Spatial cross modulation method using a random diffuser and phase-only spatial light modulator for constructing arbitrary complex fields Atsushi Shibukawa,1,* Atsushi Okamoto,1 Masanori Takabayashi,2 and Akihisa Tomita1 1
2
Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-Ku, Sapporo, Hokkaido 060-0814, Japan Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan * [email protected]
Abstract: We propose a spatial cross modulation method using a random diffuser and a phase-only spatial light modulator (SLM), by which arbitrary complex-amplitude fields can be generated with higher spatial resolution and diffraction efficiency than off-axis and double-phase computergenerated holograms. Our method encodes the original complex object as a phase-only diffusion image by scattering the complex object using a random diffuser. In addition, all incoming light to the SLM is consumed for a single diffraction order, making a diffraction efficiency of more than 90% possible. This method can be applied for holographic data storage, threedimensional displays, and other such applications. ©2014 Optical Society of America OCIS codes: (090.1760) Computer holography; (090.1970) Diffractive optics; (110.7348) Wavefront encoding; (210.2860) Holographic and volume memories.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3968
16. K. Anderson and K. Curtis, “Polytopic multiplexing,” Opt. Lett. 29(12), 1402–1404 (2004). 17. M. Takabayashi and A. Okamoto, “Self-referential holography and its applications to data storage and phase-tointensity conversion,” Opt. Express 21(3), 3669–3681 (2013). 18. K. Zukeran, A. Okamoto, M. Takabayashi, A. Shibukawa, K. Sato, and A. Tomita, “Double-referential holography and spatial quadrature amplitude modulation,” Jpn. J. Appl. Phys. 52(9S2), 09LD13 (2013). 19. L. G. Shirley and N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A 4(4), 734–745 (1987). 20. A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, and K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express 19(14), 13436–13444 (2011).
1. Introduction A method to construct arbitrary complex amplitude fields [1] was used for the first time in the field of optical information processing half a century ago [2]. Since that time, this method has been widely used in numerous applications, ranging from optical tweezers to threedimensional (3D) displays [3–5]. A well-known method for generating arbitrary complex amplitude fields is to display an off-axis computer-generated hologram (CGH) on a spatial light modulator (SLM), in which a desired complex amplitude object is encoded as amplitude or phase holograms through interference with an off-axis reference plane wave [6,7]. Using this method, however, consumes energy of the incident plane wave not only for the desired diffraction order but also unwanted diffraction orders. This considerably lowers the diffraction efficiency for the desired diffraction order. In addition, several tens of SLM pixels must be sacrificed within one data pixel to form a diffraction grating that satisfies the sampling theorem; therefore, this method is limited to obtaining a low spatial resolution in the reconstructed image. The other typical CGH used for complex amplitude modulation includes a double-phase hologram [8–10]. In this method, a given complex amplitude is decomposed into two phase values, and then the phase distribution, synthesized by letting these two values spatially and alternately arrange in a chessboard pattern, is displayed on the SLM. Subsequently, the desired complex amplitude is obtained by isolating a zeroth diffraction order spatially. However, this method and the above CGH method are alike in that a diffraction grating is used to obtain the desired complex amplitude object, significantly limiting the achievable diffraction efficiency and spatial resolution. In contrast, a kinoform [11], on-axis CGH, can diffract all incoming light into a single diffraction light like a blazed diffraction grating, avoiding all of the above-mentioned problems inherent to holograms. The kinoform encodes the original object into a spectral phase image within its Fourier domain. The use of this phase image is based on the fact that the Fourier spectrum of a diffusive object produces a homogeneous scattered wavefront and, even if only the phase in the scatted wavefront is retained, most of the important features of the original object are preserved [12,13]. However, in the kinoform, the reconstruction space of the object is restricted in the Fourier domain of the spectral phase image plane. For instance, when the Fourier transform is performed with a lens for the reconstruction of the kinoform, the size of the reconstructed object will be extremely small, on the order of millimeters [14]. In addition, the diffusive object is made by embedding a random phase pattern directly into the original object to ensure a smooth spectrum, causing large speckle noise in the reconstructed object. To solve the above problems, we propose a spatial cross modulation method (SCMM) using a random diffuser and phase-only SLM. The SCMM encodes a complex object as a scattered phase image by allowing the object to transmit through a random diffuser. Employing a random diffuser between the object and phase image planes enables us to make the scattered wavefront in an arbitrary space, which increases the degrees of freedom for the design of an optical system to reconstruct the larger object. Moreover, although the kinoform embeds the random phase directly into the object, the SCMM arranges the random diffuser with the random phase outside the object, and therefore, if the object to be reconstructed has comparatively homogeneous surface, speckle noise is not mostly caused in the reproduced object. Furthermore, like with the kinoform method, this method can consume all incoming light for a single diffraction order, so the resulting diffraction efficiency can be extremely #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3969
high. In addition, the SCMM can achieve higher spatial resolution than off-axis and doublephase hologram methods. In Sect. 2.1, the basic operation of the SCMM will be described. In Sect. 2.2, a numerical simulation to investigate the reconstruction characteristics of the SCMM is reported. In particular, we first confirm that full-complex-amplitude modulation is possible with the SCMM. Next, we discuss the behavior of the reconstruction quality when changing the extent to which the diffuser spreads the original complex object. The achievable spatial resolution and diffraction efficiency is then compared among the SCMM, off-axis hologram and doublephase hologram methods. In Sect 2.3, we present an application of the SCMM to holographic data storage (HDS) [15–17]. In particular, the basic operation in the case of SCMM applied to HDS is described. Then, we report an experiment for producing a data page with arbitrary complex amplitude modulation. The SCMM can promote the use of spatial quadrature amplitude modulation [18], in which multi-valued amplitude and phase modulations are combined in order to the increase data transfer rate and storage capacity of HDS. 2. Spatial cross modulation 2.1. Basic operation Let us begin with an explanation of the basic operation of the SCMM, as illustrated in Fig. 1. The operation is divided into two main steps: the calculation of the phase-only diffusion image in a computer (digital encode step) and the optical reconstruction of the desired complex field with the SLM (optical decode step). In Fig. 1, the reconstruction plane of the original object is set to be optically equal to the SLM plane via a 4f optical system, but this configuration of the SCMM is only one possibility. The SCMM can be also realized by a simpler configuration where the 4f optical system is removed and only the diffuser lies between the input (reconstruction) and output (SLM) planes.
Fig. 1. Conceptual diagrams of spatial cross modulation. (a) Digital encode step in a computer. (b) Optical decode step.
