Dynamic wave field synthesis: enabling the generation of field distributions with a large space-bandwidth product

Edwin N. Kamau; Julian Heine; Claas Falldorf; Ralf B. Bergmann

doi:10.1364/OE.23.028920

1. Introduction

The concept of holography has been around for more than six decades now. Since their invention in the late 1940s by Denis Gabor, holograms have motivated innumerable inventions especially in the field of modern optics with applications which include microscopy [1], data storage [2, 3], metrology [4, 5, 6] and advanced imaging [7, 8, 9]. One of the most commendable inventions was made by Lohmann [10]. He proposed a method of numerically generating holographic optical elements in the form of interference patterns using a digital computer. These elements are referred to as computer generated holograms (CGH) and are usually placed in the wider category of diffractive optical elements. One major advantage of CGHs is the fact that they can be applied to generate arbitrary wave fields. This may include complex light field distributions, e.g. optical vortex beams [11].

The task of generating as well as recording arbitrary wave fields across a given plane has been extensively studied. Thereby both classic optical elements (lenses, prisms or mirrors) and holographic optical elements (HOE) are used to manipulate a wave front impinging upon a given plane in order to create a desired wave field distribution. In the near past HOEs have replaced classic optical elements in diverse applications [12, 13, 14, 8]. This is mainly due to advances in micro and nano fabrication methods, as well as their versatile nature. For example, the iterative optimization algorithms used to design these elements offer very high flexibility. Nonetheless, owing to their inherent 2D nature, most holographic type elements like CGHs can only modulate an impinging wave across a single transverse plane. This is one of their major disadvantages and one that has led to an ever increasing demand of developing 3D optical elements. For instance, stratified CGHs in a cascaded setup have been proposed [15] and it has been shown that by adding more degrees of freedom they facilitate both angular and frequency multiplexing [16, 17].

Another example is presented by thick volume holograms [18]. These are optically recorded periodic diffractive optical elements that depict both angular and wavelength selectivity in cases where the Bragg condition is fulfilled, as opposed to their 2D counter parts. Although these two types of volume elements have found numerous applications, their ability to generate arbitrary light distributions is quite limited. This is mainly due to their comparatively low diffraction efficiency as well as their static nature. For instance, the diffraction efficiency of optically recorded volume holograms decreases as the square root of the number of multiplexed recordings [19]. This can also be attributed to the fact that since individual voxels cannot be addressed, the achievable degrees of freedom are limited by the optical recording process. Recently however, computer generated volume holograms (CGVH) were proposed [20] and their superior performance in terms of angular and wavelength multiplexing [21, 22] as well as their application in the dynamic synthesis of wave fields demonstrated [23]. CGVHs are volumetric optical elements whose complex transfer functions can be controlled and optimized in the design process. In analogy to their 2D counterparts, CGVHs are generated numerically by means of iterative optimization algorithms. They are modeled mathematically as a randomly scattering inhomogeneous medium that is characterized by a refractive index distribution - the so called scattering potential [24]. The scattering potential is assumed to be embedded in a known background potential. It therefore corresponds to a modulation of the refractive index e.g. in the bulk of a transparent dielectric material thus facilitating fabrication of CGVHs by means of 3D laser lithography.

The application of an electronically addressed CGVH in dynamic synthesis of wave fields has not yet been demonstrated in the current state of the art. Such an approach would facilitate the generation of predefined wave field distributions having a higher space bandwidth product as compared to planar or stratified HOEs. In this paper, we propose a novel approach that employs a spatial light modulator to dynamically decouple a set of far field projections from a CGHV. To achieve this goal, an advanced mathematical model that allows for the design of such a hybrid system using optimization theory was derived. Our model, as opposed to the ones described in the current state of the art, is based on the linearization of the inhomogeneous wave equation within the validity of the Rytov approximation. Therein, constraints that account for voxel based micro fabrication of non-absorbing binary index structures in a nonlinear optical material were incorporated. Initial experimental work done to characterize this system and to validate this model show that this approach facilitates a decoupling of single or a linear combination of far field projections without any detectable cross-talk between them. In the course of this work, we were able to demonstrate that a Bragg selectivity on the order of 1° can be achieved using such a system.

2. Model of light scattering within a CGVH

2.1. Problem definition and the scalar scattering theory

The problem of designing a CGVH can be stated as follows: the set of wave field distributions that is to be generated in a certain predefined domain is the known effect for which the scattering potential inside the CGVH is the sought cause. This is in essence an ill-posed inverse problem involving the mapping of the 3D field distribution scattered in the volume of the CGVH onto one or a set of transverse 2D planes [24]. The resolution of this inverse problem requires very good knowledge of the underlying forward problem, i.e. an explicit description of how light is scattered inside the volume of the hologram is needed. The first step in designing a CGVH is to come up with a method for mathematically modeling a volume hologram in terms of the dielectric properties of a given medium. In the following, a model that allows for the numerical computation of the CGVH is thus derived from the scalar theory of scattering.

