A technique using RGB color filters is proposed for creating high-quality full-color computer-generated holograms (CGHs). The fringe of these CGHs is composed of more than a billion pixels. The CGHs reconstruct full-parallax three-dimensional color images with a deep sensation of depth caused by natural motion parallax. The simulation technique as well as the principle and challenges of high-quality full-color reconstruction are presented to address the design of filter properties suitable for large-scaled CGHs. Optical reconstructions of actual fabricated full-color CGHs are demonstrated in order to verify the proposed techniques.
© 2017 Optical Society of America
Undoubtedly, computer holography is one of the candidates for ultimate digital three-dimensional (3D) imaging technology. However, the development of computer holography has suffered from the gigantic space-band product (SBP) that is required for creating high-quality 3D images with a large viewing-angle. To produce a 3D image whose screen size is several hundreds of square centimeters and viewing-angle is several tens of degrees, we need more than billions of pixels in computer holography. This fact makes it very difficult to develop holographic displays using electrical devices.
Recently, we presented some large-scale computer-generated holograms (CGHs) [1–5]. These CGHs reconstruct only a static 3D image, but the size and viewing-angle are sufficient for multiple viewers to see the 3D image simultaneously because they are composed of several billions or sometimes several tens-of-billions of pixels. We call this a high-definition CGH. These high-definition CGHs reconstruct full-parallax 3D images accompanied with natural and continuous motion parallax enabled by occlusion processing called the silhouette method . Thus, viewers can perceive a strong sensation of depth that is comparable to traditional optical holography. As a result, a CGH can be displayed in museums .
However, these high-definition CGHs can only reconstruct monochromatic images. Many techniques for full-color reconstruction are found in the computer holography literature. For example, there are techniques utilizing volume holograms. Traditional optical holograms can reconstruct a full-color image when they are recorded as a “thick” or volume hologram, because volume holograms feature wavelength selectivity and multiplexed recording. Color holographic printers are implemented using these properties . This type of printer, however, reconstructs stereograms which commonly present only a parallax view without exact accommodation cues [7,8]. Color wavefront printers are also being developed in the same vein . The structure of a wavefront printer is very similar to a holographic printer, but the wavefront printers can reconstruct object waves in the same manner as traditional optical holography. Thus, the reconstructed 3D images do not cause any vergence–accommodation conflict unlike a holographic printer. However, the image quality is not sufficient at this stage. In electro-holography, where an electrically controlled spatial light modulator (SLM) displays the holographic fringe, time-division multiplexing is a trend for full-color reconstruction [10–14]. In this case, primary color holograms are sequentially reconstructed by the SLM. However, the SBP is still too small to reconstruct high-quality holograms, as mentioned above.
When we reconstruct three monochromatic holograms corresponding to primary colors, the full-color reconstruction is easily obtained using optical combiners [15–17]. In practice, excellent full-color reconstruction can be achieved by combining three 3D images of high-definition CGHs using dichroic mirrors and a white LED . However, the optical combiner has no portability, i.e., it is too heavy, large, and delicate to display the CGH at exhibitions and museums. A technique using three stacked CGHs has also been proposed . In this technique, three layers of the RGB-CGH fringe are printed on corresponding dichroic thin films and deposited on a single substrate. This is an excellent technique, but requires a very sophisticated film formation technology.
Techniques employing a single hologram instead of three holograms have also been proposed for full-color reconstruction of CGHs [19,20] and 2D image projection [21,22]. These methods use techniques of space-division of an SLM  and depth-division of reconstructed images . Deep phase modulation  and special phase encoding  are also used in phase-only SLMs. These techniques are for small SBP holograms.
The use of color filters has also been proposed to obtain full-color reconstruction from a single CGH [23,24]. Kajiki et al. attempted to produce full-color images from a CGH using RGB color filters and a miniature light bulb . Iwami and Sakamoto also proposed a glasses-type color CGH using RGB color filters and reported on the optimization of the filter pattern . Although both methods are for small SBP CGHs, and thus the technique cannot be directly applied to large SBP CGHs like our high-definition CGHs, the idea is very useful. Holographic display using an SLM, color filters, and white LED was also reported . However, because coarse color filters are installed in the Fourier plane in the reported system, the color image without color shifts may be obtained only at the vertically centered viewpoint. Thus, the technique cannot be used for full-parallax CGHs.