During the digital encode step completed within the computer in Fig. 1(a), an arbitrary complex object A(x,y)exp[jφ(x,y)] is first prepared in an input plane in order to be reconstructed in the subsequent decode step. The complex object is not restricted to either two or three dimensions, but the following explanation proceeds from a two-dimensional (2D) object. Next, the Fourier transformation of the object F{A(x,y)exp[jφ(x,y)]} is calculated with a fast Fourier transform (FFT). Here, F{·} denotes the operator of two-dimensional Fourier
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3970
transform. Then, the spatial phase distribution of a digital random diffuser (a virtual random diffuser) exp[jhd(x,y)] and the spatial spectrum of the object F{A(x,y)exp[jφ(x,y)]} are multiplied in a spatial Fourier plane. Through an inverse Fourier transformation of the multiplied spectrum distribution F−1{exp[jh(x,y)]·F{A(x,y)exp[jφ(x,y)]}} by inverse FFT (IFFT), a diffusion image (scattered wavefront) of the original object S(x,y)exp[jξ(x,y)] results within an output plane. Subsequently, amplitude distribution in the diffusion image S(x,y) are uniformed and the phase conjugation exp[-jξ(x,y)] is calculated, which is mathematically equivalent to inverting the sign of the phase distribution in the image. The elimination of the amplitude is based on the assumption that the scattered phase contains more of the important features of the original complex object than the scattered amplitude does. In other words, both the original amplitude A(x,y) and the original phase exp[jφ(x,y)] are spatially cross-modulated into a phase-only diffusion image exp[jξ(x,y)]. This importance of the phase has been demonstrated in a number of different contexts, including optical and acoustical holograms [12,13]. The phase-only image exp[-jξ(x,y)] is referred to as cross-modulated image herein. In regard to the complexity of the calculations, the above encoding step includes 2D FFT, 2D IFFT, and the multiplication of 2D distributions. The total complexity is therefore given by 2N2log2N + N2, where the SLM has N2 pixels in the x-y plane. If a three-dimensional (3D) object A(x,y,z)exp[jφ(x,y,z)] is treated with the SCMM, the 3D object A(x,y,z)exp[jφ(x,y,z)] is prepared as an assembly of many 2D images on the input plane. Each 2D image An(x,y,zn)exp[jφn(x,y,zn)] is changed to a cross-modulated image exp[-jξn(x,y)] with the above encode step, and then each cross-modulated image exp[-jξn(x,y)] is superimposed on the output plane to obtain the final cross-modulated image exp[-jξ(x,y)] to be displayed on the SLM. During the optical decode step, as shown in Fig. 1(b), the cross-modulated image exp[jξ(x,y)] calculated in step (a) is first displayed onto a phase-only SLM. After incident light falls on the SLM, the modulated light is Fourier-transformed via the first lens. Then, the light F{exp[-jξ(x,y)]} propagates through the optical diffuser with a phase distribution exp[jho(x,y)], and the reconstructed complex object A'(x,y)exp[jφ'(x,y)] is obtained with a second lens. Note that the reconstructed object A'(x,y)exp[jφ'(x,y)] is extremely similar to the original object A(x,y)exp[jφ(x,y)] but have slightly random noise due to the elimination of the scattered amplitude. The spatial phase distributions exp[jhd(x,y)] and exp[jho(x,y)] of the digital and optical diffusers must be matched for correctly working the SCMM. Two ways for matching these distributions are possible: one measures the distribution of the optical diffuser by a phase detection method for the decoding and subsequently uses the measured phase distribution as the digital diffuser in the encode step; another uses an electrically addressable phase-only SLM for the optical diffuser. The latter method appears to be more practical. As stated above, the kinoform uses the Fourier spectrum of the object to produce a scattered wavefront (encoded image), so the reconstruction space of the object is restricted to the Fourier domain of the encoded image plane. In contrast, the SCMM places a random diffuser between the object and encoded image planes, which produces the scattered wavefront without restriction, meaning that the object can be reconstructed in arbitrary spaces such as the Fresnel region and the optically equivalent (conjugate) region to the SLM plane, via a 4f system. This feature increases the degree of freedom for designing the optical system, so allows larger objects to be reconstructed as compared to the kinoform based on a Fourier transform lens. In addition, the kinoform adds a random phase into the object to make it diffusive for the smooth Fourier spectrum, and so the reconstructed object includes speckle noise. On contrast, the SCMM places the diffuser with the random phase outside the object, which eliminates most of speckle noises in the reproduced object if the object to be prepared in the computer has homogeneous surface. In common with the kinoform, this method can use all the incoming light energy for the desired complex object, so the resulting diffraction efficiency is much higher than that obtained with conventional CGHs.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3971
2.2. Simulation 2.2.1. Models and flows In Sect. 2.3, we report a numerical simulation that was performed in order to understand behaviors of the SCMM and to compare the performance among the SCMM, double-phase hologram, and off-axis amplitude hologram. Thus, in this section, we describe simulation conditions for each of the modulation methods. The simulation parameters are summarized in Table 1. The specifications of the phase-only SLM modeled in the simulation are as follows; the number of pixels is 1024 × 1024, the modulation range is [0, 2π], the number of gray levels is 256 (8 bit), and the fill factor and reflectivity (transmittance) are each 100%. Simulation models and flows are illustrated in Figs. 2 and 3, respectively. The complex object treated in this simulation is assumed to be a two-dimensional complex image for simplicity. For each modulation method, the reconstruction space of the original object is arranged on a plane optically equivalent to the SLM plane via a 4f imaging system. In the simulation flow for the double-phase hologram in Fig. 3(b), the decomposition of the given complex value A(x,y)exp[jφ(x,y)] into two phase θ1 and θ2 is calculated as follows: A( x, y ) , 2
(1)
A( x, y ) . 2
(2)
θ1 ( x, y ) = ϕ ( x, y ) + cos −1
θ 2 ( x, y ) = ϕ ( x, y ) − cos −1
In the simulation flow for the off-axis amplitude hologram in Fig. 3(c), the amplitude hologram I(x,y) to be displayed on the SLM is given by: I ( x, y ) = I 0 + A( x, y ) cos [kx sin θin − ϕ ( x, y )] ,
(3)
where I0 is the bias component to keep the cosine term positive and θin is the incident angle of the off-axis reference plane wave interfering with the object. Table 1. Summarized simulation parameters. Common parameters Wavelength of laser source, λ (nm)
Parameters for spatial cross modulation 532
Diffusion angle by diffuser, θdiff (degree)
200
Zero padding rate, N0
Focal length of Fourier transform lens, Lf (mm) Number of data pixels, Ndx × Ndy
32 × 32
Pitch of data pixel, Ldx × Ldy
Variable
Nyquist aperture size, DN (µm)
λLf / Ldx
Gray level for amplitude image, Gp (level)
8
Gray level for phase image, Ga (level)
8
Parameters for SLM Gray level of SLM, Gslm(level) Pitch of SLM pixel, Lpx × Lpy (µm2) Number of SLM pixel, Nx × Ny Modulation depth of phase-only SLM, Md
Gray level of random diffuser, Gdiff (level)
10 × 10 1024 × 1024 2π
32 2048
Parameters for double-phase hologram Number of subpixels per data pixel, Nux × Nuy Size of spatial filter in Fourier plane ( × DN), Sf
Variabl e Variabl e
Parameters for off-axis amplitude hologram Number of SLM pixels per grating, Ns
256
Variabl e
Number of gratings per data pixel, Ng Incident angle of reference light, θin (degree) Size of spatial filter in Fourier plane ( × DN), Sf
Variabl e Variabl e λ/ (NsLdx) Variabl e
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3972
Fig. 2. Simulation models.
Fig. 3. Simulation flows.
Pixel layouts on the SLM for each complex amplitude modulation method are depicted in Fig. 4. On the SLM plane for the SCMM in Fig. 4(a), the original complex image consists of Ndx × Ndy SLM pixels where Ndx and Ndy are the number of data pixels along the x- and y-axis, respectively. Then, the cross-modulated image is calculated by spreading the original image over a wider region with the diffuser at a diffusion ratio Ndiff and composed of NdxNdiff × NdyNdiff SLM pixels. Here, the diffusion ratio Ndiff is defined as the ratio of the original and diffusion image (encoded image) sizes and it is calculated as follows: N diff =
θ diff + cos−1 ( N dx Ldx / 2 L f ) cos −1 ( N dx Ldx / 2 L f
)
.
(4)
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3973
Here, arccosine term represents the maximum angular spectrum of the original complex image, θdiff is the diffusion angle of the diffuser, Ldx is the pitch of the data pixels, and Lf is the focal length of the Fourier transform lens. The entire area on the SLM is expressed as NdxN0 × NdyN0 where N0 is zero padding rate for the extra area that spreads out the original complex image. Here, if Ndiff is equal to N0, the diffusion image (encoded image) occupies the entire area on the SLM. On the SLM plane for the double-phase hologram method in Fig. 4(b), a phase image, similar to a chessboard pattern, is composed of NdxNux × NdyNuy pixels and it is displayed on the SLM. Here, Nux and Nuy are the number of subpixels (SLM pixel) along the x- and y-axis, respectively. In this method, one data pixel is expressed as an assembly of alternately arranged subpixels (SLM pixel). On the SLM plane for an off-axis amplitude hologram, represented in Fig. 4(c), the amplitude hologram consists of NdxNsNg × NdyNsNg pixels, where Ns is the number of SLM pixels per grating period along the x- or y-axis and Ng is the number of gratings within one data pixel along x- or y-axis. The region of one data pixel then includes NsNg × NsNg SLM pixels.
