Complex amplitude modulation method is presented theoretically and performed experimentally for three-dimensional (3D) dynamic holographic display with reduced speckle using a single phase-only spatial light modulator. The determination of essential factors is discussed based on the basic principle and theory. The numerical simulations and optical experiments are performed, where the static and animated objects without refinement on the surfaces and without random initial phases are reconstructed successfully. The results indicate that this method can reduce the speckle in reconstructed images effectively; furthermore, it will not cause the internal structure in the reconstructed pixels. Since the complex amplitude modulation is based on the principle of phase-only hologram, it does not need the stringent alignment of pixels. This method can be used for high resolution imaging or measurement in various optical areas.
© 2013 Optical Society of America
Dynamic holographic three-dimensional (3D) display is one of the most promising 3D display techniques for providing all depth cues of human visual system. It is well known that the basic principle of holographic display is the reconstruction of the wavefront of objects . With the combination of laser technique, optoelectronic technology and computer technology, 3D dynamic holographic display can be realized via the reconstruction of wave-front by computer-generated holograms (CGHs). Different algorithms have been proposed to obtain CGHs. According to 3D physical model, it is usually classified into two groups: point source methods , and polygon-based methods . Point source methods deal with each point from 3D object as a simple spherical wave while they lack solid sense of 3D reconstructed object. Polygon-based methods are a little bit more complicated but easier to achieve reasonable 3D rendering effect.
The reconstructed image quality is influenced by the CGH and the performance of the optoelectronic device. As is well known, owing to the basic feature of liquid crystal materials , modulation plot for the optoelectronic device, normally it is the Spatial Light Modulator (SLM) (it is described by a polar plot showing the amplitude transmission versus the phase modulation) where the polar plot is a line segment or a curve rather than the whole region; hence one cannot achieve the modulation of the amplitude and the phase simultaneously and independently by loading holograms on a single SLM directly. Kinoform is widely used because of its high diffraction efficiency, but there exist several problems: firstly, it requires the objects are diffusers with random phase on the surface, which causes speckle noise ; secondly, the profile structure of the 3D objects is reconstructed whereas the precise structure is blurred or lost if analytical polygon-based methods  are employed in calculating a kinoform. According to the diffraction integral, the amplitude distribution on hologram plane for each polygon with constant initial phase cannot be regarded as homogeneous one . Therefore, if the CGH is a kinoform, the texture on the smooth object surface will be lost. One solution is to refine each polygon, which can be achieved by necessary subdivision of each one , but it is too time-consuming to be applied in dynamic display.
Another solution to realize the reconstruction is to achieve the complex amplitude modulation. Several methods have been reported in literatures [8–16]. Two phase-only CGHs , used as two phase filters, were employed to achieve complex holograms reconstruction by aligning them together. Afterwards, J. Amako and L. G. Neto [9,10] controlled wavefront with amplitude and phase modulation by the coupled mode SLMs respectively according to the basic principle in Ref . It is obvious that it makes the system much bigger and more complex than others for holographic display. The superpixel methods [11–13] are presented where a few pixels of a SLM are combined together to act as one “superpixel” for complex amplitude modulation. According to the Whittaker-Shannon sampling theorem , the basic ideas of those methods are resampling the hologram on SLM, which are realized by a “4f” filtering architecture. However, owing to the combination of neighboring pixels to form a superpixel, the number of pixels on SLM is decreased. Meanwhile, although the holograms on SLM are resampled, there is an internal structure in each “superpixel”(because according to the sampling theorem, resampling will not change the original function and the shifting information of each “sub-pixel” is remained), which may result in spatial shift, affecting the quality of display. Furthermore, only a single phase-only SLM  is used where half of the SLM realizes the amplitude modulation and the other half realizes the phase modulation. J. Liu and H. Song synthesized complex optical field by the sum of two holograms [15,16]. Those techniques require a stringent alignment condition of the system for pixel-to-pixel superimposition.
In this paper, we use a single SLM to perform the complex amplitude modulation where the 3D objects with constant initial phase and the dynamic holographic display of the object with smooth surface are reconstructed successfully. The spatial shift of the modulated light wave and the stringent alignment of pixels are avoided.
