our system, we can adjust the focal length of the digital orthogonal cylindrical lenses encoded to the hologram loaded on the SLM. According to the above equations, we can adjust the vertical magnification of the reconstructed image.
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Experiment and discussion We have a basic experiment to confirm that a digital cylindrical lens can just expand the size of the image in only one direction. With the purpose of illustrating the capabilities of the digital cylindrical lens we performed some experiences just using a blue laser, a filter, a SLM, two solid lenses and a receiving screen. The blue laser, the filter and the solid lens are used to generate a collimated light source. Then the light
0 0 and 90 0 are shown in Fig. 4. It is easy to see that if
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cylindrical lens with
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source illuminates the SLM. The digital cylindrical lens loaded on the SLM modulates the light by effecting its phase. The light passes through the solid lens sitten behind the SLM to focus on the receiving screen. The holograms of the digital
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00 , we recover the expression for a horizontal cylindrical lens. If 900 , we recover the expression for a vertical cylindrical lens. Fig. 4(a) is the hologram of the
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horizontal cylindrical lens with focal length 50cm, and Fig. 4(b) is the hologram of the vertical cylindrical lens with focal length 50cm.
(a)
(b)
Fig. 4 Holograms of the (a) horizontal and (b) vertical cylindrical lenses.
The digital horizontal and vertical cylindrical lenses are loaded on the SLM. Three different results are shown in Fig. 4. Fig. 5(a) is the reconstructed image
(b)
(c)
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without any digital lens loaded on the SLM and Fig. 5(b) is the reconstructed image with the digital horizontal cylindrical lens. Fig. 5(c) is the reconstructed image with the digital vertical digital cylindrical lens.
Fig. 5 Experimental images shown on the receiving screen by means of the digital cylindrical lens: (a) No digital lens loaded on the SLM; (b) digital horizontal cylindrical lens; (c) digital vertical cylindrical lens.
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The vertical stretching is shown in Fig. 5(b) and the horizontal stretching is shown in Fig. 5(c). So, the digital horizontal cylindrical lens can make images stretching vertically. The digital vertical cylindrical lens can make images stretching horizontally. To demonstrate a holographic magnification system based on a cylindrical lens, we adopt the reflective SLM with the pixel number 1920×1080 and with pixel pitch 8 μm. The solid horizontal cylindrical lens is placed behind the SLM. The distance between the SLM and the solid cylindrical lens is 10cm. The focal length of the solid cylindrical lens is 40cm, because only one solid cylindrical lens is available. We first calculate the phase-only hologram of the image by Gaschberg-Saxton (GS) algorithm. Fig. 6 (a) is the phase-only hologram of the original image, and Fig. 6(b) is the hologram of the digital orthogonal cylindrical lenses which have a positive focal length f1 in the x direction and a different positive focal length f2 in the y direction, so a new phase hologram can be generated by combining the hologram of the origin image with that of the digital orthogonal cylindrical lenses, as shown in Fig. 6(c). Then the new phase hologram is loaded on the SLM.
(a)
(b)
(c) Fig. 6 Process of the holograms: (a) Hologram of the original image, (b) hologram of the
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digital orthogonal cylindrical lenses, (c) combined hologram.
We put corresponding parameters into Eq. (7). We get d2=10cm and d3=f1-d2. According to Eqs. (8)-(9), we can get the following equation: d 2 d 3 (d 2 d 3 ) f 3 40( f1 10) 10 . d3 f 3 f1 50
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f2
(18)
(a)
(b)
(c)
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When f1 is given, we can get f2 from Eq. (10) and f1= d2+d3. When the focal length of the digital horizontal cylindrical lens changes, the focal length of the digital vertical cylindrical lens can be calculated correspondingly according to Eq. (18). The focal length of the digital horizontal cylindrical lens can range from 20cm to 80cm, and we can change it by programming. Fig. 6 is the reconstructed result when the digital orthogonal cylindrical lenses with different focal lengths are loaded on the SLM. From the results we can see that the images can be changed. We can see the full images from Figs. 7(a)-(f).
(d)
(e)
(f)
Fig. 7 Results of the holographic display system using a solid lens with (a) f1=35cm, M’x=1, and results of the proposed holographic magnification system with, (b) f1=40cm, M’x=3, (c) f1=20cm, M’x=4, (d) f2=80cm, M’y=4, (e) f2=70cm, M’y=4.5, (f) f1=60cm, M’y=5.
From the results we can see that by changing the focal lens of the digital orthogonal cylindrical lenses loaded on the SLM, the magnification of the reconstructed image changes. By measuring the change of the width of the image in the x direction or in the y direction, we can measure the magnification M x' and M y' .
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We define unit magnification of the observed image when the focal length of the digital horizontal cylindrical lens is 90cm and the focal length of the digital vertical cylindrical lens is 50cm, respectively. The zoom ratio is defined as the ratio of the magnification to be the unit magnification. The proposed system still has some unsolved issues. Since rings cannot be encoded when the ring spacing is less than the pixel spacing [26], the minimum focal
length is f N . Where N is the smaller number of pixels in the x or y direction, 2
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=8μm and λ=473nm, the minimum focal length is f=14.2cm.
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pixels with
N
is the pixel spacing, and is the wavelength. For an SLM that has 1920 1080
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Programming focal lengths shorter than this Nyquist limit directs energy into replica images that are formed because of diffraction by the SLM. Finally, the number of
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pixels on the SLM is limited, which affects the resolution of the optical element. In the other hand, in order to adjust the vertical focus plane to the horizontal image field we use a solid cylindrical lens, so the quality of the reconstructed image decreases. And we discard the vertical parallax. So, in the next work, we will use lenticular sheet to expand the vertical viewing zone. We can revise the aspect ratio of the data to get the desired reconstructed holographic image. Then we will try to set up multiple SLMs continuously in the horizontal direction to expand the size and the viewing zone of the reconstructed image in the horizontal direction and use the proposed holographic magnification system combined with SLMs loaded with digital orthogonal cylindrical lenses and solid cylindrical lens to expand the size of
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reconstructed image in the vertical direction. We will try to solve the problems mentioned above, which can make our system better. Conclusion We propose a holographic magnification system based on a cylindrical lens. In the proposed system, the SLM and the solid horizontal cylindrical lens constitute the zoom modules. We can control the vertical magnification in the holographic
magnification system very quickly by tuning the focal length of the two digital orthogonal cylindrical lenses loaded on the SLM. The magnification in the vertical direction can increase from 1 to 5 times. In short, the proposed system is simple and flexible, which has wide applications in holographic displays. Acknowledgements
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This work is supported by National Key R&D Program of China under Grant No. 2017YFB1002900 and the Equipment Research Program in Advance of China under Grant No. JZX2016-0293/Y031-1.
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