Fig. 4. Pixel layouts on SLM plane for each complex modulation. (a) Zero padding rate N0 provides extra area to diffuse the original image. (b) One data pixel consists of Nux × Nuy subpixels (SLM pixels). (c) One data pixel consists of NsNg × NsNg SLM pixels.
Examples of encoded images for each complex modulation method are summarized in Figs. 5(a)–5(c). Parameters for the original image are as stated in Table 1. Figure 5(a) shows a cross-modulated image with Ndiff = 8. This image shows that the original image spreads across wide area and the original pixel structure totally collapses. Figure 5(b) shows a chessboardtype phase image with Nux × Nuy = 8 × 8. In this pattern, it can be observed that two phase values, θ1 and θ2, decomposed from a given complex value, are alternately arranged.
Fig. 5. Example of encoded images displayed on the SLM. These are used for reconstructing the original complex object.
Figure 5(c) shows an amplitude hologram when Ns is 4 and Ng is 2. It can be seen that one grating period consists of four SLM pixels and within one data pixel there are two gratings including totally 8 × 8 SLM pixels. #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3974
For a diffuser model, we assume a Gaussian random diffuser because most commercially available wide-angle diffusers have a Gaussian-like autocorrelation [19]. The transmission function t(x,y) of the diffuser is represented as 2π t ( x, y ) = exp i ( ndiff − 1) h ( x, y ) , λ
(5)
where ndiff is the refractive index of the diffuser, λ is the laser wavelength, and h(x,y) is the diffuser surface profile. Details for calculating the diffuser profile are described in ref [19]. 2.2.2. Results At first, it will be demonstrated that the SCMM performs appropriately as full-complex amplitude modulation method. In Figs. 6(a) and 6(b), the original amplitude and phase images (arranged at the input plane in Fig. 1(a)) are shown respectively. These images are expressed by 256 gray level and contains all levels properly, thus the combination of these images includes full complex values. Amplitude and phase images (at the SLM plane in Fig. 1(b)), scattered by the random diffuser having surface profile of Fig. 6(c), are shown in Figs. 6(d) and 6(e). Here, it is important to note that this scattered phase image has the spatial cross modulation of the original amplitude and phase images. Reproduced results using the phaseonly image of Fig. 6(e) are presented in Figs. 6(f) and 6(e). It was found that the original phase image is reproduced without degradation, whereas the amplitude image includes slight background noises. This is due to the assumption where the scattered amplitude in Fig. 6(d) is to be constant. We can reduce this background noise by improving inhomogeneity of the scattered amplitude wavefront with higher diffusion ratio Ndiff. However, note that, when increasing diffusion ratio Ndiff, the achievable spatial resolution of the image is to be low.
Fig. 6. Full complex amplitude modulation by spatial cross modulation method. Number of data pixel in used image is 256 × 256 and diffusion ratio by diffuser is 4.0. Note that the number of data pixel is different from that shown in Table 1 just in this case. (a) Original amplitude image; (b) Original phase image; (c) Spatial profile of random diffuser; (d) Diffused amplitude image; (e) Diffused phase image (Cross-modulated image); (f) Reproduced amplitude image; (g) Reproduced phase image.
Next, we discuss the behavior of amplitude and phase components in the reproduced image when increasing the spread extent with a random diffuser. Figure 7 shows the dependence of the reconstruction quality on the diffusion ratio Ndiff. Here, to evaluate the quality of reproduced complex images, we use signal to noise ratio (SNR) defined as follows: #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3975
N dx −1 N dy −1
SNR ( dB ) = 10 log10
O ( mL
m=0
n =0
{O ( mL
N dx −1 N dy −1 m=0
n =0
dx
dx
, nLdy )
2
, nLdy ) − R ( mLdx , nLdy ) 2
2
}
,
(6)
where (mLdx,nLdy) denotes the spatial pixel coordinate in x-y plane, O(mLdx,nLdy) denotes amplitude or phase distributions of the original image, and R(mLdx,nLdy) denotes amplitude or phase distribution of the reproduced image. The phase distribution is expressed in the range [0, 2π] and the reproduced phase distribution is not phase-unwrapped. In Fig. 7, it is shown that the SNR of the amplitude image is improved with increasing the diffusion ratio Ndiff, as expected. The reason for this is that with increasing Ndiff, more information from the original amplitudes is transferred to the scattered-phase wavefront. It is also interesting to note that the SNR of the phase decreases as Ndiff reaches 3 and then turns to increase from Ndiff = 3. We think that this behavior is due to the homogeneity in the scattered amplitude distribution. In particular, while Ndiff is low, the scattered amplitude distribution is highly inhomogeneous and the resulting gap between the true value in the scattered amplitude and the approximated value of the constant amplitude deteriorates the phase image. On the other hand, while Ndiff is high, since the inhomogeneity in the scattered amplitude is reduced and the gap described above appears to be small, the SNR of the phase image is improved. For another possible explanation for a the SNR decay, the original phase information may have been transferred to the scattered amplitude by the diffuser. However, if this reason is true, the SNR of the phase should continue to decrease when Ndiff is greater than 3.0, and so we can rule it out.
Fig. 7. Reconstruction quality vs. diffusion ratio of the random diffuser, Ndiff.
A comparison between the SCMM and conventional methods in terms of the achievable spatial resolution was performed, as shown in Fig. 8. Here, the code rate for the horizontal axis is defined as the ratio of the size of the original image and the encoded image on the SLM and it is used for evaluating the achievable spatial resolution. If the code rate is large, the spatial resolution is low, and vice versa. A more detailed definition of the code rate is given in the caption of Fig. 8. Let us consider the result below. As shown in Fig. 8, the SNR improves in proportion to the code rate for each of the each modulation method. Remarkably, this result shows that across every code rate the SCMM keeps better SNR than conventional methods. Thus, the SCMM can obtain the same SNR using a lower code rate than conventional methods. If the SNR threshold is set to be 10(dB), the SCMM requires a code rate between 2 and 3, whereas conventional methods require a code rate of 6. This shows that the SCMM is superior to conventional methods in terms of
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3976
achievable spatial resolution. Table 2 shows examples of reproduced complex images when the code rate is 8. With conventional methods, the edges of the data pixel blur due to spatial filtering. The lack in uniformity within each data pixel can also be observed. In contrast, the SCMM preserves high spatial frequency and no exhibits no inhomogeneity.
Fig. 8. Comparison of the achievable spatial resolutions for SCMM and conventional methods. The code rate, on the horizontal axis, is defined as the ratio of the size of original image to the encoded image on the SLM, and is used for evaluating the achievable spatial resolution. For example, in the SCMM, if the phase-only diffusion image is displayed on the SLM in the region being 4 × 4 times larger than the original image, the code rate is 4. As well, in the case of the double-phase hologram, if the number of subpixels Nux × Nuy within one data pixel is 4 × 4, the code rate is 4. In the case of the off-axis amplitude hologram, if the number of SLM pixels NsNg × NsNg within one data pixel is 4 × 4, the code rate is regarded as 4. The size of the square aperture used in the off-axis amplitude hologram and double-phase hologram is optimized to obtain the highest SNR in order for a fair comparison. Table 2. Example of reproduced complex images for each complex modulation methods when code rate is 8. Spatial cross modulation
Double-phase hologram
Off-axis amplitude hologram
Amplitude
Phase
Another comparison in terms of the diffraction efficiency was conducted, as shown in Fig. 9. In the double-phase hologram, the obtained diffraction efficiency is less than 20%. This is due to the elimination of unwanted high diffraction orders to make the complex modulation #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3977
(Fig. 10(b)). In the off-axis amplitude hologram, a diffraction efficiency of nearly 5% is obtained. Even if the off-axis hologram is operated in phase-mode instead of amplitude-mode, the obtainable diffraction efficiency will only be about 20%. The power of the incident light is dispersed into several diffraction orders with each conventional modulation method (Figs. 10(b) and 10(c)), severely limiting the resulting diffraction efficiency. In contrast, the SCMM achieves a diffraction efficiency of nearly 90%, as shown in Fig. 9. In the SCMM, the illuminated phase-only image on the SLM yields a single diffraction order (Fig. 10(a)) and, ideally, all the incident light is used to reconstruct one image like a blazed grating. From these results, it is clear that the SCMM exceeds conventional methods in diffraction efficiency.