For the derivation of this model we initially presumed that a CGVH can be described in terms of the holographic medium’s refractive index distribution $n (\vec{r})$ and hence scalar scattering theory can be applied [25]. The basic theory behind our model for the design of a given CGVH can be derived by considering the geometry depicted in Fig. 1(a). Here, a plane wave field $U_{o} (\vec{r}, ν)$ with wave vector ${\vec{k}}_{o}$ and of unit amplitude (i.e. ${\vec{k}}_{o} = k {\vec{q}}_{o}$ where k = c/ν = 2π/λ is the wave number) is incident upon the volume V of a given linear, isotropic and non-magnetic dielectric material from the direction specified by the unit vector ${\vec{q}}_{o}$ . This field is scattered such that a predefined field $U_{s} (\vec{r}, ν)$ is projected into the signal window W across the plane ${\vec{ξ}}$ , in the far-field domain of the hologram. We start by noting that the total field behind the hologram can be defined as the sum of the scattered and the un-scattered portion of the incident plane wave as follows

U (\vec{r}, ν) = U_{o} (\vec{r}, ν) + U_{s} (\vec{r}, ν) .

Fig. 1 (a) Schematic of the geometry considered for a general scattering problem. The gray area corresponds to the volume V of the volume hologram, which is defined by the scattering potential $F (\vec{r}, ν)$ while the intermediate region corresponds to the volume of the base dielectric material with a refractive index n_o. A wave field incident (wave vector, ${\vec{k}}_{o} = k {\vec{q}}_{o}$ ) upon this volume will be scattered such that a predefined scattered field (wave vector, ${\vec{k}}_{s} = k {\vec{q}}_{s}$ ) is projected into the signal window W across the plane ${\vec{ξ}}$ , in the far-field domain of the hologram. (b) Frequency space representation of the scattering problem where values of the scattered field are seen to lie on a portion (solid line) of the Ewald’s sphere (broken line). These values are projected onto a planar detector, whereby each pixel on this plane integrates the light scattered within a solid angle $Δ Ω ({\vec{k}}_{s})$ .

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Born and Wolf have shown that once $U (\vec{r^{'}}, ν)$ at points within the scattering medium is known, the solution $U (\vec{r}, ν)$ at points exterior of V can be obtained from integral equation of potential scattering [25, pp. 698]

U (\vec{r}, ν) = U_{o} (\vec{r}, ν) + \int_{V} F ({\vec{r}}^{'}, ν) U ({\vec{r}}^{'}, ν) G (| \vec{r} - {\vec{r}}^{'} |, ν) d^{3} {\vec{r}}^{'} .

Here, the second term on the rhs of this equation is the sought scattered wave field $U_{s} (\vec{r}, ν)$ and

G (| \vec{r} - {\vec{r}}^{'} |, ν) = \frac{e^{i k} | \vec{r} - {\vec{r}}^{'} |}{| \vec{r} - {\vec{r}}^{'} |}

is the outgoing free-space Green’s function. This integral can not be solved directly using analytical methods, but a solution can be formulated using either the so called Born or Rytov approximations. It has been shown that these two approximations are valid under different conditions (c.f. Sect. 2.2). However, the form of the resulting solutions is quite similar [26]. In the current state of the art, the Born approximation was applied in Designing CGVHs. Here, the Rytov approximation will be adopted.

2.2. First born approximation of the inhomogeneous wave equation

The scattered field in Eq. (2) is however written here in terms of the total field as it is seen from Eq. (1); since $U ({\vec{r}}^{'}, ν) = U_{o} ({\vec{r}}^{'}, ν) + U_{s} ({\vec{r}}^{'}, ν)$ . Thus we still have to solve this equation for the scattered field. Let us start by assuming that the volume of the CGVH can be described as a weak scattering but continuously inhomogeneous dielectric medium. Given the fact that we are considering U_s in the far-field of the CGVH for the case of plane wave illumination, we can introduce a few assumptions in order to determine the far-field approximation of expressions (2) and (3). We can start by setting $\vec{r} = r {\vec{q}}_{s}$ (where ${\vec{q}}_{s}$ is a unit vector in the direction of a given scattered plane wave) and making use of the asymptotic approximation to the expression in Eq. (3), i.e.

G (| \vec{r} - {\vec{r}}^{'} |, ν) ≅ \frac{e^{i k r}}{r} e^{- i k {\vec{q}}_{s} \cdot {\vec{r}}^{'}} .

This approximation can be inferred directly from Fig. 1 since evidently $\vec{r} ≫ {\vec{r}}^{'}$ for all points that contribute to the integral in (2). Furthermore, we assume that the change in refractive index is very small, i.e. it is on the order of δn ≈ 10⁻³. Therefore, a portion of incident plane wave field transverses through the volume unscattered and therefore the total field within the volume $U ({\vec{r}}^{'}, ν)$ in Eq. (2) can be replaced with the incident plane wave field. Making these three assumptions is analogous to invoking the so called weak scattering approximation to arrive at the following expression for the total field [25, 26]

U (r {\vec{q}}_{s}, ν) ≅ U_{o} ({\vec{r}}^{'}, ν) + \frac{e^{i k r}}{r} \int_{V} F ({\vec{r}}^{'}, ν) e^{- i k ({\vec{q}}_{s} - {\vec{q}}_{o}) \cdot {\vec{r}}^{'}} d^{3} {\vec{r}}^{'},

where the scattering potential is given by

F ({\vec{r}}^{'}, ν) = \frac{1}{4 π} k^{2} [n^{2} ({\vec{r}}^{'}, ν) - n_{o}^{2}] .