2. Basic principle
It is well-known that the wavefront can be obtained by complex amplitude reconstruction, which can be encoded into a complex CGH. Only the necessary terms in hologram equations are taken into account. The diffraction efficiency of phase-only hologram is higher than others , so that it is an appropriate candidate to be applied to holographic display. After obtaining the complex amplitude distribution of object beam on hologram plane, the complex amplitude transmittance of phase-only hologram can be described as :17]17], and only the first term is picked up generally, so that the diffraction beam of −1 order will be given as
Since is variable, as stated in the distribution of Bessel functions, a perfect elimination of zero order and high orders of diffraction cannot be achieved, but an optimized coefficient can limit other orders effectively (high orders especially), so that the main orders of diffraction beam are ± 1 order and zero order (including that of SLM). If the reference light is tilted, −1 order beam can be picked up and reproduced, and the filtering architecture (including “4f” filtering architecture, set-up based on Abbe's theory of image formation, and etc .) can be introduced to filter out + 1 order and zero order; then a 3D high quality object image can be obtained since the complex amplitude is reconstructed by −1 order modulation. In addition, the quality of complex amplitude modulation is normally influenced by the zero-order beam introduced by a pixelated SLM, which causes a strong background noise . This problem is automatically avoided by our proposed method. The required diffraction order of reconstruction beam has been departed from the zero order which is superposed on the zero order of SLM. The influence can be eliminated by filtering other orders of the reconstructed light wave.
3. Assessment and verification
3.1 The simulated assessment
To evaluate the ability of complex amplitude modulation by the proposed method, first we perform the numerical simulation to demonstrate the amplitude and the phase modulation individually and simultaneously. At first, the target complex amplitude distribution on the hologram plane is calculated. Then considered as the complex amplitude distribution of object beam on hologram plane, it is encoded into a phase-only hologram according to Eq. (1). Based on the analysis in Sec. 2, the target complex amplitude will be reconstructed by our proposed method through the filtering system.
In simulated evaluation, the resolution of the CGH is set as 1080 × 1080 while the pixel pitch is 8μm, and the wavelength of light is 671nm. The optimal angle between the reference beam and the object beam is 1.2°. The target complex amplitude distribution is designed as the table shown in Fig. 1, where the amplitude range varies from 0 to 1 along horizontal direction and phase from to along vertical one. Then we compute the CGH based on the Eq. (1). After the calculation employing filtering function, the numerical simulation result is acquired and shown in Fig. 1, in which a random row of the amplitude result and a column of the phase one are plotted to illustrate the modulation numerically. It is clear that both the phase and the amplitude can be modulated linearly, independently and simultaneously. It is noticed that the amplitude modulation range is from 0 to 0.5, where the intensity decrease is caused by the filtering, which can be pre-compensated by the incident light. The phase modulation is achieved successfully, where the phase shift is caused by a complex constant “” shown in Eq. (3). The numerical results demonstrate the validity and the capability of actualizing the simultaneous and independent modulation of amplitude and phase.
3.2 Preliminary experimental verification
To verify experimentally the capability of amplitude modulation, we first test an image with half binary stripe image and half 256-gray-level image with constant phase respectively. The experimental set-up is shown in Fig. 2. The linearly polarized laser beam is collimated and illuminated to the SLM where the light wave is modulated to reconstruct the object complex amplitude. Then the −1st order of the reflected reconstructive beam is picked up after the beam goes through a “4f” filtering architecture, where the focal lengths of and are and respectively. 3D reconstructed light wave appears on the back focal plane of the second Fourier transform lens and recorded by the CCD. The light source is a solid laser with wavelength of 671nm; the reflective phase-only SLM (Holoeye PLUTO-VIS), with a pixel pitch of 8μm and a pixel number of 1920 × 1080, is employed to load the phase-only CGH with 256-level; and a CCD camera (INFINITY 4-11C) is used to record the results. The focal lengths = = 543mm and their apertures are both 120mm. In addition, the angle of tilt is maintained at 1.2°.
The reconstructed results are exhibited in Fig. 3. Figure 3(a) is the intensity distribution recorded by CCD directly, where the upper half result illuminates the modulation ability of 256-gray-level image and the lower half result examines that of binary stripe image. To exhibit clearly, we plot the average results of the rows where the upper part and lower one are plotted in Figs. 3(b) and 3(c) respectively, where the red dashed lines stand for the ideal objects and the blue solid ones denote the experimental results, the vertical coordinates represent the normalized amplitude. As it is easily seen, the outcome of the amplitude modulation matches the ideal one generally when the normalized amplitude is more than 0.5, while there exists fluctuation when less than 0.5 (Fig. 3(b)), which might be caused by the dynamic range of CCD and the system noise. Moreover the binary stripe image is produced successfully (Fig. 3(c)).