Fig. 9. Comparison of achievable diffraction efficiency of SCMM and conventional methods. The size of the square aperture used in the off-axis amplitude hologram and double-phase hologram is the same as used in Fig. 8.
Fig. 10. Spatial Fourier spectrum for each complex modulation method. (a) No unwanted diffraction order appears. (b) The zeroth diffraction order corresponds to the desired complex signal. (c) The first diffraction order corresponds to the desired complex signal.
2.3. Application to holographic data storage The spatial cross modulation method (SCMM) has many applications such as threedimensional displays, biomedical imaging, and beam shaping. Its application to holographic data storage (HDS) is particularly beneficial. For example, spatial quadrature amplitude modulation (SQAM) [18], the combination of amplitude and phase modulations, becomes available using SCMM, improving the data transfer rate and storage capacity. Moreover, this method may reduces the use of optical elements and helps eliminate optical distortions. In this
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3978
section, we report the experimental demonstration of the applicability of SCMM in the HDS without additional optical elements and its ability to generate a SQAM signal. 2.3.1. Basic operation Herein, we explain the basic operation when the SCMM is introduced in the HDS system. In conventional HDS systems, amplitude-only and phase-only SLMs are combined to generate a SQAM signal. Even if CGHs with a single SLM such as off-axis hologram and double-phase hologram are used to generate the SQAM signal, an optical decoding processing using a spatial filtering system must be added to the conventional HDS system in order to remove unwanted diffracted light. This consequently complicates the overall HDS system. In contrast, in the SCMM, since only the desired image, the cross-modulated image including the original information, is diffracted from the SLM, the cross-modulated image can be directly recorded in a recording medium without optical decoding processing. The processing to decode the cross-modulated image can be digitally performed through phase detection after reading the recorded hologram. Hence, the SCMM can be applied in conventional HDS systems without adding optical elements such as optical diffusers and lenses that would be necessary for optical decoding. Figure 11 illustrates an overview of the new HDS system including three main steps: (a) Digital diffusion step; (b) Optical hologram recording/reading step; (c) Digital phase conjugate reconstruction step. In the digital diffusion step (Encode step) in Fig. 11(a), the SQAM signal A(x,y)exp[jφ(x,y)] is diffused by a digital random diffuser with the phase distribution exp[jhd(x,y)], as described in Fig. 1(a). Then, the diffused amplitude distribution S(x,y) is made uniform based on the cross modulation effect of the diffuser to obtain the phase-only diffusion (cross-modulated) image exp[-jξ(x,y)]. In the optical hologram recording/reading step in Fig. 11(b), the phase-only image exp[-jξ(x,y)] made during step (a) is projected onto the phase-only SLM and is recorded as a hologram in the same manner as a conventional HDS system. The conventional HDS system directly records the binary amplitude signal in a recording medium, whereas in the new HDS system with the SCMM, the phase-only image exp[-jξ(x,y)] including most of features about the SQAM signal A(x,y)exp[jφ(x,y)] is recorded alternatively in the medium. During reading, cross-modulated image exp[-jξ'(x,y)] is read out from the recorded hologram and is measured through the phase detection method. Here, if we assume that the holographic recording/readout is distortion-free, the image exp[-jξ(x,y)] displayed on the SLM and the image exp[-jξ'(x,y)] measured by the phase detection method will be the same. In the digital phase conjugate reconstruction step (Decode step) in Fig. 11(c), the detected phase-only diffusion image exp[jξ'(x,y)] is passed through the digital diffuser with the phase distribution exp[jhd(x,y)] again in the phase conjugate optical system of Fig. 11(a). The recovered SQAM signal A'(x,y)exp[jφ'(x,y)] is then obtained from the phase-only image.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3979
Fig. 11. Conceptual diagrams of our system with spatial cross modulation introduced in holographic data storage.
2.3.2. Experimental setup and results Experimental setup for our proposed system is shown in Fig. 12 and the original SQAM signal used in the experiment is shown in Fig. 13. The signal has two intensity and four phase levels, and each data pixel is comprised of 16 × 16 SLM pixels. The phase-only diffusion image, encoded by the random diffuser with diffusion ratio Ndiff = 4.0, is shown in Fig. 14(a). It is displayed on a phase-only LCoS-SLM in Fig. 12. The diffusion image, modulated by the SLM, propagates through a recording medium. The phase-only diffusion image is then detected by a holographic diversity interferometer (HDI) composed of two CCDs [20]. The phase image detected by the HDI is shown in Fig. 14(b). This phase image is decoded through digital phase conjugate reconstruction, and the resulting reproduced image is shown in Fig. 15. As shown in Fig. 15(a), the reproduced intensity image appears to be slightly blurred, but the binary structure can be observed clearly. In the reproduced phase image shown in Fig. 15(b), enough contrast is obtained to readily distinguish four phase levels. These result indicate that the SCMM can appropriately generate the SQAM signal while improving the data transfer rate and storage capacity in the HDS. Finally, the reproduced SQAM signals depending on the diffusion ratio are presented in Table 3. It can be seen that the quality of reproduced images is greatly increased with increasing the diffusion ratio Ndiff. This result is similar to the simulated result in Fig. 7.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3980
Fig. 12. Experimental setup. Reference 1 is for optical hologram recording/reading. Reference 2 is for the phase detection method. In this experiment, holographic recording/reading is not conducted for simplicity, so the reference 1 is not used in actuality. The SLM used in our experiment is a phase-only LCoS-SLM (Hamamatsu, x10468). The recording medium arranged in the focal plane of L2 is a nanogel photopolymer with thickness of 400µm, provided by Kyoeisha Chemical Co. The phase of the diffusion image propagated from the SLM is detected by a holographic diversity interferometer (HDI) composed of two CCDs.
Fig. 13. Original SQAM signal used in our experiment. Each pixel in the intensity image randomly takes either of two values, 1 and 0.25. Each pixel in the phase image has either of four values: π/2, π, 3π/2, and 2π. Therefore, a 8-level SQAM is to be used. Each data pixel consists of 16 × 16 SLM pixels.
Fig. 14. (a) Cross-modulated image displayed on SLM. LCoS-SLM used in our experiment can modulate phase delay of 2π when the gray level is 85. This image was displayed on a region of 512 × 512 SLM pixels. (b) Cross-modulated image measured with holographic diversity interferometer using two CCDs. This image is also detected on a region of 512 × 512 CCD pixels.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3981
Fig. 15. SQAM signals recovered through digital phase conjugate reconstruction in Fig. 11(c). Table 3. Experimental results for reconstruction quality vs. diffusion ratio Ndiff. Diffusion ratio, Ndiff
2.5
3.0
3.5
4.0
Intensity
Phase
3. Conclusion We have proposed the spatial cross modulation method (SCMM), in which the complex object can be encoded as a phase-only diffusion image using a random diffuser. This distinguishes SCMM from the conventional method of kinoform where the encoding is performed via Fourier transformation. Simulation results clarified that the SCMM can achieve higher spatial resolution and diffraction efficiency, as compared with off-axis hologram and double-phase hologram. In the experiment examining possible application to HDS, we succeeded in generating an 8-level SQAM signal without the combination of amplitude-only SLM and phase-only SLM. Future experimental work will focus on the optical reconstruction of the complex object. We expect that optical reconstruction is realized readily by using the phase-only SLM as the optical random diffuser.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3982
2
Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-Ku, Sapporo, Hokkaido 060-0814, Japan Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan * [email protected]
Abstract: We propose a spatial cross modulation method using a random diffuser and a phase-only spatial light modulator (SLM), by which arbitrary complex-amplitude fields can be generated with higher spatial resolution and diffraction efficiency than off-axis and double-phase computergenerated holograms. Our method encodes the original complex object as a phase-only diffusion image by scattering the complex object using a random diffuser. In addition, all incoming light to the SLM is consumed for a single diffraction order, making a diffraction efficiency of more than 90% possible. This method can be applied for holographic data storage, threedimensional displays, and other such applications. ©2014 Optical Society of America OCIS codes: (090.1760) Computer holography; (090.1970) Diffractive optics; (110.7348) Wavefront encoding; (210.2860) Holographic and volume memories.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6(10), 1739–1748 (1967). A. W. Lohmann and D. P. Paris, “Computer generated spatial filters for coherent optical data processiing,” Appl. Opt. 7(4), 651–655 (1968). A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a singlephase-only spatial light modulator,” Opt. Lett. 29(3), 295–297 (2004). Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. 28(24), 2518–2520 (2003). A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10(3), 619–624 (1971). V. Arrizón, G. Méndez, and D. Sánchez-de-La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express 13(20), 7913–7927 (2005). C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. 17(24), 3874–3883 (1978). J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991). Z. Göröcs, G. Erdei, T. Sarkadi, F. Ujhelyi, J. Reményi, P. Koppa, and E. Lorincz, “Hybrid multinary modulation using a phase modulating spatial light modulator and a low-pass spatial filter,” Opt. Lett. 32(16), 2336–2338 (2007). L. B. Lesem, P. M. Hirch, and J. A. Jordan, Jr., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969). A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981). J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23(6), 812–816 (1984). M. Stanley, M. A. Smith, A. P. Smith, P. J. Watson, S. D. Coomber, C. D. Cameron, C. W. Slinger, and A. Wood, “3D electronic holography display system using a 100Mega-pixel spatial light modulator,” Proc. SPIE 5249, 297–308 (2004). E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5(8), 1303–1311 (1966).
