FTh4G.2.pdf
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Fabrication of Multiplexed Computer Generated Volume Holograms in Photosensitive Glass 1 , Claas Falldorf1 , Ralf B. ¨ Edwin N. Kamau∗1 , Vijay V. Parsi Sreenivas2 , Mike Bulters Bergmann1,2 1
BIAS-Bremer Institut f¨ur Angewandte Strahltechnik, Klagenfurter Strasse 2, D-28359 Bremen Germany 2 Applied Optics, University of Bremen, Otto-Hahn-Allee 1, D-28359 Bremen Germany Corresponding author:
[email protected]∗
Abstract: We present a new approach for the fabrication of volume holograms in an optical nonlinear material with voxel sizes on the order of 1µ m i.e., with increased degrees of freedom and thus improved multiplexing functionality. OCIS codes: (090.4220) Mult. holography, (050.6875) 3D fabrication, (100.3190) Inv. Problems, (050.1970) DOEs.
1.
Introduction
In the recent past, major focus has been on 3D diffractive and holographic optical elements, which offer more degrees of freedom as opposed to their 2D counterparts [1, 2]. For instance, computer generated volume holograms (CGVH) with angular and frequency multiplexing capabilities have recently been demonstrated [3]. In a CGVH several projections of given far-field intensities can be encoded in the design process and by subsequently illuminating it with the corresponding plane reference waves, single or a linear combination of these projections can be decoded. In the current state of the art, such CGVHs are fabricated by direct laser writing in the bulk of glass using a tightly focused femtosecond (fs) laser beam. This process relies on highly localized light excitation for the modification of material through laser induced breakdown. One major limitation of this approach is the achievable focal volume which directly limits the voxel size ∆v. In order to fabricate devices with advanced functionalities, e.g. for applications in wavefield synthesis [2], voxel sizes on the order of ∆v ≈ 1µ m are necessary [3] since this leads to devices with more degrees of freedom and to reduced cross talk between single projections in multiplexed devices. Such focal volume sizes are however beyond the current capabilities of linear absorption based laser direct writing, whereby voxel sizes on the order of ∆v = 2µ m have been reported [3]. In this article we propose a new approach for the design and fabrication of CGVHs in the bulk of commercially available photosensitive Foturan glass by laser induced 3D modification of the refractive index through a nonlinear six-photon absorption process [4]. In such a nonlinear optical material the absorption profile is narrower than the beam profile. This allows us to design and realize CGVHs with much smaller voxels as compared to the current state of the art. Furthermore, we propose an iterative simulated annealing algorithm (SAA) for the design of the CGVHs, whereby a binary constraint which is inherently imposed by the fabrication process is integrated directly into the design. This global approach, as opposed to the commonly applied projection onto convex sets (POCS) method [2], helps avoid convergence to an arbitrary local minimum. To verify the refractive index change induced in the glass, a novel computational shear interferometry based differential interference contrast (DIC) microscope was applied [5]. 2.