Equation (5) represents an integral form of the inhomogeneous wave equation within the validity of the Born approximation, which is essentially a weak potential approximation. Hereby, the function $F ({\vec{r}}^{'}, ν)$ is the scattering potential of the medium. The scattered field

U_{B} (r {\vec{q}}_{s}, ν) = \frac{e^{i k r}}{r} \int_{V} F ({\vec{r}}^{'}, ν) e^{- i k ({\vec{q}}_{s} - {\vec{q}}_{o}) \cdot {\vec{r}}^{'}} d^{3} {\vec{r}}^{'},

behind the CGVH is thus solely attributed to the presence of inhomogeneities in the volume. In this work we assumed that these inhomogeneities correspond to the refractive index distribution given by

n (\vec{r}, v) = n_{o} + Δ n ((\vec{r})),

where n_o is the refractive index of the dielectric material and

Δ n ((\vec{r}))

represents localized modulation of n_o. The term on the rhs of (7) is commonly referred to as the scattering amplitude. It clearly represents the Fourier transform of the scattering potential, i.e.

\tilde{F} (\vec{K}, ν) = \int_{V} F ({\vec{r}}^{'}, ν) e^{- i \vec{K} \cdot {\vec{r}}^{'}} d^{3} {\vec{r}}^{'} .

From Eq. (9) and as shown in Fig. 1(b) it is clear that for a given incident field, the scattered plane wave in the far-field of CGVH in the direction specified by ${\vec{q}}_{s}$ depends entirely on only one Fourier component of the scattering potential, since [25]

\vec{K} = k ({\vec{q}}_{s} - {\vec{q}}_{o}) .

Equation (10) in turn implies that Eq. (5) defines a mapping between the 3D scattering potential in Eq. (6) and the scattered field U_s. This physical implication becomes apparent when one considers the scattering process within the bulk of the hologram in frequency space. Here, the values of the scattered field are seen to lie on a portion of a sphere with a radius of 1/λ -the so called Ewald’s sphere. Eq. (10) is a restatement of the Bragg condition for holograms designed using this approach. Thus, Eq. (5) presents the direct problem of designing a multiplexed CGVH whose corresponding inverse problem can be stated as follows: the set of wave field distributions that is to be generated in a certain predefined domain is the known effect for which the scattering potential inside the CGVH is the sought cause.

Unfortunately, this is an ill-posed inverse problem particularly since the existence of a unique solution is not guaranteed. For instance, for a fixed value of k, a complete determination of $F ({\vec{r}}^{'}, ν)$ is only possible if U_s is known for various values ${\vec{q}}_{s}$ and ${\vec{q}}_{o}$ . However, given a set of constraints, numerical optimization based methods can be applied to compute an optimal solution. In the current design approach [21, 23], using Eqs. (5) and (10) optimization theory is applied to compute an optimal solution iteratively. Thereby, assuming that for every target far-field projection U_t there exists an optimal binary index contrast $\hat{n} = Δ n (\vec{r})$ such that $\hat{n} = \arg \min_{\hat{n}} (L)$ , i.e. it minimizes the objective function

L ({\tilde{n}}_{j}) = {‖ U_{t} - U ({\tilde{n}}_{j}) ‖}^{2}

where

{\tilde{n}}_{j}

is the approximated index contrast at the j–th iteration. It is however crucial that the index change must be constrained to values on the order of Δn ≈ 10⁻³, i.e that lie within the validity of the Born approximation. This leads to a limitation on the thickness d of the designable CGVHs, since as d increases a considerable phase change is introduced to the incident wave. This means that the assumption that the total field can be replaced with the incident wave in Eq. (2) is not valid for large values of d. This in turn translates to an inherent limitation on the SBWP and thus also to the achievable optical functionality of the holograms [27]. To avoid this limitation the Rytov approximation, which does not presume this limitation on Δn, was adopted in this work.

3. Design and fabrication of CGVHs within the Rytov approximation

3.1. First Rytov approximation of the inhomogeneous wave equation

To arrive at the Rytov approximation, we start by expressing the total and the incident fields in Eq. (2) in terms of its complex phase form as follows, i.e. $U (\vec{r}, ν) = e^{Φ (\vec{r}, ν)}$ and $U_{o} (\vec{r}, ν) = e^{Φ_{o} (\vec{r}, ν)}$ . Then the phase $Φ (\vec{r}, ν)$ also admits a perturbation expansion similar to the one given in Eq. (5) for the Born approximation, i.e. we assume that $U ({\vec{r}}^{'}, ν)$ can be replaced with $U_{o} (\vec{r}, ν)$ . The total field within validity of the Rytov approximation is then given by [25]:

Φ (r {\vec{q}}_{s}, ν) ≅ Φ_{o} (\vec{r}, ν) + \frac{1}{U_{o} (\vec{r}, ν)} \frac{e^{i k r}}{r} \int_{V} F (\vec{r^{'}}, ν) e^{- i k ({\vec{q}}_{s} - {\vec{q}}_{o}) \cdot \vec{r^{'}}} d^{3} \vec{r^{'}},

The second term on the rhs of this equation is a complex phase representation of the scattered field which can be expressed in terms of the Born approximation as follows as

Φ_{R} (\vec{r}, ν) = \frac{U_{B} (\vec{r}, ν)}{U_{o} (\vec{r}, ν)}

It is therefore apparent that both approximations have the same overall functional form. They are however valid under totally different conditions. A model derived under the validity of Rytov approximation imposes a less stringent constraint on the thickness of the hologram and is thus superior to a model based on the Born approximation. The condition necessary for the validity of the Rytov approximation have been derived to be [26, 28]

Δ n < {[\frac{\nabla Φ_{R} λ}{2 π}]}^{2} .

From this we can directly infer that it is the change in scattered phase ϕ_s over one wavelength that is more important and not the change in phase over d as in the case of Born approximation.

3.2. Design algorithm

The algorithm developed in the course of this work is based on Eq. (12) and involves optimizing a multivariable system as defined by the functional in Eq. (11). It is a Bi-directional algorithm which is similar to the iterative Fourier transform based Gerchberg-Saxton algorithm [29]. The design algorithm can be summarized in five steps as follows:

In the initial step, a discrete scattering potential $F_{j} (\vec{r}, ν)$ for a desired CGVH composed of M ×N ×O number of voxels is initialized by generating a random index contrast $Δ n (\vec{r})$ as an initial guess.
To attain the resulting scattered fields, a 3D forward FFT of the scattering Potential is computed and the far field projections are extracted from the corresponding portion of the Ewald’s sphere for each illumination direction. From Eqs. (9), (12) and (13) the scattered field for a given incident field q_o can be expressed in frequency space as ${\tilde{U}}_{j} (\vec{K}) = \exp [\frac{e^{i k r}}{r} \tilde{F} (\vec{K}) / U_{o} (\vec{r}, ν)] .$
Each of these single fields is then modified to better approximate the target intensity $I_{t} (\vec{K}, ν)$ across a given plane in the far field of the CGVH by imposing the following amplitude constraint: ${\tilde{U}}_{j} (\vec{K}, ν) = {\begin{array}{l} c_{k} \sqrt{I_{t} (\vec{K}, ν)} \cdot e^{[i ϕ_{j} (\vec{K}, ν)]} & \forall (\vec{K}, ν) \in W \\ {\tilde{U}}_{j} (\vec{K}, ν) & otherwise, \end{array}$ where $ϕ_{j} (\vec{K}, ν) = \arg {{\tilde{U}}_{j} (\vec{K})}$ is the unmodified phase. We hereby assume that $I_{t} (\vec{K}, ν)$ is only given within the limited signal window W (c.f. Fig. 1). The weighting factor c_k is introduced here to ensure the convergence of the algorithm and was determined here heuristically by means of a parameter scan.
The modified and un-modified sections of the Ewalds’s spheres are recombined to obtain ${\tilde{F}}_{j} (\vec{K}, ν)$ and an inverse 3D FFT is used to compute a new scattering potential, i.e. ${\overset{´}{F}}_{j} (\vec{r}, ν) = ℱ_{3 D}^{- 1} {{\tilde{F}}_{j} (\vec{K}, ν)}$ .
To obtain an approximated binary index contrast ${\tilde{n}}_{j}$ , a fabrication constraint is imposed on this new scattering potential as follows $F_{j + 1} (\vec{r}, ν) = {\begin{array}{l} \frac{1}{4 π} k^{2} [n_{j}^{2} (\vec{r}, ν) - n_{o}^{2}] & \forall {\overset{´}{F}}_{j} (\vec{r}, ν) \geq \frac{1}{8 π} k^{2} [n_{j}^{2} (\vec{r}, ν) - n_{o}^{2}] \\ 0 & otherwise . \end{array}$ Finally, $F_{j + 1} (\vec{r}, ν)$ is set as the new scattering potential and steps (2 5) are repeated until a the algorithm converges to an optimum solution $\hat{n}$ . To monitor the convergence of the algorithm, both diffraction efficiency (c.f. Eq. (26)) the average value of Δn (see Fig. 2) were computed at each iteration.

Fig. 2 Comparison of the convergence behavior of the design algorithm for both the model based on the Rytov approximation (broken line) and the conventional method which is based on the Born approximation. It is apparent from these results that in the former, the algorithm converges to higher Δn values as it was discussed in Sect. 3.1.