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3968
16. K. Anderson and K. Curtis, “Polytopic multiplexing,” Opt. Lett. 29(12), 1402–1404 (2004). 17. M. Takabayashi and A. Okamoto, “Self-referential holography and its applications to data storage and phase-tointensity conversion,” Opt. Express 21(3), 3669–3681 (2013). 18. K. Zukeran, A. Okamoto, M. Takabayashi, A. Shibukawa, K. Sato, and A. Tomita, “Double-referential holography and spatial quadrature amplitude modulation,” Jpn. J. Appl. Phys. 52(9S2), 09LD13 (2013). 19. L. G. Shirley and N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A 4(4), 734–745 (1987). 20. A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, and K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express 19(14), 13436–13444 (2011).
1. Introduction A method to construct arbitrary complex amplitude fields [1] was used for the first time in the field of optical information processing half a century ago [2]. Since that time, this method has been widely used in numerous applications, ranging from optical tweezers to threedimensional (3D) displays [3–5]. A well-known method for generating arbitrary complex amplitude fields is to display an off-axis computer-generated hologram (CGH) on a spatial light modulator (SLM), in which a desired complex amplitude object is encoded as amplitude or phase holograms through interference with an off-axis reference plane wave [6,7]. Using this method, however, consumes energy of the incident plane wave not only for the desired diffraction order but also unwanted diffraction orders. This considerably lowers the diffraction efficiency for the desired diffraction order. In addition, several tens of SLM pixels must be sacrificed within one data pixel to form a diffraction grating that satisfies the sampling theorem; therefore, this method is limited to obtaining a low spatial resolution in the reconstructed image. The other typical CGH used for complex amplitude modulation includes a double-phase hologram [8–10]. In this method, a given complex amplitude is decomposed into two phase values, and then the phase distribution, synthesized by letting these two values spatially and alternately arrange in a chessboard pattern, is displayed on the SLM. Subsequently, the desired complex amplitude is obtained by isolating a zeroth diffraction order spatially. However, this method and the above CGH method are alike in that a diffraction grating is used to obtain the desired complex amplitude object, significantly limiting the achievable diffraction efficiency and spatial resolution. In contrast, a kinoform [11], on-axis CGH, can diffract all incoming light into a single diffraction light like a blazed diffraction grating, avoiding all of the above-mentioned problems inherent to holograms. The kinoform encodes the original object into a spectral phase image within its Fourier domain. The use of this phase image is based on the fact that the Fourier spectrum of a diffusive object produces a homogeneous scattered wavefront and, even if only the phase in the scatted wavefront is retained, most of the important features of the original object are preserved [12,13]. However, in the kinoform, the reconstruction space of the object is restricted in the Fourier domain of the spectral phase image plane. For instance, when the Fourier transform is performed with a lens for the reconstruction of the kinoform, the size of the reconstructed object will be extremely small, on the order of millimeters [14]. In addition, the diffusive object is made by embedding a random phase pattern directly into the original object to ensure a smooth spectrum, causing large speckle noise in the reconstructed object. To solve the above problems, we propose a spatial cross modulation method (SCMM) using a random diffuser and phase-only SLM. The SCMM encodes a complex object as a scattered phase image by allowing the object to transmit through a random diffuser. Employing a random diffuser between the object and phase image planes enables us to make the scattered wavefront in an arbitrary space, which increases the degrees of freedom for the design of an optical system to reconstruct the larger object. Moreover, although the kinoform embeds the random phase directly into the object, the SCMM arranges the random diffuser with the random phase outside the object, and therefore, if the object to be reconstructed has comparatively homogeneous surface, speckle noise is not mostly caused in the reproduced object. Furthermore, like with the kinoform method, this method can consume all incoming light for a single diffraction order, so the resulting diffraction efficiency can be extremely #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3969
high. In addition, the SCMM can achieve higher spatial resolution than off-axis and doublephase hologram methods. In Sect. 2.1, the basic operation of the SCMM will be described. In Sect. 2.2, a numerical simulation to investigate the reconstruction characteristics of the SCMM is reported. In particular, we first confirm that full-complex-amplitude modulation is possible with the SCMM. Next, we discuss the behavior of the reconstruction quality when changing the extent to which the diffuser spreads the original complex object. The achievable spatial resolution and diffraction efficiency is then compared among the SCMM, off-axis hologram and doublephase hologram methods. In Sect 2.3, we present an application of the SCMM to holographic data storage (HDS) [15–17]. In particular, the basic operation in the case of SCMM applied to HDS is described. Then, we report an experiment for producing a data page with arbitrary complex amplitude modulation. The SCMM can promote the use of spatial quadrature amplitude modulation [18], in which multi-valued amplitude and phase modulations are combined in order to the increase data transfer rate and storage capacity of HDS. 2. Spatial cross modulation 2.1. Basic operation Let us begin with an explanation of the basic operation of the SCMM, as illustrated in Fig. 1. The operation is divided into two main steps: the calculation of the phase-only diffusion image in a computer (digital encode step) and the optical reconstruction of the desired complex field with the SLM (optical decode step). In Fig. 1, the reconstruction plane of the original object is set to be optically equal to the SLM plane via a 4f optical system, but this configuration of the SCMM is only one possibility. The SCMM can be also realized by a simpler configuration where the 4f optical system is removed and only the diffuser lies between the input (reconstruction) and output (SLM) planes.
Fig. 1. Conceptual diagrams of spatial cross modulation. (a) Digital encode step in a computer. (b) Optical decode step.