We have proposed a technique to create full-parallax full-color high-definition CGHs using RGB color filters, whose fringes are composed of more than a billion pixels . These large-scaled CGHs reconstruct full-parallax high-quality 3D images with continuous motion parallax. The detailed technique is presented in this paper. The principle of the use of color filters is presented and discussed in Section 2. The simulation technique necessary for designing the properties of filters suitable for high-definition CGHs is described in Section 3. Optical reconstructions of actual full-color high-definition CGHs fabricated by the proposed techniques are demonstrated in Section 4. We conclude this paper in Section 5.
2. Principles of full-color high-definition CGH using RGB color filters
2.1 High-definition CGHs and its fabrication by laser lithography
We define a high-definition CGH to be a CGH that is composed of roughly more than several billion pixels with pitches of less than 1 μm. High-definition CGHs can be fabricated using several techniques such as special printers designed for printing CGHs. However, one of the most useful ways to fabricate high-definition CGHs is to utilize conventional microfabrication technology. For example, the photomask used for photolithography is itself a CGH if the mask pattern is of the hologram fringe [1,2].
A high-definition CGH fabricated as a photomask features high contrast and reflectivity of a binary amplitude fringe pattern, because the fringe pattern is formed by chromium thin films coated on a glass substrate. As a result, the CGH can be a reflection-type hologram as well as a transmission-type hologram. In practice, a CGH fabricated by this technique can be reconstructed by illumination placed in front of it. An example of optical reconstruction of the CGH made of a photomask is shown in Fig. 1. In this case, the red LED placed in front of the CGH illuminates the surface of the photomask. The 3D image appears behind the area of the fringe pattern, which looks as if it is a virtual window, while the chromium film outside the fringe pattern simply works as a mirror.
2.2 Full-color reconstruction using RGB color filters
Figure 2 shows the principle of full-color CGH reconstruction using RGB color filters. The CGH is reconstructed by reflected illumination, where a broadband white light source is used for the illumination. If any filter is not used in reconstruction, this kind of illumination commonly generates severe color smears caused by chromatic aberration because the printed CGH is generally a thin hologram that has no wavelength selectivity.
In the proposed technique, however, RGB color filters are closely attached to the fringe of the CGH in order to filter the illumination light wavelength. Each fringe pattern behind the RGB filters is calculated in a wavelength that corresponds to the RGB primary colors. Thus, the three monochromatic 3D images are independently reconstructed in the RGB colors. Viewers are able to see a full-color image by combining these RGB color images.
However, several problems remain before this type of full-color CGH can be realized. The following main issues need to be considered:
- a. Low spatial resolution of the RGB color filters
- b. Alignment error of the RGB color filters with the fringe pattern
- c. The broadband properties of the RGB color filters
2.2.1 Filtering of illumination light on fringe blocks
Because the resolution of the color filter is commonly lower than that of the fringe pattern, it is difficult to attach a color filter to each individual fringe pixel as is done in image sensors and LCD panels. Therefore, the fringe pattern is divided into RGB blocks. Each block is composed of many pixels, as shown in Fig. 3. The fringe pattern for each block is generated in the wavelength representing either the R, G, or B color, and the corresponding color filter is attached to each fringe block. When this CGH is irradiated with white illumination, light filtered by the attached color filter illuminates the correct fringe block. As a result, a full-color 3D image that is a combination of the RGB images reconstructed by RGB blocks emerges from the CGH.
The layered structure of a full-color CGH is shown in Fig. 4. The fringe pattern is formed by chromium thin films, as mentioned above, and the color filter is attached to it.
2.2.2 Guard gap
Because the resolution of the fringe pattern is on the scale of sub-microns, tolerance for alignment of RGB color filters is also in sub-microns. In addition, there may be color transition areas along the border between the colors of the RGB color filters. Thus, we create a “guard gap” along the border between RGB blocks of the fringe pattern, as shown in Figs. 3 and 4. The fringe pattern is not formed in the guard-gap area.