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It is important to note that since the different far-field projections are to be superposed across a planar surface, e.g. of a CCD sensor, each pixel on this plane then integrates the light scattered within a solid angle $Δ Ω ({\vec{k}}_{s})$ . Unfortunately, $Δ Ω ({\vec{k}}_{s})$ decreases with the increase in spatial frequency as it is shown in Fig. 1(b). This in turn means that projections to the middle part of the Ewald’s sphere tend to be larger than those towards the polar regions, which leads to artifacts in the synthesized fields. To remedy this problem a bi-cubic interpolation, whereby $Δ Ω ({\vec{k}}_{s})$ - see Fig. 1(b) - is considered during this computation, was implemented. Moreover, the real size of the feasible axial and lateral voxel sizes in the nonlinear material used in this were determined experimentally in terms of the point spread function of the laser lithography system used as discussed in Sect. 3.3 and were introduced as constraint in steps 2 and 4 of the design algorithm.

From our preliminary investigations, we could infer that imposing these extra constraints dramatically improves the convergence of the algorithm to an optimum solution, as compared to the current state of the art. In order to compare the results of the model based on the Rytov approximation, a CGVH was designed also using the conventional model. As it can be clearly seen in Fig. 2 the algorithm converges to higher Δn values in case of the design model based on the Rytov model as compared to the conventional one as it was discussed in Sect. 3.1. In the course of this work we designed and fabricated different CGVHs using the Rytov model with Δn values on the order of Δn = 10⁻³ – 10⁻², which shows the flexibility of this design approach in terms of the feasible range in achievable δn values and hologram thickness. As we have shown in our previous work [22], this increase in thickness leads to an increased space-band-width-product (SBWP). Additionally, the algorithm depicts a sub-linear convergence in the case of the proposed model and does not stagnate after a few iterations as in the case of the conventional approach. This can be attributed to the introduction of further constraints and the introduction of a more precise nonlinear data interpolation. The computational work at each iteration is dominated by the 3D FFT operation and the extraction of the projections from the Ewald’s sphere. It follows that the overall computational complexity, with respect to the hologram size, of our proposed algorithm is logarithmic - $O (n^{2} \log n)$ . This in turn implies that both the proposed Rytov and the Born approximation based algorithms have a comparable overall computational complexity. For instance, the computation of a 128×128×128 voxels CGVH on a standard PC (2.6 GHz dual core processor, 8 GB RAM and 250 iterations) takes approximately 215 seconds for both algorithms.

3.3. Fabrication of CGVHs in photostructurable glass-ceramics

The choice of a suitable dielectric material for the fabrication of CGVHs is key to the optimization of their optical functionality. The photostructurable Glass-Ceramics (GC) material used in this work was Foturan™ glass. This is a nonlinear optical material which mainly consists of silica, stabilizing admixtures, a nucleating agent and a photoactive component. We have recently shown that since in such a nonlinear material the absorption profile is narrower than the beam profile, CGVHs with much smaller voxels than in the current state of the art can be realized [23]. Glass-Ceramics are generally manufactured by adding dopant compounds in a base material like glass. This provides the potential to tailor a specific functional property either on the local or global scale. Foturan glass and other GCs such as chalcogenide glasses are specifically manufactured in the amorphous glass state and contain a photoactive component that permits the controlled modification of the refractive index of the material following exposure to a given radiation. A further advantage of this GCs is the fact that they can be transformed to a composite material in a subsequent heat treatment through a controlled nucleation and crystallization of the GC constituents [30, 31]. In Table 1, typical concentrations of the GC used in this work are summarized.

Table 1. Overview of typical constituents and concentrations (in mass percentage - wt%) found in a commercially available Foturan photostructurable glass ceramic [30].

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An infrared femtosecond laser system that emits mode-locked pulses with a pulse length τ < 800 fs at wavelength of 1550 nm was used for our study. Thereby a beam with a diameter of 5 mm was focused using a microscope objective (40x; NA = 0.75) into the bulk of the GC material. The photo-induced process in this material during a direct laser writing process was analyzed closely prior to the fabrication of the holograms. The goal was hereby to determine the peak power density needed to induce photomodification within the volume of a single voxel. The laser intensity of the fs-laser had to be appropriately chosen in order to achieve conditions where nonlinear absorption is induced in the focal region only. This absorption would lead to the generation of non equilibrium charge carriers in the area localized within the focal spot. The size of the focal spot thus dictates the achievable resolution. In multi-photon absorption processes (MPA), the focal region is usually elongated in the beam’s propagation direction and the intensity distribution within it depicts an elliptic form [30]. This distribution represents the point spread function (PSF) in the focal spot of a laser beam focused using an objective of a given numerical aperture. To experimentally approximate the PSF for our system, the lateral (Δx) and axial (Δz) dimensions of the focal spot was determined by means of microscopy for different pulse energies and number of pulses. These values were then incorporated into the design algorithm as discussed in Sect. 3.2.

4. Dynamic wave field synthesis

4.1. Optoelectronical wave field modulation

To facilitate the dynamic decoupling of single or a set of far field projections from one CGVH, an electronically addressable spatial light modulator (SLM) was used in this work. SLMs can modulate the phase of an optical beam with just a few microsecond response time and a large angle scan range. The feasibility of wave field synthesis using a CGVH and an SLM in a setup like the one shown in Fig. 3(a) is discussed in this section. This configuration makes use of a phase only SLM, which allows for the adaptive manipulation of the impigent wave. The SLM is regarded here as a non mechanical multiple-angle beam-splitting device. For instance, the 2D Fourier shift theorem can be applied to shift a wave field in the frequency space. The spectrum of the shifted field, with a shift s, is in this case the Fourier transform of the original field modulated by the linear transfer function

H_{s} (\vec{ν}) = e^{- i 2 π \vec{ν} s} .