During the digital encode step completed within the computer in Fig. 1(a), an arbitrary complex object A(x,y)exp[jφ(x,y)] is first prepared in an input plane in order to be reconstructed in the subsequent decode step. The complex object is not restricted to either two or three dimensions, but the following explanation proceeds from a two-dimensional (2D) object. Next, the Fourier transformation of the object F{A(x,y)exp[jφ(x,y)]} is calculated with a fast Fourier transform (FFT). Here, F{·} denotes the operator of two-dimensional Fourier
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3970
transform. Then, the spatial phase distribution of a digital random diffuser (a virtual random diffuser) exp[jhd(x,y)] and the spatial spectrum of the object F{A(x,y)exp[jφ(x,y)]} are multiplied in a spatial Fourier plane. Through an inverse Fourier transformation of the multiplied spectrum distribution F−1{exp[jh(x,y)]·F{A(x,y)exp[jφ(x,y)]}} by inverse FFT (IFFT), a diffusion image (scattered wavefront) of the original object S(x,y)exp[jξ(x,y)] results within an output plane. Subsequently, amplitude distribution in the diffusion image S(x,y) are uniformed and the phase conjugation exp[-jξ(x,y)] is calculated, which is mathematically equivalent to inverting the sign of the phase distribution in the image. The elimination of the amplitude is based on the assumption that the scattered phase contains more of the important features of the original complex object than the scattered amplitude does. In other words, both the original amplitude A(x,y) and the original phase exp[jφ(x,y)] are spatially cross-modulated into a phase-only diffusion image exp[jξ(x,y)]. This importance of the phase has been demonstrated in a number of different contexts, including optical and acoustical holograms [12,13]. The phase-only image exp[-jξ(x,y)] is referred to as cross-modulated image herein. In regard to the complexity of the calculations, the above encoding step includes 2D FFT, 2D IFFT, and the multiplication of 2D distributions. The total complexity is therefore given by 2N2log2N + N2, where the SLM has N2 pixels in the x-y plane. If a three-dimensional (3D) object A(x,y,z)exp[jφ(x,y,z)] is treated with the SCMM, the 3D object A(x,y,z)exp[jφ(x,y,z)] is prepared as an assembly of many 2D images on the input plane. Each 2D image An(x,y,zn)exp[jφn(x,y,zn)] is changed to a cross-modulated image exp[-jξn(x,y)] with the above encode step, and then each cross-modulated image exp[-jξn(x,y)] is superimposed on the output plane to obtain the final cross-modulated image exp[-jξ(x,y)] to be displayed on the SLM. During the optical decode step, as shown in Fig. 1(b), the cross-modulated image exp[jξ(x,y)] calculated in step (a) is first displayed onto a phase-only SLM. After incident light falls on the SLM, the modulated light is Fourier-transformed via the first lens. Then, the light F{exp[-jξ(x,y)]} propagates through the optical diffuser with a phase distribution exp[jho(x,y)], and the reconstructed complex object A'(x,y)exp[jφ'(x,y)] is obtained with a second lens. Note that the reconstructed object A'(x,y)exp[jφ'(x,y)] is extremely similar to the original object A(x,y)exp[jφ(x,y)] but have slightly random noise due to the elimination of the scattered amplitude. The spatial phase distributions exp[jhd(x,y)] and exp[jho(x,y)] of the digital and optical diffusers must be matched for correctly working the SCMM. Two ways for matching these distributions are possible: one measures the distribution of the optical diffuser by a phase detection method for the decoding and subsequently uses the measured phase distribution as the digital diffuser in the encode step; another uses an electrically addressable phase-only SLM for the optical diffuser. The latter method appears to be more practical. As stated above, the kinoform uses the Fourier spectrum of the object to produce a scattered wavefront (encoded image), so the reconstruction space of the object is restricted to the Fourier domain of the encoded image plane. In contrast, the SCMM places a random diffuser between the object and encoded image planes, which produces the scattered wavefront without restriction, meaning that the object can be reconstructed in arbitrary spaces such as the Fresnel region and the optically equivalent (conjugate) region to the SLM plane, via a 4f system. This feature increases the degree of freedom for designing the optical system, so allows larger objects to be reconstructed as compared to the kinoform based on a Fourier transform lens. In addition, the kinoform adds a random phase into the object to make it diffusive for the smooth Fourier spectrum, and so the reconstructed object includes speckle noise. On contrast, the SCMM places the diffuser with the random phase outside the object, which eliminates most of speckle noises in the reproduced object if the object to be prepared in the computer has homogeneous surface. In common with the kinoform, this method can use all the incoming light energy for the desired complex object, so the resulting diffraction efficiency is much higher than that obtained with conventional CGHs.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3971
2.2. Simulation 2.2.1. Models and flows In Sect. 2.3, we report a numerical simulation that was performed in order to understand behaviors of the SCMM and to compare the performance among the SCMM, double-phase hologram, and off-axis amplitude hologram. Thus, in this section, we describe simulation conditions for each of the modulation methods. The simulation parameters are summarized in Table 1. The specifications of the phase-only SLM modeled in the simulation are as follows; the number of pixels is 1024 × 1024, the modulation range is [0, 2π], the number of gray levels is 256 (8 bit), and the fill factor and reflectivity (transmittance) are each 100%. Simulation models and flows are illustrated in Figs. 2 and 3, respectively. The complex object treated in this simulation is assumed to be a two-dimensional complex image for simplicity. For each modulation method, the reconstruction space of the original object is arranged on a plane optically equivalent to the SLM plane via a 4f imaging system. In the simulation flow for the double-phase hologram in Fig. 3(b), the decomposition of the given complex value A(x,y)exp[jφ(x,y)] into two phase θ1 and θ2 is calculated as follows: A( x, y ) , 2
(1)
A( x, y ) . 2
(2)
θ1 ( x, y ) = ϕ ( x, y ) + cos −1
θ 2 ( x, y ) = ϕ ( x, y ) − cos −1
In the simulation flow for the off-axis amplitude hologram in Fig. 3(c), the amplitude hologram I(x,y) to be displayed on the SLM is given by: I ( x, y ) = I 0 + A( x, y ) cos [kx sin θin − ϕ ( x, y )] ,
(3)
where I0 is the bias component to keep the cosine term positive and θin is the incident angle of the off-axis reference plane wave interfering with the object. Table 1. Summarized simulation parameters. Common parameters Wavelength of laser source, λ (nm)
Parameters for spatial cross modulation 532
Diffusion angle by diffuser, θdiff (degree)
200
Zero padding rate, N0
Focal length of Fourier transform lens, Lf (mm) Number of data pixels, Ndx × Ndy
32 × 32
Pitch of data pixel, Ldx × Ldy
Variable
Nyquist aperture size, DN (µm)
λLf / Ldx
Gray level for amplitude image, Gp (level)
8
Gray level for phase image, Ga (level)
8
Parameters for SLM Gray level of SLM, Gslm(level) Pitch of SLM pixel, Lpx × Lpy (µm2) Number of SLM pixel, Nx × Ny Modulation depth of phase-only SLM, Md
Gray level of random diffuser, Gdiff (level)
10 × 10 1024 × 1024 2π
32 2048
Parameters for double-phase hologram Number of subpixels per data pixel, Nux × Nuy Size of spatial filter in Fourier plane ( × DN), Sf
Variabl e Variabl e
Parameters for off-axis amplitude hologram Number of SLM pixels per grating, Ns
256
Variabl e
Number of gratings per data pixel, Ng Incident angle of reference light, θin (degree) Size of spatial filter in Fourier plane ( × DN), Sf
Variabl e Variabl e λ/ (NsLdx) Variabl e
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3972
Fig. 2. Simulation models.
Fig. 3. Simulation flows.
Pixel layouts on the SLM for each complex amplitude modulation method are depicted in Fig. 4. On the SLM plane for the SCMM in Fig. 4(a), the original complex image consists of Ndx × Ndy SLM pixels where Ndx and Ndy are the number of data pixels along the x- and y-axis, respectively. Then, the cross-modulated image is calculated by spreading the original image over a wider region with the diffuser at a diffusion ratio Ndiff and composed of NdxNdiff × NdyNdiff SLM pixels. Here, the diffusion ratio Ndiff is defined as the ratio of the original and diffusion image (encoded image) sizes and it is calculated as follows: N diff =
θ diff + cos−1 ( N dx Ldx / 2 L f ) cos −1 ( N dx Ldx / 2 L f
)
.
(4)
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3973
Here, arccosine term represents the maximum angular spectrum of the original complex image, θdiff is the diffusion angle of the diffuser, Ldx is the pitch of the data pixels, and Lf is the focal length of the Fourier transform lens. The entire area on the SLM is expressed as NdxN0 × NdyN0 where N0 is zero padding rate for the extra area that spreads out the original complex image. Here, if Ndiff is equal to N0, the diffusion image (encoded image) occupies the entire area on the SLM. On the SLM plane for the double-phase hologram method in Fig. 4(b), a phase image, similar to a chessboard pattern, is composed of NdxNux × NdyNuy pixels and it is displayed on the SLM. Here, Nux and Nuy are the number of subpixels (SLM pixel) along the x- and y-axis, respectively. In this method, one data pixel is expressed as an assembly of alternately arranged subpixels (SLM pixel). On the SLM plane for an off-axis amplitude hologram, represented in Fig. 4(c), the amplitude hologram consists of NdxNsNg × NdyNsNg pixels, where Ns is the number of SLM pixels per grating period along the x- or y-axis and Ng is the number of gratings within one data pixel along x- or y-axis. The region of one data pixel then includes NsNg × NsNg SLM pixels.