This guard gap allows us to increase the positioning tolerance of the RGB color filters and avoid the problem of the color transition areas. In color LCD panel technology, a black matrix (BM) is usually formed on the color filter to avoid similar problems. However, in the case of CGHs, the guard gap is a simpler solution than a BM because it makes the structure of the RGB color filters very simple.
Note that the reflectivity or transmittance in the guard-gap area of the fringe pattern is dependent on whether the CGH is reconstructed by reflection or transmission. When the color CGH is reconstructed as a reflection-type hologram, the fringe pattern in the guard-gap area must transmit the illumination light, whereas the guard gap must shield the illumination light in the transmittance-type color CGHs, as shown in Fig. 4.
2.2.3 Use of multi-chip white LEDs
One of the most difficult challenges of applying RGB color filters to the full-color reconstruction of CGHs is their broadband filter properties. Color filters are commonly used for LCD panels or image sensors. Narrow spectral transmittance is not required in these applications. However, broad spectral illumination causes severe chromatic aberration in high-definition CGHs.
Therefore, it is essential to use multi-chip type white LEDs as the illumination light source instead of other types of white light sources, such as single-chip white LEDs using phosphor or halogen lamps. Here, a multi-chip white LED is an LED in which three LED chips corresponding to the RGB primary colors are integrated into a single package. This type of LED gives a narrower spectrum for each color than single-chip LEDs and other white light sources.
Figure 5(a) shows examples of the emission spectrum of single- and multi-chip LEDs, where the drive current of the single-chip LED is adjusted so that the peak intensities have the same value. For reflection-type CGHs, effective illumination is obtained from the product of LED intensity and the square of the filter transmittance because the illumination light passes through the filter twice. Figure 5(b) shows examples of the spectra of the effective illuminations estimated by the spectra of LED outputs and filter properties in Fig. 5(a), where the peak values of the effective intensities are normalized to unity. The multi-chip white LED produces a narrower spectra than the single-chip LED. Therefore, chromatic aberration can be reduced by adopting multi-chip type white LEDs as the illumination source.
Here, note that the position of the light source slightly shifts depending on the RGB colors in the multi-chip white LEDs, i.e., the positions of three chips are not the same. However, this can be easily taken into account when generating the fringe patterns. We simply change reference waves for each RGB fringe block in its numerical interference.
3. Simulated reconstruction for designing RGB color filters
There are many types of color filters that have various structures and are made of various materials. The shapes, sizes, and spectral properties of the filters must be determined to design the RGB color filters most appropriate for full-color high-definition CGHs.
The brightness of reconstructed images, as compared with zeroth-order light, is especially important for creating high-quality 3D images. In our early study of full-color CGHs, we suffered from this problem. Many high-brightness LEDs are available on the market. The reconstructed image is bright when these high-intensity LEDs are used for illumination, but zeroth-order light is also very bright. The non-diffractive light was, in practice, too bright to see the 3D image. This meant that the efficiency of the hologram was too low.
To overcome this problem, the parameters of the color filters should be carefully chosen by simulating the reconstructed images, based on wave optics.
3.1 Binary fringe pattern of CGH
Suppose that and are monochromatic object and reference wave fields at wavelength , where C refers to either R, G, or B. The wavelength corresponding to the three primary RGB colors is referred to as the design wavelength of the CGH for color C.
The fringe pattern printed by laser lithography is commonly a binary transmittance or reflectance pattern. Therefore, the fringe intensity is obtained by
For simplicity, we discuss only the reflection-type CGH below. The simulation model is composed of two planes, the fringe and filter planes, as shown in Fig. 6. In the fringe plane, the incident wave field is reflected by the fringe pattern given in Eq. (2). The fringe pattern is made of chromium thin film coated on a glass substrate. Thus, a pixel with fringe intensity reflects the incident illumination light in the reflectance of the chromium thin film, while a pixel with reflects it in the glass’s surface reflectance. Therefore, supposing that the incident wave field is represented by in the fringe plane, the reflected wave field is given as follows:
3.2 Color filters
Suppose that the illumination wave field of a color CGH is represented by immediately before the filter plane in Fig. 6, where the illumination light is not monochromatic. Note that illumination wavelength is different from CGH design wavelength . In a reflection CGH, the illumination light passes through the color filter first, and then reaches the fringe plane. We assume that the fringe plane does not closely contact the filter plane in order to take into account the thickness of the color filters or a small gap between the filter and fringe planes. Thus, we represent the incident wave field in the fringe plane byFig. 6. Note that the propagation is backward propagation in this case. We use the band-limited angular propagation technique for this numerical propagation in practice .