Fig. 3 (a) A conceptual sketch of a 4 f-setup that allows for the decoupling of individual far-field projections Ψ_i from a CGVH, which is defined by the scattering potential $S (\vec{r})$ , by inscribing the transfer function $H (\vec{ν})$ on an SLM. (b) A schematic representation of angular multiplexing, whereby for each angle α_i a far field projection ψ_i is decoupled. (c) Bragg effect is shown in terms of the relative diffraction efficiency η/η_o of a test CGVH. This shows how the efficiency of a single projection from a CGVH decreases as the angle of the reference wave deviates from the Bragg angle θ_B.

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Such a modulation can be achieved optically with the help of a 4 f -setup. Hereby, the wave field is first Fourier-transformed using a lens with a focal length f. An SLM is then placed in the Fourier plane of this lens. Across this plane there will be a field $U_{f} (\vec{r^{'}})$ , whose complex valued amplitude at the position defined by $\vec{r^{'}}$ is proportional to the Fourier transform of the field $U_{i} (\vec{r^{'}})$ incident at the front focal point of the lens at the position $\vec{ν} = {\vec{r}}^{'} / λ f$ , i.e. [32]:

U_{f} ({\vec{r}}^{'}) = \frac{1}{i λ f} \cdot ℱ {U_{i} ({\vec{r}}^{'})} (\frac{{\vec{r}}^{'}}{λ f}),

where ℱ denotes the Fourier transform operator. Given Eq. (18), if the complex transmittance

t_{s} ({\vec{r}}^{'}) = H_{d} (\frac{{\vec{r}}^{'}}{λ f}) = e^{- i 2 π \frac{{\vec{r}}^{'}}{λ f} s}

is generated by the SLM and an identical lens is introduced at a distance of 2 f, the shifted wave field exiting the 4 f -setup can be written as

\begin{array}{l} U_{f} ({\vec{r}}^{″}) = ℱ {ℱ {U_{i} (\vec{r})} t_{s}} \\ = - U_{i} (- ({\vec{r}}^{″} - s)), \end{array}

which is clearly a copy of the translated field U_i - but one that is rotated by 180°. A set of shifted planes waves k_i, which correspond to individual reference waves impinging on the CGVH at different angles, can then be coupled into the hologram thereby dynamically synthesizing a given set of far-field projections ψ_i as shown in Fig. 3(a). Moreover, a set of arbitrary orthogonal elementary waves can be coherently superposed in order to attain the complex valued field

U (\vec{r}) = \sum_{i = 1}^{I} A_{i} e^{i ϕ_{i}} \cdot ψ_{i} (\vec{r}),

where each

ψ_{i} (\vec{r})

and therefore

U (\vec{r})

has a large SBWP. Owing to the birefringence property of the SLM, only light that is polarized along its slow axis is modulated, while light polarized along the fast axis remains unchanged. This property can be used additionally for multiplexing as it will be discussed in the next section.

4.2. Angular multiplexing and superposition of complex wave fields

We first validated the concept developed here by investigating the Bragg selectivity properties of the CGVHs. Since in general CGVHs have a limited size, we expect a slight deviation from the Bragg condition. The size of the CGVH effectively imparts a multiplicative rect-function and this translates to a convolution of all far-field components with a sinc-function:

rect (\frac{z}{L}) F (\vec{r}, ν) \overset{ℱ}{\to} \frac{z}{π} \sin c (\frac{z}{2} k_{z}) * \tilde{F} (\tilde{K}, ν) .

Let us assume that we have a finite hologram with a scattering potential with the general form of

S (\vec{r}) = F (\vec{r}) \cdot rect [\frac{x}{L_{x}}, \frac{y}{L_{y}}, \frac{z}{L_{z}}],

where (L_x, Ly) determine the lateral extensions and L_z the thickness of the medium; whereas (k_x, k_y, k_z) are components of the frequency space vector

\vec{k}

. From Eq. (9) we can compute the scattering amplitude

\tilde{S} (\vec{K}, ν) = \tilde{F} (\vec{K}, ν) \otimes L_{x} L_{y} L_{z} \sin c [\frac{L_{x} k_{x}}{2 π}, \frac{L_{y} k_{y}}{2 π}, \frac{L_{z} k_{z}}{2 π}] .