Fig. 4. Pixel layouts on SLM plane for each complex modulation. (a) Zero padding rate N0 provides extra area to diffuse the original image. (b) One data pixel consists of Nux × Nuy subpixels (SLM pixels). (c) One data pixel consists of NsNg × NsNg SLM pixels.
Examples of encoded images for each complex modulation method are summarized in Figs. 5(a)–5(c). Parameters for the original image are as stated in Table 1. Figure 5(a) shows a cross-modulated image with Ndiff = 8. This image shows that the original image spreads across wide area and the original pixel structure totally collapses. Figure 5(b) shows a chessboardtype phase image with Nux × Nuy = 8 × 8. In this pattern, it can be observed that two phase values, θ1 and θ2, decomposed from a given complex value, are alternately arranged.
Fig. 5. Example of encoded images displayed on the SLM. These are used for reconstructing the original complex object.
Figure 5(c) shows an amplitude hologram when Ns is 4 and Ng is 2. It can be seen that one grating period consists of four SLM pixels and within one data pixel there are two gratings including totally 8 × 8 SLM pixels. #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3974
For a diffuser model, we assume a Gaussian random diffuser because most commercially available wide-angle diffusers have a Gaussian-like autocorrelation [19]. The transmission function t(x,y) of the diffuser is represented as 2π t ( x, y ) = exp i ( ndiff − 1) h ( x, y ) , λ
(5)
where ndiff is the refractive index of the diffuser, λ is the laser wavelength, and h(x,y) is the diffuser surface profile. Details for calculating the diffuser profile are described in ref [19]. 2.2.2. Results At first, it will be demonstrated that the SCMM performs appropriately as full-complex amplitude modulation method. In Figs. 6(a) and 6(b), the original amplitude and phase images (arranged at the input plane in Fig. 1(a)) are shown respectively. These images are expressed by 256 gray level and contains all levels properly, thus the combination of these images includes full complex values. Amplitude and phase images (at the SLM plane in Fig. 1(b)), scattered by the random diffuser having surface profile of Fig. 6(c), are shown in Figs. 6(d) and 6(e). Here, it is important to note that this scattered phase image has the spatial cross modulation of the original amplitude and phase images. Reproduced results using the phaseonly image of Fig. 6(e) are presented in Figs. 6(f) and 6(e). It was found that the original phase image is reproduced without degradation, whereas the amplitude image includes slight background noises. This is due to the assumption where the scattered amplitude in Fig. 6(d) is to be constant. We can reduce this background noise by improving inhomogeneity of the scattered amplitude wavefront with higher diffusion ratio Ndiff. However, note that, when increasing diffusion ratio Ndiff, the achievable spatial resolution of the image is to be low.
Fig. 6. Full complex amplitude modulation by spatial cross modulation method. Number of data pixel in used image is 256 × 256 and diffusion ratio by diffuser is 4.0. Note that the number of data pixel is different from that shown in Table 1 just in this case. (a) Original amplitude image; (b) Original phase image; (c) Spatial profile of random diffuser; (d) Diffused amplitude image; (e) Diffused phase image (Cross-modulated image); (f) Reproduced amplitude image; (g) Reproduced phase image.
Next, we discuss the behavior of amplitude and phase components in the reproduced image when increasing the spread extent with a random diffuser. Figure 7 shows the dependence of the reconstruction quality on the diffusion ratio Ndiff. Here, to evaluate the quality of reproduced complex images, we use signal to noise ratio (SNR) defined as follows: #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3975
N dx −1 N dy −1
SNR ( dB ) = 10 log10
O ( mL
m=0
n =0
{O ( mL
N dx −1 N dy −1 m=0
n =0
dx
dx
, nLdy )
2
, nLdy ) − R ( mLdx , nLdy ) 2
2
}
,
(6)
where (mLdx,nLdy) denotes the spatial pixel coordinate in x-y plane, O(mLdx,nLdy) denotes amplitude or phase distributions of the original image, and R(mLdx,nLdy) denotes amplitude or phase distribution of the reproduced image. The phase distribution is expressed in the range [0, 2π] and the reproduced phase distribution is not phase-unwrapped. In Fig. 7, it is shown that the SNR of the amplitude image is improved with increasing the diffusion ratio Ndiff, as expected. The reason for this is that with increasing Ndiff, more information from the original amplitudes is transferred to the scattered-phase wavefront. It is also interesting to note that the SNR of the phase decreases as Ndiff reaches 3 and then turns to increase from Ndiff = 3. We think that this behavior is due to the homogeneity in the scattered amplitude distribution. In particular, while Ndiff is low, the scattered amplitude distribution is highly inhomogeneous and the resulting gap between the true value in the scattered amplitude and the approximated value of the constant amplitude deteriorates the phase image. On the other hand, while Ndiff is high, since the inhomogeneity in the scattered amplitude is reduced and the gap described above appears to be small, the SNR of the phase image is improved. For another possible explanation for a the SNR decay, the original phase information may have been transferred to the scattered amplitude by the diffuser. However, if this reason is true, the SNR of the phase should continue to decrease when Ndiff is greater than 3.0, and so we can rule it out.
Fig. 7. Reconstruction quality vs. diffusion ratio of the random diffuser, Ndiff.
A comparison between the SCMM and conventional methods in terms of the achievable spatial resolution was performed, as shown in Fig. 8. Here, the code rate for the horizontal axis is defined as the ratio of the size of the original image and the encoded image on the SLM and it is used for evaluating the achievable spatial resolution. If the code rate is large, the spatial resolution is low, and vice versa. A more detailed definition of the code rate is given in the caption of Fig. 8. Let us consider the result below. As shown in Fig. 8, the SNR improves in proportion to the code rate for each of the each modulation method. Remarkably, this result shows that across every code rate the SCMM keeps better SNR than conventional methods. Thus, the SCMM can obtain the same SNR using a lower code rate than conventional methods. If the SNR threshold is set to be 10(dB), the SCMM requires a code rate between 2 and 3, whereas conventional methods require a code rate of 6. This shows that the SCMM is superior to conventional methods in terms of
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3976
achievable spatial resolution. Table 2 shows examples of reproduced complex images when the code rate is 8. With conventional methods, the edges of the data pixel blur due to spatial filtering. The lack in uniformity within each data pixel can also be observed. In contrast, the SCMM preserves high spatial frequency and no exhibits no inhomogeneity.
Fig. 8. Comparison of the achievable spatial resolutions for SCMM and conventional methods. The code rate, on the horizontal axis, is defined as the ratio of the size of original image to the encoded image on the SLM, and is used for evaluating the achievable spatial resolution. For example, in the SCMM, if the phase-only diffusion image is displayed on the SLM in the region being 4 × 4 times larger than the original image, the code rate is 4. As well, in the case of the double-phase hologram, if the number of subpixels Nux × Nuy within one data pixel is 4 × 4, the code rate is 4. In the case of the off-axis amplitude hologram, if the number of SLM pixels NsNg × NsNg within one data pixel is 4 × 4, the code rate is regarded as 4. The size of the square aperture used in the off-axis amplitude hologram and double-phase hologram is optimized to obtain the highest SNR in order for a fair comparison. Table 2. Example of reproduced complex images for each complex modulation methods when code rate is 8. Spatial cross modulation
Double-phase hologram
Off-axis amplitude hologram
Amplitude
Phase
Another comparison in terms of the diffraction efficiency was conducted, as shown in Fig. 9. In the double-phase hologram, the obtained diffraction efficiency is less than 20%. This is due to the elimination of unwanted high diffraction orders to make the complex modulation #203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3977
(Fig. 10(b)). In the off-axis amplitude hologram, a diffraction efficiency of nearly 5% is obtained. Even if the off-axis hologram is operated in phase-mode instead of amplitude-mode, the obtainable diffraction efficiency will only be about 20%. The power of the incident light is dispersed into several diffraction orders with each conventional modulation method (Figs. 10(b) and 10(c)), severely limiting the resulting diffraction efficiency. In contrast, the SCMM achieves a diffraction efficiency of nearly 90%, as shown in Fig. 9. In the SCMM, the illuminated phase-only image on the SLM yields a single diffraction order (Fig. 10(a)) and, ideally, all the incident light is used to reconstruct one image like a blazed grating. From these results, it is clear that the SCMM exceeds conventional methods in diffraction efficiency.