The reflection light in the fringe plane forward propagates to the filter plane and again passes through the color filters. The reflected wave field immediately after the filter plane is described by
3.3 Color image formation
The output wave field of the color CGH is calculated from illumination wave field using Eqs. (3)–(5). The finally reconstructed image is calculated by image formation using virtual optics. The output field is propagated to a plane in which the pupil and lens are located. Here, the center position of the field is changed using off-axis propagation techniques [28,29]. Thus, the incident field on the pupil and lens is written as30].
The propagated and rotated field passes through a lens and forms the image on the screen. The monochromatic image in the screen is given by the distribution of optical intensity as follows:
We calculate many monochromatic images from illumination wave field in discrete wavelengths as shown in Fig. 7. These monochromatic images are then combined into a XYZ color image using the CIE XYZ color matching function. Finally, the XYZ image is converted to the sRGB color image.
4. Design of RGB color filters and reconstruction of full-color CGHs
4.1 Design parameters of the RGB color filters
We adopt a simple stripe pattern as the shape for the RGB color filters in order to make it easier to adjust the position of the RGB color filters to the corresponding fringe block. It is important to optimize the widths of the color stripe and guard gap because the color filters may be perceived by viewers if these widths are unnecessarily wide. The reconstructed 3D images will be degraded if the filter is detected. On the contrary, narrower widths make alignment more difficult. By repeating the simulation mentioned in Section 3, we finally adopted a stripe width of 80 μm in the color filters and a guard-gap width of 20 μm in the fringe blocks, as shown in Table 1.
Here, note that we do not evaluate the visibility of the filter in practice. Since the reconstructed images are well reproduced by the simulation, we can assess how much the filter may be detected. Therefore, by using the simulation, we decided the widths that may be small enough for the filters to be invisible, but not too small to adjust the filter by hand. Figure 8(b) shows the simulated reconstruction in the adopted widths. The widths are mainly determined by the fact whether we can adjust the filter positon in a given time or not. Fortunately, the simulation shows that the filter with the adopted widths is most likely invisible, or at lease fairly better than the other filters having larger widths.
Furthermore, the spectral properties of the filters must be determined by choosing the materials and thickness of the RGB color filters. We repeated the simulation for various materials and thicknesses of the color filters to optimize them. The designed color filter is actually fabricated using conventional techniques used for color LCD panels, i.e., photolithography using photomasks and color photoresist. Figure 9(a) shows the spectral transmittances of the RGB color filters that we finally adopted and fabricated. Here, the total intensity of the multi-chip LED in Fig. 5(a) is again indicated for reference. (The drive currents are again adjusted so that the peaks have the same intensity.) The spectra of the effective illuminations estimated from the spectra of the LED intensity and filter transmittance are shown in Fig. 9(b). Several examples of filter properties that were not adopted in this study are also shown in Fig. 10. These have an advantage of being high transmittance in G and B colors, but definitely have a problem of color separation.
4.2 Comparison of simulated and optical reconstruction
The parameters and 3D scene used for creating a CGH called “Color Cube” are shown in Table 1 and Fig. 11, respectively. The object wave of Color Cube is calculated using the polygon-based method  and silhouette method with the switch-back technique .
Figures 12 (a) and (b) show the simulated and optical reconstruction when the color filters are attached to the fringe pattern as horizontal stripes. The parameters used for the simulated reconstructions are summarized in Table 2. It is verified that the reconstructed images (a) and (b) are similar to each other. It is especially important that the simulation predicts the zeroth-order diffraction properly, because if the filter property is not appropriate, the zeroth-order light tends to be too bright to see the reconstructed image in the proposed full-color CGH, as mentioned in Section 3.
The optical reconstruction may look better than the simulation. This is because speckle noises appear in the simulation image. Whereas the illumination light source used in the simulation is a theoretical point source, the actual illumination is not a dimensionless point source, i.e. it has a given size. This spatial incoherency a little blurs the image (b) of optical reconstruction, and makes it better.