For further demonstrating the feasibility of the proposed method, the random 2D image: two random Chinese characters “北理” with high contrast are used as a binary object, which located on the back focal plane of the second Fourier transform lens, so the target complex amplitude distribution can be obtained directly according to the intensity distribution of the image . All the other parameters are kept. The experimental reconstructed image is shown in Fig. 4(a). It is evidently seen that the two Chinese characters are reconstructed successfully. Another random 256-gray-level image: a picture, cameraman, is tested as the target. The reconstructed result is recorded by CCD shown in Fig. 4(b). It is easily observed that the details of the columns and that of the tripod in the picture are displayed with high image quality. It is noted that we obtain the scene without random initial phase and iteration for calculating CGH. The zero-order noise produced by pixel structure of SLM is eliminated by the filtering architecture automatically. Although the some similar methods (such as that in Ref .) can be used to filter out the zero-order noise where only the Fourier kinoforms are applicable, our proposed method can achieve the modulation of both amplitude and phase for reconstructing the objects by complex holograms.
It is clear that the accurate reconstruction of amplitude distribution with high quality can be realized by using complex amplitude modulation based on phase-only hologram, which could be employed in 3D holographic display. Compared with traditional holographic project, the method is more suitable to be applied to 3D dynamic display for improving the image quality and the calculation speed.
3.3 3D holographic display by complex amplitude modulation
The complex amplitude modulation method can realize the 3D image reconstruction without refinement based on CGH calculated by the analytical polygon-based method. We now reconstruct 3D objects located at different positions, as shown in Fig. 5. The distances between two objects is , and their surfaces are smooth with constant initial phase. Firstly, two Chinese characters are set as the target objects, and the size of each character is 6mm × 6mm and = 300mm. The characters are reconstructed on the different planes, as shown in Fig. 6, where Fig. 6(a) and 6(b)) are the numerical simulation results and Fig. 6(c) and 6(d)) are the optical experiment ones. It is clear that 3D image can be reconstructed by the complex amplitude modulation, when one is clear, the other is blurred. The result of further verification is shown in Fig. 7, where 7(a) and 7(b) are simulation results while 7(c) and 7(d) are experimental ones. There are two cubes with Chinese characters and letters respectively and the side length of 4.1mm, and the cube with Chinese characters locates in front while = 300mm. It is easy observed that when the cube with letters becomes clear, the other is defocused, and vice versa. Meanwhile, it is noted that when the illuminating light source is in front, it causes the 3D scene surfaces with “A” and “右” the brightest and surfaces with “C” and “上” the darkest. To realize the 3D dynamic holographic display using a single SLM by the complex amplitude modulation, we perform an animation with a rotational cube, as shown in Fig. 8, which verifies that the proposed method can perform 3D dynamic display.
The complex amplitude modulation can also be realized by Abbe's theory of image formation. The typical set-up is shown in Fig. 9, where the focal length and wave length are maintained unchanged: = 543mm and = 671nm respectively. Also, the angle of tilt in the experiment is 1.2°. The predesigned 3D objects are composed by two objectives with same size located at different positions. According to the basic principle of image formation and Abbe's theory, there is a quadratic phase factor, (represents the magnification of the system), in the complex amplitude distribution on the image plane . Consequently, the distribution can be considered as a complex hologram recording the interference pattern from the object wave and a spherical wave. Considering the image location and magnification of a hologram , one can determine the change of the distance and magnification of the image and where is the target distance between the constructing object and the complex hologram. In our experiment, we set = 200mm and = 150mm. When = 10cm, measured in the experiment, the ideal distances are = 1002mm and = 1208mm respectively, while the measured distances are = 100cm and = 120cm. The experimental results are in good agreement with the desired ones. The primary results are shown in Fig. 10(a) and 10(b). It is easily observed that two objectives are not the same size. The size of the front object is smaller than that of the back one, where = 4mm, = 5mm. We pre-compensate the magnification and the distortions using the similar method to literature , then the difference of the image size can be corrected in the objective construction, and the experimental results are exhibited in Fig. 10(c) and 10(d), where = = 5mm.