Theory, Design and Fabrication
The proposed design is based on inverse scattering theory, which is itself anchored on the angular spectrum expansion of the scattered field within the weak scattering approximation. The experimental configuration and the principle of decoding a set of far-field projections from a CGVH are depicted in Fig. 1 (a). The key problem here is determining the structure of a weakly scattering volume element, given a set of known far-field intensity distributions. If we consider a CGVH defined by a function F(xx), it can be shown that the 3D frequency spectrum A(k) =
−iko 8π 2 w
ZZZ
V
F(xx)Ui (xx) exp −iko (ux2 + vy2 + wz2 ) dx2 dy2 dz2 ,
(1)
of the wave field scattered in the hologram is confined to the Ewald’s sphere in reciprocal space [6]. Here, ko = 2π /λ is the wave number with wavelength λ ; x = (x2 , y2 , z2 )T denotes a location vector in the CGVH space and
FTh4G.2.pdf
FiO/LS 2014 © OSA 2014
1
k = (u, v, w)T a vector in reciprocal space with the variables w = (1 − u2 − v2 ) 2 , whereby only values that fulfill u2 + v2 ≤ 1 are considered. The function F(xx) = −ko2 [n2 (xx) − n2o ] is the scattering potential of the CGVH for a given ko , where n(xx) = no + ∆n(xx) is its refractive index distribution, no is the refractive index of the photosensitive glass at the the illumination wavelength and ∆n(xx) is a slight increase of this index at the position x . By taking into account the aforementioned weak scattering assumption, the term Ui (xx) in Eq. 1 is taken to represent the unscattered portion of the illuminating plane wave within the volume of the hologram. Thus a plane wave incident on the CGVH at an angle θ as shown in Fig. 1 (a) decodes a far-field projection Us (ss; z). Therefore, the 2D wave field scattered in the positive z-direction z away from the CGVH across the plane aperture Σ is given by Us (ss; z) = and recorded a distance RR T Σ A(k; θ ) exp iko (ux1 + vy1 + wz) du dv, where s = (x1 , y1 ) denotes a vector in the image space. (a)
(b)
Us
(c)
no no+ Δn
z Ui
30 µm
CGVH
{ s}
Design
Fabricated
Fig. 1. (a) Schematic of an experimental setup whereby several angular projections of given farfield projections Us are encoded in a CGVH. These projections can be decoded by illuminating the hologram with the corresponding plane waves Ui . (b) Shows the refractive index distribution n(xx) of the top layer of a binary CGVH with dimensions of 64x64x96 µ m and (c) a differential interference contrast microscope (DIC) measurement of the index contrast in (b) recorded after fabrication. Therefore, in this article we seek an inverse solution for a general forward imaging problem of the form A(k; θ ) = (−iko /8π 2 w)F {F(xx)Ui (xx)}, where F {•} denotes the 3D Fourier transform (FT). However, it is apparent that knowledge of A(k; θ ) for a given single or a few illumination angles only determines A(k) on a portion of the Ewald’s sphere and therefore a unique inverse solution does not exist [6]. On the other hand, by applying optimization theory an optimal or near optimal inverse solution can be computed iteratively. To do this, we start by assuming that for every desired far-field projection Ud there exists an optimal binary index contrast nˆ = ∆n(xx) such that nˆ = arg minnˆ {Φ} minimizes the objective function Φ(n˜j ) = kUd −Us (n˜ j )k2 , (2) where n˜j is the approximated index contrast at the j-th iteration. In our proposed method this inverse solution is computed using Eq. 2 in 4 simple steps. In step (i), a discrete scattering potential for a CGVH composed of V 3 number of voxels is initialized by generating a random binary index contrast n˜ ( j=1) = b( j=1) · δ n where b j is a binary function with values [0, 1] and δ n = 1x10−3 is a constant index difference, which was chosen in this work in order to fulfill the weak scattering assumption. In the second step (ii) for each of the m−illumination conditions the corresponding farfield projections Us (n˜j ) are computed. At iteration j = 1 a smart initial guess n˜ ( j=1) can be attained by implementing a single POCS iteration [3]. In step (iii) an average sum of squared error (SSE) is calculated i.e, Φ j = M1 ∑m Φm . In the last step (iv), a set of q-voxels is randomly chosen and their probability, Pj = [1.0 − exp(−(Φ( j−2) − Φ j )/Φ( j−1) )], of minimizing Φ j is evaluated [7]. The parameter Pj is initially assumed to be unity. The binary function b of these voxels is then flipped between the states [0, 1] to generate a new CGVH. If the new index contrast n˜ j minimizes Φ below a pre-set threshold Rt the new CGVH is accepted i.e. if Φ( j+1) − Φ j > Rt and steps (ii) to (iv) are repeated until convergence. Otherwise, b j is restored and step (iv) is repeated. At each iteration, each voxel is marked every time it is tested to avoid concurrent testing. The annealing threshold Rt is a real and positive regularization parameter which ensures that a unique approximate solution which minimizes Eq. 2 is found, by avoiding convergence into an arbitrary local minimum. If in step (iv), no single b can be generated, Rt is reduced and steps (ii) to (iv) are repeated. In an initial experiment a CGVH of voxel volume 643 , voxel sizes 1x1x1.5 µ m and an illumination wavelength of λ = 632.8 nm was designed to project the far-field intensities shown in the inlay of Fig. 2 (a). To monitor the optimization process the average sum of squared errors was calculated using Eq. 2 and as it can be seen in Fig. 2 (a) the proposed algorithm has a sublinear convergence behavior. To fabricate the designed structure, laser pulses of an infrared (IR) fs-laser system with a wavelength of λ = 1550 nm and a pulse length τ = 800 fs were focused into the glass using an objective lens (40x; NA = 0.75). Since activating the photosensitivity of Foturan glass with
FTh4G.2.pdf
FiO/LS 2014 © OSA 2014
an IR fs-laser allows for the manipulation of the refractive index in a subsequent postbake process [4], the threshold of the fabrication energy below which an index difference δ n < 1x10−3 could be induced was first experimentally determined. The corresponding parameters were found to be a pulse energy of E = 5 µ J at a repetition rate of 10 Hz. With the glass samples mounted on a translation stage, which was synchronized to the laser pulse output, the designed structure was then inscribed voxel by voxel. Finally, the samples were postbaked at a constant temperature of 550◦ C for a period of 3 hours in order to achieve an index difference of δ n = 1x10−3 . (b)
(a)
Ф [norm.]