Thus we arrive at a scattering amplitude that is convolved with a sinc function whose width is scaled by a positive-valued scale factor b = L_i/2π along the i – th direction. Note that from from Eq.(25), changing the angle of incidence e.g. in the y-direction leads to a deviation in the reciprocal space scattering vector which can be written as k_y = 1/λ sin(θ + Δθ)sin(α). And in turn this leads to a change in the diffraction efficiency which from Eq. (15) can be expressed as

η = \frac{\int {| \tilde{U} (k_{x}, k_{y}) |}^{2} I_{t} (k_{x}, k_{y}) d k_{x} d k_{y}}{\int {| \tilde{U} (k_{x}, k_{y}) |}^{2} d k_{x} d k_{y}}

The result of this convolution is a blurring of the far-field projection around the Bragg angle θ_B as it’s apparent from Fig. 3(c). Here, the Bragg effect is shown in terms of the relative diffraction efficiency as a function a certain deflection Δθ from the Bragg angle. These results therefore show very good agreement between the theoretical case predicted by Eq. (25) and the experimentally measured data since the efficiency of reconstruction decreases as the angle of the reference wave deviates from the Bragg angle θ_B. To minimize cross-talk, a certain Δθ_m has to be achieved. In the course of this project, values of Δθ_m on the order of Δθ_m ≤ 1 were achieved.

In the next step a setup was realized and the experimental validity of the model derived in Sect. 3.1 and the concept described in Sect. 4.1 were assessed. Figure 4(a) shows the setup realized for wave field synthesis through angular multiplexing. Hereby, an SLM (Holoeye Pluto, 1920 × 1080 pixels, pixel pitch of 8 μm, frame rate 60 Hz) is located in the Fourier plane of the first objective lens (f = 105 mm). The SLM is illuminated with a collimated laser beam (λ = 532 nm) from a diode laser, thereby generating one or a set of reference waves. Each reference beam is then Fourier-transformed by a second objective lens. An aperture is placed in the Fourier plane to block unwanted higher diffraction orders resulting from the SLM. Each beam behind the second objective lens is then collimated using a lens (f = 10 mm), which allows the set of feasible reference beams to sweep through an angle range of approximately ±10°. A second aperture is placed behind this lens making the beam as small as possible. The size of this aperture is chosen such that it matches the dimensions of the effective exit pupil of the CGVH in order to reduce the magnitude of the zero order in the far-field. A motorized rotation (Thorlabs CR1-Z7, 5 arcmins resolution) and two linear stages (Newport FMS linear stage and XPS-C8 controller, on-axis resolution 1 μm) are used to precisely position the CGVH along the optical axis. This setup was used to investigate multiplexing capabilities of a CGVH that projects far-field target intensities of the initials bias (see Fig. 4(b)). Each projection was offset by an angle step of 2° from each other. Figure 4(c) shows results whereby the feasibility of cross-talk free decoding of these four far-field projections ψ_i from a single CGVH could be demonstrated. The far-field intensities of all four letters could clearly be decoded without any detectable cross-talk. The presence of speckle noise on these projections can be attributed mainly to the application binary index constraint (cf. Eq. (17)). This is a common problem that is encountered while dealing with coherently illuminated discreet CGHs. It can in this be ameliorated for instance by generating each given far-field projection using different reference waves which are incident from slightly different directions. Thereby, the resulting superposed signal for each projection will have an increased signal to noise ration due to spatial speckle averaging.

Fig. 4 (a) An experimental setup which comprises in a 4f-setup consisting of a fiber coupled diode laser, 2 lenses with a focal length of f₁ = 105 mm, a λ/2 wave plate and a polarizer (not drawn to scale). This part is used to implement a non-mechanical beam modulation unit with the help of an SLM. The other part, high precision positioning stages are used to position the CGVH and a lens is used to collimate and couple in single or a set of reference waves at different angles θ into the CGVH. For instance light that is polarized along its slow (dashed line) axis can be modulated to generate a given reference wave, while light polarized along the fast axis (solid line) represents a second reference wave. With this setup the target intensities in (b) were decoded experimentally from a single CGVH with dimensions of 683×683×100 μm³ for four different angles. The resulting far-field projections ψ_i (false color coded) are shown in (c). Each projection was offset by an angle step of 2° from each other.

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In a proof of principle study, a CGVH for the target intensities in Fig. 4(b) was also used to assess the feasibility of dynamic wave field synthesis. In an initial test, the birefringence property of the SLM was used [33]. In Fig. 5(a) the projection ψ₂ corresponding to a single reference wave that was generated using light of a single polarization is shown. Alternatively, using the concept described above, a deflected reference wave ${\vec{k}}_{θ}$ can be generated so that the projection ψ₃ is decoded as shown in Fig. 5(b). If however orthogonally polarized light is chosen, then the SLM can be used to generate two reference waves with k-vectors ${\vec{k}}_{2}$ and ${\vec{k}}_{3}$ concurrently. Two such reference waves can be coupled simultaneously into the CGVH thereby decoding two different far-field projections and thus enabling synthesis of wave fields with a higher SBWP within the signal window W as shown in Fig. 5(c). By comparing these two images, this superposition is quite apparent since two projections are decoupled simultaneously and are superposed leading to the synthesis of a new field ψ₂ + ψ₃.