In brief, 3D dynamic holographic display can be achieved by the complex amplitude modulation method, where the phase-only hologram encoding technique and a single SLM are employed. Owing to the basic principle of the method, one can maintain more information of the object beam in complex holograms, improve the quality of reconstructed image.
It is noted that phase modulation is very useful, and its measurement needs an additional interferometer with high stability. On one hand, to measure the phase distribution on the plane, the accuracy of experimental system is a significant factor, because the imprecise planar wave illuminating SLM, inaccurate modulation of SLM, imprecise position of optical elements and other error in the optical system can lead to unexpected optical path length, which introduces noise to the phase distribution. On the other hand, the performance of the phase modulation can be confirmed by 3D reconstructions. Also, human eyes can only detect the amplitude information of the object, and cannot detect the phase distribution of the object, so within the human visual perceptibility, the phase information of the object is not considered in 3D holographic display area at present. Of course, our proposed method can be employed into other applications, if the phase information is required, it can be analyzed by a special interferometer with high precision.
To realize the method, what we need is encoding the CGH based on the parameters of target objects and that of the system for complex amplitude modulation. One of the most important steps for CGH calculation is determining the parameters in .When reference beam is a plane wave, can be given by where represents the wavelength of the illuminating light, is the angle between the reference beam and the object beam, and is the coordinates along the optical axis. As is well known, for off-axis hologram, carrier frequency must satisfy where is the cutoff frequency of object along the axis of . However, considering the multi-order effect of SLM which causes the spectrum of hologram appears frequently, we cannot choose matching merely. As the results of the zero and ± 1 order given by Eq. (2), their bandwidth presumed to be the same is acceptable. In addition, based on the Whittaker-Shannon sampling theorem, the half of bandwidth of SLM is only equal to, which is equal to the maximum of carrier frequency. According to the grating equation where stands for the pixel pitch of SLM and is the diffraction angle. To avoid crosstalk of each orders, we can obtain which meets the demand of bandwidth of SLM. Then the range of can be written as
Based on geometric optics , the longitudinal magnification, , and lateral one,, of the system are given by19]. Therefore, the 3D target scene can be reconstructed and magnified.
One of disadvantages of filtering architecture is the diminution of efficiency for luminous energy utilization. An aperture as large as possible can lessen the influence, which is obtained when the carrier frequency is maximum, According to the definition of space frequency, where is the coordinate in frequency domain and is the focal length of Fourier lens, the maximum diameter of the aperture is given byEq. (5).
Another disadvantage can be abstracted that complex amplitude modulation methods are realized with the reduction of the space-bandwidth product (SBP); for example, the region in space domain – size of SLM – is not change, while the bandwidth of spectrum is lessened by filtering. The Whittaker-Shannon sampling theorem implies that the SBP is equivalent to the resolution of SLM , which predicates that the effective resolution decreases. It is apparent that the amount of information of a complex amplitude distribution doubles that of an amplitude-only or a phase-only one. Therefore, the essence of complex amplitude modulation, no matter how to realize, is sacrificing the SBP to achieve the expression of the amplitude and the phase simultaneously and respectively. Despite the fact that the complex amplitude modulation is facing the restriction of finite SBP, it is meaningful to introduce it into holographic display. As it mentioned in the Sec. 1, the method avoids not only the speckle noise but also the loss of the precise structure, which are also proven in the simulation and experimental results. Furthermore, the limitation is also a common problem in holographic display, which can be remitted by some approaches, such as horizontal parallax only hologram, time division or space division method, and so on [21,22].
We achieve 3D dynamic holographic display by modulating the phase and the amplitude of the light wave simultaneously and respectively where the noise is eliminated. Encoded by the analytical polygon-based method, 3D objects without subdivision of surfaces and without random initial phases are reconstructed by a pixelated phase-only SLM successfully based on the complex amplitude modulation method which doesn’t introduce any time consuming process. This method makes the stringent alignment of the pixels in optical realization unnecessary, and it avoids the internal structure of the light wave and improves the quality of 3D images. The range of tilted angle and the maximal size of aperture are analyzed. Our further investigation will simplify and minimize the system and make the complex amplitude modulation more suitable for 3D dynamic holographic display. It is noted that the proposed method can be employed in various optical systems where the complex amplitude needs to be modulated.
This work was supported by the National Basic Research Program of China (973 Program Grant No. 2013CB328801 and No. 2013CB328806), the National Natural Science Founding of China (61235002).
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