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θ = 0° θ = 7° θ = 14°
0 0.8
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(c)
0.6
θ = 7°
θ = 14°
Experiment (z = 40 cm)
0.4 x 10 0
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Fig. 2. (a) Convergence plot of the SAA algorithm and (inlay) the schematic of the design object showing a set Ud which corresponds to 3 different Ui incident at the CGVH with angles θ = 0◦ , 7◦ and 14◦ . Subfigures (b) and (c) show a comparison between numerical and experimental reconstructions (false color coded). The simulation uses a beam propagation method (BPM). Using a DIC microscope, the distribution of the induced refractive-index change was characterized and a good agreement between designed and fabricated structures could be observed (c.f. Fig. 1 (a) and (b). From these results it is clear that index distributions having smooth internal contours and with voxel dimensions on the order of 1 µ m can be fabricated. In a further step the propagation of light through the CGVH was simulated using a beam propagation method and the results were compared to those attained from optical reconstruction using the setup shown in Fig. 1 (a). Likewise, a good agreement between numerical and experimental results could be observed. As it is evident from Fig. 2 (b) and (c), both far field projections could be decoded with little or no cross-talk. 3.
Conclusion
For the first time, we have demonstrated fabrication of CGVHs in photosentive glass by nonlinear laser induced 3D modification of refractive index. We have fabricated functional devices having smooth internal contours and with smaller voxel dimensions (∆v = 1µ m) than in the current state of the art (∆v = 2µ m). Furthermore, we have achieved cross-talk free decoding of single projections from such a CGVH. Future work will consist in assesing the fesibility of fabricating CGVHs with sub-micrometer voxel sizes and their potential in multiplexed wave field synthesis. Acknowledgment: We wish to thank the Deutsche Forschungsgemeinschaft for funding this work under grant No. BE 1924/3-1 (DynaHolo). References 1. J. W. Goodman, Introduction to Fourier optics (Roberts & Company Publishers, 2005). 2. E. N. Kamau, C. Falldorf, and R. B. Bergmann, “A new approach to dynamic wave field synthesis using computer generated volume holograms,” in “12th Workshop on information Optics,” (IEEE, 2013), pp. 1–3. 3. T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nature Photonics 4, 188–193 (2010). 4. T. Hongo, K. Sugioka, H. Niino, Y. Cheng, M. Masuda, I. Miyamoto, H. Takai, and K. Midorikawa, “Investigation of photoreaction mechanism of photosensitive glass by femtosecond laser,” J. Appl. Phys. 97, eid. 063517 (2005). 5. C. Falldorf, C. von Kopylow, and R. B. Bergmann, “Wave field sensing by means of computational shear interferometry,” J. Opt. Soc. Am., A 30, 1905–1912 (2013). 6. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, 1999), pp. 605–703. 7. B. B. Chhetri, S. Yang, and T. Shimomura, “Stochastic approach in the efficient design of the direct-binarysearch algorithm for hologram synthesis,” Applied Optics 39, 5956–5964 (2000).