It is also confirmed in Figs. 12 (a) and (b) that there are color differences between the simulated and optical reconstruction. This is most likely caused by the fact that we assume the RGB illumination lights have the same peak intensity in the simulation. This is not a severe problem because, in practice, we can easily change the peak intensities by changing the drive current if the color reproduction of the object model is important in some application.
4.3 Horizontal or vertical stripes
The fabricated RGB color filters can be attached to the fringe pattern as vertically striped filters as well as horizontally striped filters. In general, we supposed that the horizontal stripes give better results than the vertical stripes. Because the horizontal viewing angle and thus horizontal resolution is important in practical 3D images, we expected that a vertically striped filter that interrupts the fringe in a horizontal direction will degrade the reconstructed image.
However, as shown in Fig. 12, the horizontally striped filters cause optical diffraction in a vertical direction, i.e., the light diffracted by the horizontal structure of the filters spreads vertically in reconstruction. In the CGH in Fig. 12, the center of a spherical reference wave is placed in the lower-left of the CGH, and thus, there is a bright spot of zeroth-order light in the lower-left of the reconstructed image. This bright spot is diffracted by the filter and extended in the vertical direction in the reconstructed images of Fig. 12. Figure 12 (c) shows another photograph of optical reconstruction of the same CGH. In this case, the photograph is taken from a different angle, i.e. the viewpoint is different from those in (a) and (b). It is verified that the light diffracted by the filter overlaps the object image and hinders the view. This is more undesirable in many other cases, because it is better that the center of the spherical reference wave is arranged right below the object rather than the lower-left of the object in order to gain larger diffraction efficiency and reconstruct bright 3D images.
In addition, we do not detect significant degradation by the vertically striped filters, as mentioned in the following section. This may be because of the very large resolution of our high-definition CGHs. As a result, we currently conclude that the vertically-striped color filters are the best choice.
4.4 Optical reconstruction of full-color CGHs with vertically-striped RGB filters
Figure 13 shows photographs of the optical reconstruction of Color Cube. The setup for optical reconstruction is shown in Fig. 14. Here, the vertically-striped RGB filters are attached to the CGH. The size of the CGH is approximately 5 × 5 cm2. The viewing-angle is approximately 45° for red and 33° for blue in both the horizontal and vertical directions. The location of light-emitting parts of the multi-chip white LED differs depending on the color, as shown in Fig. 14. However, this difference can be easily compensated in the numerical interference step, as mentioned in Section 2.2.3. The actual positions of the light emitting parts are shown as the centers of the spherical reference waves in Table 1.
Unfortunately, 2D medium such as photographs in Fig. 13 cannot convey the true impression of 3D images reconstructed by the high-definition CGH. In this hologram, viewers can see the reconstructed objects as if real objects are placed at a deep position behind the CGH. A video is at least better than still images. Visualization 1 confirms that the color objects in the 3D scene are well reconstructed with continuous motion parallax as well as accommodation. It is also shown that the vertically-striped filters do not notably degrade the reconstructed image. If a horizontal stripe is used, unwanted diffraction light expands vertically and overlaps the reconstructed objects.
The optical reconstruction of a bigger CGH named “Casino Chips” is shown in Fig. 15 and Visualization 2. This CGH is composed of more than 16 G pixels and the size is approximately 10 × 10 cm2. The parameters are also summarized in Table 1.
We emphasize again that degradation caused by vertical splitting of the fringe pattern is not detected in either of the reconstructed images. Furthermore, the structure of the striped RGB filters and guard gaps are not perceived in the optical reconstruction because these widths are sufficiently narrow.
We proposed a technique using RGB color filters for reconstructing high-definition CGHs in full-color. Several challenges for realizing full-color reconstruction using the color filters were discussed and then practical solutions were presented. Furthermore, a simulation technique was proposed for predicting the full-color 3D images reconstructed by high-definition CGHs with RGB color filters. Using the simulation results, the properties of color filters suitable for high-definition CGHs were determined and actually fabricated. Optical reconstructions of the high-definition CGHs using the fabricated color filters verify that the proposed technique makes it possible to reconstruct high-quality and bright full-color 3D images.