Fig. 5 Dynamically synthesized fields. (a) a single projection ψ₂ corresponding to a single reference wave is decoupled by light polarized in the fast axis. (b) a second projection ψ₃ corresponding to a single reference wave is decoupled by light polarized in the slow axis and modulated using the SLM in order to generate a deflected reference wave ${\vec{k}}_{θ}$ . (c) projections ψ₂ and ψ₃ are decoupled simultaneously and are superposed within the same signal window W leading to the synthesis of a new field ψ₂ +ψ₃. (d) Vectorial representation of the reconstruction process, whereby the same signal window W is chosen and hence both reference waves ${\vec{k}}_{2}$ and ${\vec{K}}_{2}$ have the same scattering vector ${\vec{k}}_{s}$ . This leads to the superposition of ψ₂ and ψ₃ even though the corresponding reference waves are offset by an angle θ.

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In a similar manner, by employing the SLM as an electronic diffraction grating several diffraction orders which depend on the shape of the grating can be generated thus facilitating a simultaneous superposition of N far-field projections in a given signal window W. This generation of several projections from a single CGVH was discussed in details in our previous work [22]. Note that if for N-reference waves the same signal window W is chosen (see Fig. 1(a)), the resulting CGVH contains N-holograms each having a grating vector ${\vec{K}}_{n}$ . Consequently, according to Eq. (10) these holograms have the same scattering vector ${\vec{k}}_{s}$ . Such a vectorial representation of the scattering process for the results attained in this work is shown in Fig. 5(d). This representation shows how the superposition of ψ₂ and ψ₃ is achieved even though the corresponding reference waves are offset by an angle θ.

The information capacity of a CGVH having refractive index values that are constrained to C discreet levels can be expressed in terms of the degrees of freedom of the feasible field distribution that it can synthesize. To do this let us start by first considering a wave field scattered within the CGVH to represent a 3D analytic function $U_{s} ({\vec{r}}^{'}, ν)$ with a compact support in a give region. Suppose that this function is sampled in a uniform manner in the (x, y, z)-directions, which is indicated by

U_{s} (x, y, z, ν) \to U_{s} (m Δ x, n Δ y, p Δ z, ν)

where the sample intervals are (Δx, Δy, Δz) the x–, y– and z– directions respectively and (m, n, p) are integer valued indices of the samples. Now let us assume that this sampling is done in accord to the sampling theory, i.e. with spacings Δx ≤ 1/2B_x, Δy ≤ 1/2B_y and Δz ≤ 1/2B_z - where B_i is the bandwidth of U_s in the i–th direction. The the total number of significant samples required to represent U_s can be expressed in terms of the SBWP as follows

S B W P = \frac{C}{8} [L_{x} L_{y} L_{z} B_{x} B_{y} B_{z}] .

From Eq. (28) it is clear that the results in Fig. 4(b) were obtained from a CGVH with a SBWP = 8.7 × 10⁶. This is three orders of magnitude higher than the current state of the art in cascaded CGH systems [16], where a system with SBWP = 5.1 × 10³ was demonstrated. Moreover, since our system allows for a simultaneous coupling of a set of reference waves, the achievable SBWP is much larger than that of a single CGVH. Hence we can conclude that our approach facilitates the generation of field distributions with a large space-bandwidth product, i.e. SBWP ≥ 8.7 × 10⁶.

5. Conclusion and outlook

We present a new approach for the dynamic synthesis of wave fields with a higher SBWP as compared to the current state of the art in the design of computer generated holograms. We have outlined the derivation of the mathematical model which is based of the first Rytov approximation and was used in the design process. In particular, we have discussed how the incorporation of physical constraints into the design process improves the convergence of the iterative algorithm used in this work. From the initial experimental work done to characterize this system and to validate this model we could show that this approach facilitates a dynamic decoupling of single or a linear combination of far field projections without any detectable cross-talk between them. In the course of this work, we were able to demonstrate that a Bragg selectivity on the order of 1° can be achieved using such a system. This in turn allows for the superposition of a set of wave fields and thus also the dynamic synthesis of fields having a space-bandwidth product of SBWP ≥ 8.7 × 10⁶.

Acknowledgments

The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for funding this work within the frame of the project DynaHolo under the grant BE 1924/3-1.

References and links

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Constituents	wt%	Description
Primary substrate:
SiO₂	75 – 85	Base glass
Admixtures:
Li₂O	7 – 11	Stabilizer
K₂O and Al₂O₃	3 – 6	Stabilizer
Al₂O₃	1–2	Stabilizer
Zn_O	< 2	Stabilizer
Ce₃O₃	0.01 – 0.04	Photoactive component
Ag₂O	0.05 – 0.15	Nucleating agent

Dynamic wave field synthesis: enabling the generation of field distributions with a large space-bandwidth product

Abstract

1. Introduction

2. Model of light scattering within a CGVH

2.1. Problem definition and the scalar scattering theory

2.2. First born approximation of the inhomogeneous wave equation

3. Design and fabrication of CGVHs within the Rytov approximation

3.1. First Rytov approximation of the inhomogeneous wave equation

3.2. Design algorithm

3.3. Fabrication of CGVHs in photostructurable glass-ceramics

4. Dynamic wave field synthesis

4.1. Optoelectronical wave field modulation

4.2. Angular multiplexing and superposition of complex wave fields

5. Conclusion and outlook

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (28)

Optics Express