Fig. 9. Comparison of achievable diffraction efficiency of SCMM and conventional methods. The size of the square aperture used in the off-axis amplitude hologram and double-phase hologram is the same as used in Fig. 8.
Fig. 10. Spatial Fourier spectrum for each complex modulation method. (a) No unwanted diffraction order appears. (b) The zeroth diffraction order corresponds to the desired complex signal. (c) The first diffraction order corresponds to the desired complex signal.
2.3. Application to holographic data storage The spatial cross modulation method (SCMM) has many applications such as threedimensional displays, biomedical imaging, and beam shaping. Its application to holographic data storage (HDS) is particularly beneficial. For example, spatial quadrature amplitude modulation (SQAM) [18], the combination of amplitude and phase modulations, becomes available using SCMM, improving the data transfer rate and storage capacity. Moreover, this method may reduces the use of optical elements and helps eliminate optical distortions. In this
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3978
section, we report the experimental demonstration of the applicability of SCMM in the HDS without additional optical elements and its ability to generate a SQAM signal. 2.3.1. Basic operation Herein, we explain the basic operation when the SCMM is introduced in the HDS system. In conventional HDS systems, amplitude-only and phase-only SLMs are combined to generate a SQAM signal. Even if CGHs with a single SLM such as off-axis hologram and double-phase hologram are used to generate the SQAM signal, an optical decoding processing using a spatial filtering system must be added to the conventional HDS system in order to remove unwanted diffracted light. This consequently complicates the overall HDS system. In contrast, in the SCMM, since only the desired image, the cross-modulated image including the original information, is diffracted from the SLM, the cross-modulated image can be directly recorded in a recording medium without optical decoding processing. The processing to decode the cross-modulated image can be digitally performed through phase detection after reading the recorded hologram. Hence, the SCMM can be applied in conventional HDS systems without adding optical elements such as optical diffusers and lenses that would be necessary for optical decoding. Figure 11 illustrates an overview of the new HDS system including three main steps: (a) Digital diffusion step; (b) Optical hologram recording/reading step; (c) Digital phase conjugate reconstruction step. In the digital diffusion step (Encode step) in Fig. 11(a), the SQAM signal A(x,y)exp[jφ(x,y)] is diffused by a digital random diffuser with the phase distribution exp[jhd(x,y)], as described in Fig. 1(a). Then, the diffused amplitude distribution S(x,y) is made uniform based on the cross modulation effect of the diffuser to obtain the phase-only diffusion (cross-modulated) image exp[-jξ(x,y)]. In the optical hologram recording/reading step in Fig. 11(b), the phase-only image exp[-jξ(x,y)] made during step (a) is projected onto the phase-only SLM and is recorded as a hologram in the same manner as a conventional HDS system. The conventional HDS system directly records the binary amplitude signal in a recording medium, whereas in the new HDS system with the SCMM, the phase-only image exp[-jξ(x,y)] including most of features about the SQAM signal A(x,y)exp[jφ(x,y)] is recorded alternatively in the medium. During reading, cross-modulated image exp[-jξ'(x,y)] is read out from the recorded hologram and is measured through the phase detection method. Here, if we assume that the holographic recording/readout is distortion-free, the image exp[-jξ(x,y)] displayed on the SLM and the image exp[-jξ'(x,y)] measured by the phase detection method will be the same. In the digital phase conjugate reconstruction step (Decode step) in Fig. 11(c), the detected phase-only diffusion image exp[jξ'(x,y)] is passed through the digital diffuser with the phase distribution exp[jhd(x,y)] again in the phase conjugate optical system of Fig. 11(a). The recovered SQAM signal A'(x,y)exp[jφ'(x,y)] is then obtained from the phase-only image.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3979
Fig. 11. Conceptual diagrams of our system with spatial cross modulation introduced in holographic data storage.
2.3.2. Experimental setup and results Experimental setup for our proposed system is shown in Fig. 12 and the original SQAM signal used in the experiment is shown in Fig. 13. The signal has two intensity and four phase levels, and each data pixel is comprised of 16 × 16 SLM pixels. The phase-only diffusion image, encoded by the random diffuser with diffusion ratio Ndiff = 4.0, is shown in Fig. 14(a). It is displayed on a phase-only LCoS-SLM in Fig. 12. The diffusion image, modulated by the SLM, propagates through a recording medium. The phase-only diffusion image is then detected by a holographic diversity interferometer (HDI) composed of two CCDs [20]. The phase image detected by the HDI is shown in Fig. 14(b). This phase image is decoded through digital phase conjugate reconstruction, and the resulting reproduced image is shown in Fig. 15. As shown in Fig. 15(a), the reproduced intensity image appears to be slightly blurred, but the binary structure can be observed clearly. In the reproduced phase image shown in Fig. 15(b), enough contrast is obtained to readily distinguish four phase levels. These result indicate that the SCMM can appropriately generate the SQAM signal while improving the data transfer rate and storage capacity in the HDS. Finally, the reproduced SQAM signals depending on the diffusion ratio are presented in Table 3. It can be seen that the quality of reproduced images is greatly increased with increasing the diffusion ratio Ndiff. This result is similar to the simulated result in Fig. 7.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3980
Fig. 12. Experimental setup. Reference 1 is for optical hologram recording/reading. Reference 2 is for the phase detection method. In this experiment, holographic recording/reading is not conducted for simplicity, so the reference 1 is not used in actuality. The SLM used in our experiment is a phase-only LCoS-SLM (Hamamatsu, x10468). The recording medium arranged in the focal plane of L2 is a nanogel photopolymer with thickness of 400µm, provided by Kyoeisha Chemical Co. The phase of the diffusion image propagated from the SLM is detected by a holographic diversity interferometer (HDI) composed of two CCDs.
Fig. 13. Original SQAM signal used in our experiment. Each pixel in the intensity image randomly takes either of two values, 1 and 0.25. Each pixel in the phase image has either of four values: π/2, π, 3π/2, and 2π. Therefore, a 8-level SQAM is to be used. Each data pixel consists of 16 × 16 SLM pixels.
Fig. 14. (a) Cross-modulated image displayed on SLM. LCoS-SLM used in our experiment can modulate phase delay of 2π when the gray level is 85. This image was displayed on a region of 512 × 512 SLM pixels. (b) Cross-modulated image measured with holographic diversity interferometer using two CCDs. This image is also detected on a region of 512 × 512 CCD pixels.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3981
Fig. 15. SQAM signals recovered through digital phase conjugate reconstruction in Fig. 11(c). Table 3. Experimental results for reconstruction quality vs. diffusion ratio Ndiff. Diffusion ratio, Ndiff
2.5
3.0
3.5
4.0
Intensity
Phase
3. Conclusion We have proposed the spatial cross modulation method (SCMM), in which the complex object can be encoded as a phase-only diffusion image using a random diffuser. This distinguishes SCMM from the conventional method of kinoform where the encoding is performed via Fourier transformation. Simulation results clarified that the SCMM can achieve higher spatial resolution and diffraction efficiency, as compared with off-axis hologram and double-phase hologram. In the experiment examining possible application to HDS, we succeeded in generating an 8-level SQAM signal without the combination of amplitude-only SLM and phase-only SLM. Future experimental work will focus on the optical reconstruction of the complex object. We expect that optical reconstruction is realized readily by using the phase-only SLM as the optical random diffuser.
#203577 - $15.00 USD Received 26 Dec 2013; revised 7 Feb 2014; accepted 7 Feb 2014; published 12 Feb 2014 (C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003968 | OPTICS EXPRESS 3982