JSPS KAKENHI Grant Number 15K00512; MEXT Strategic Research Foundation at Private Universities (2013-2017).
We would like to thank Prof. Sakamoto at Hokkaido University for providing useful information on the fabrication of color filters using color reversal films in the early stages of our study. We also would like to thank Prof. Nakahara for his overall suggestions about printing fringe patterns.
References and links
3. K. Matsushima, H. Nishi, and S. Nakahara, “Simple wave-field rendering for photorealistic reconstruction in polygon-based high-definition computer holography,” J. Electron. Imaging 21(2), 023002 (2012). [CrossRef]
4. K. Matsushima and S. Nakahara, “Stepping closer to the perfect 3D digital image,” SPIE Newsroom (2012).
5. K. Matsushima, M. Nakamura, and S. Nakahara, “Silhouette method for hidden surface removal in computer holography and its acceleration using the switch-back technique,” Opt. Express 22(20), 24450–24465 (2014). [CrossRef] [PubMed]
8. H. Zhang, Y. Zhao, L. Cao, and G. Jin, “Fully computed holographic stereogram based algorithm for computer-generated holograms with accurate depth cues,” Opt. Express 23(4), 3901–3913 (2015). [CrossRef] [PubMed]
9. Y. Kim, E. Stoykova, H. Kang, S. Hong, J. Park, J. Park, and J. Hong, “Seamless full color holographic printing method based on spatial partitioning of SLM,” Opt. Express 23(1), 172–182 (2015). [CrossRef] [PubMed]
10. M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shiraki, N. Masuda, and T. Ito, “Time-division color electroholography using one-chip RGB LED and synchronizing controller,” Opt. Express 19(13), 12008–12013 (2011). [CrossRef] [PubMed]
11. H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014). [PubMed]
12. H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4, 6177 (2014). [CrossRef] [PubMed]
13. W. Zaperty, M. Kujawinska, T. Kozacki, and B. Wisniowski, “Wide angle, color holographic 3D display with multi-source based holographic content,” Proc. SPIE 9385, 93850E (2015). [CrossRef]
16. H. Nakayama, N. Takada, Y. Ichihashi, S. Awazu, T. Shimobaba, N. Masuda, and T. Ito, “Real-time color electroholography using multiple graphics processing units and multiple high-definition liquid-crystal display panels,” Appl. Opt. 49(31), 5993–5996 (2010). [CrossRef]
17. T. Miyaoka, K. Matsushima, and S. Nakahara, “Optimization of design-wavelength for unobtrusive chromatic aberration in high-definition color computer holography,” Proc. SPIE 9386, 93860N (2015). [CrossRef]
18. T. Kämpfe, E. B. Kley, A. Tünnermann, and P. Dannberg, “Design and fabrication of stacked, computer generated holograms for multicolor image generation,” Appl. Opt. 46(22), 5482–5488 (2007). [CrossRef] [PubMed]
20. G. Xue, J. Liu, X. Li, J. Jia, Z. Zhang, B. Hu, and Y. Wang, “Multiplexing encoding method for full-color dynamic 3D holographic display,” Opt. Express 22(15), 18473–18482 (2014). [CrossRef] [PubMed]
21. M. Makowski, M. Sypek, I. Ducin, A. Fajst, A. Siemion, J. Suszek, and A. Kolodziejczyk, “Experimental evaluation of a full-color compact lensless holographic display,” Opt. Express 17(23), 20840–20846 (2009). [CrossRef] [PubMed]
23. Y. Kajiki, M. Okamoto, T. Ando, K. Yamasaki, and E. Shimizu, “Computer-generated color holograms using RGB color filters,” Proc. SPIE 3293, 125–130 (1998). [CrossRef]
24. S. Iwami and Y. Sakamoto, “A study of glasses-type color CGH using a color filter considering reduction of blurring,” Proc. SPIE 7233, 723317 (2009). [CrossRef]
26. Y. Tsuchiyama, K. Matsushima, S. Nakahara, and Y. Sakamoto, “Full-color high-definition CGH using color filter and filter design based on simulation,” in Imaging and Applied Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper DW5I.4. [CrossRef]
27. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009). [CrossRef] [PubMed]
30. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20(9), 1755–1762 (2003). [CrossRef] [PubMed]