1. Introduction

Volume holographic imaging (VHI) is a multi-dimensional imaging technique utilizing wavefront information stored in volume holograms [1]. Volume holograms are recorded by two or more mutually coherent wavefronts, whose interference pattern modulates the refractive index of a photo sensitive material. The modulation of the refractive index generates volumetric diffraction when the volume hologram is exposed to a probe wavefront, reaching maximum diffraction efficiency if the entire probe wavefront satisfies the Bragg condition [3]. VHI systems make use of the volume hologram as a 3D optical element, exploiting the strong shift variance due to the Bragg condition to achieve depth selective imaging and hyperspectral imaging [4, 5]. We have evaluated the system response of various VHI systems [6] and demonstrated multi–dimensional imaging systems such as the volume holographic telescope and microscope [1, 7, 8].

To design a VHI system, Bragg diffraction from volume holograms has to be analyzed. Analytical models of volume holograms have already been reported [1, 6, 9]. However, they are based on wave optics and not convenient to use in design process. In this paper, we propose a ray–tracing implementation of volume hologram modeling. In modern optical system design, ray tracing software packages are frequently used because they provide more functionality and flexibility with various analysis and optimization features, which makes the design process much more efficient and reliable.

In section 2, we revisit the geometrical representation of volumetric diffraction, the k-sphere. In section 3, we describe its implementation in a Zemax ^® [2] UDS. In section 4, we investigate the consistency and accuracy of the approach and compare the simulation results with our conventional method. We present illustrative examples to demonstrate the design capabilities, focusing on the optimization of aberrated systems. Section 5 discusses advantages and limitation of our ray–tracing implementation and addresses future developments.

2. k–sphere formulation

A volume hologram is considered as a phase grating, but being significantly thicker than conventional diffraction gratings, it is subject to volumetric diffraction and shows strong wavefront selectivity when exposed to a probe beam. We have previously derived a mathematical model of volumetric diffraction of scalar fields assuming Born’s first order approximation [9]. The k–sphere formulation is the geometrical representation of the above model; it originates from a method used in crystallography and solid-state physics [10] and has the potential to simplify the calculation of volumetric diffraction by geometric construction.

 

Fig. 1. Geometric construction of volumetric diffraction using the k-sphere

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In Fig. 1, we explain the concept of the k–sphere formulation. In this particular case, we assume that the volume hologram is recorded by plane waves for simplicity. Later we will describe how to use the k–sphere formulation for volume holograms recorded by non–plane waves.

We denote by λ _f and λ _p the wavelengths of the recording and probe beams, respectively. Let k _f, k _s denote the reference and signal wavevectors, respectively (|k _f|=|k _s|=2π/λ _f); while k _p denotes the probe wavevector (|k _p|=2π/λ _p). As shown in Fig. 1(a), the grating vector K _g is obtained as

To determine the diffracted field in the Bragg regime, we need to examine the vector k _p+K _g [9]. The direction of the wavevector k _d of the diffracted field is specified by demanding that the lateral component of k _d equals the lateral component of k _p+K _g, and that the tip of k _d lies on the k-sphere (i.e., the condition for propagating field); that is,

These constraints are obtained assuming that the lateral size of the hologram is much larger than the hologram thickness. If k _p+K _g satisfies the above conditions, then k _d≡k _p+K _g, and the probe is said to be Bragg–matched as shown in Fig. 1(b). If the Bragg matching condition is not satisfied, k _d is still obtained from Eqs. (3)–(4) but the diffraction efficiency is decreased according to

where L is the hologram thickness. The vector

is referred to as the Bragg mismatch vector; clearly, Bragg matching requires δ k _d=0, whereas if δ k _d≠0 the diffraction efficiency is determined directly by the magnitude of δ k _d (Fig. 1(c)). It is evident from the geometrical construction that the Bragg condition can be fulfilled at an infinite number of combinations of probe beam wavevectors k _p and probe wavelengths λ _p. The locus of the k _p vector’s tip as function of λ _p constitutes the wavelength degeneracy curve of the volume hologram (Fig. 1(d)).

It should be noted that the k–sphere formulation can be applied either to slanted (θ _s≠θ _f) or unslanted holograms (θ _s=θ _f). Moreover, reflection holograms (|θ _s-θ _f|>90°) as well as transmission holograms (|θ _s-θ _f|<90°) can be described by the k–sphere formulation.

Typically, the k–sphere formulation is not frequently used to analyze Bragg diffraction of holograms recorded by non–plane waves, because the direction of the grating vector (K _g in Fig. 1) changes locally over the hologram surfaces. However, ray tracing software can handle this situation. Once the local grating vector where the incident ray hits the volume hologram is known, the diffracted ray can be computed by the k–sphere formulation. Thus we implement two options to compute grating vectors; 1) holograms recorded by either planes waves or spherical waves and 2) holograms that have arbitrary grating vectors. In the first case, once the angle and/or the position of a point source are known, then the grating vector can be computed analytically. In the second case, we discretize the hologram and assign local grating vectors. A similar approach has been used to model non–plane wave reference holograms using coupled mode theory [13, 14]. Our implementation interpolates the assigned grating vectors to compute local grating vectors.

 

Fig. 2. Layouts of several VHI systems with different volume hologram. Note that images are produced by Zemax ^®) with a realistic bi–convex lens (BK7, f=30 mm). d _r and d _s are distances from a volume hologram to point sources. They are used to specify the origin of spherical wavefronts.

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Figure 2 shows various geometries of VHI systems with different volume holograms. These figures are generated by Zemax ^®with our volume hologram model. Figures 2(a), (b), and (d) are transmission holograms and Figs. 2(c), (e) and (f) are reflection holograms. Figure 2(a) uses an unslanted hologram and the rest of them use slanted ones. Not only plane waves but also spherical waves can be used as shown in Figs. 2(d), (e), and (f).

3. Implementation of the k–sphere formulation in a Zemax^®User Defined Surface

We implemented the k–sphere formulation in a User Defined Surface (UDS) of Zemax ^®. The UDS feature allows making a user’s own special optical elements, where users can define special refraction, reflection, diffraction laws. A UDS is represented by a Microsoft Windows ^® Dynamic Link Library (DLL), the file which contains functions invoked by Zemax ^® at design and simulation time. Besides Zemax ^®, other commercial ray tracing software suites also provide customizable features. Hence, our approach can be easily integrated in other software packages with proper modifications.

Figure 3 describe the overall process. First, we consider a volume hologram as a surface. The thickness of the volume hologram is taken into account when the diffraction efficiency is computed. Zemax ^®chooses a single ray arriving on the hologram UDS. Then Zemax ^®computes the local grating vector K _g and the k-vector of the incident ray by using direction cosines (l,m,n) and wavelength as

 

Fig. 3. Schematic of volume hologram UDS implementation in Zemax ^®, where K _g is the local grating vector and η ₀ is the local diffraction efficiency at (x,y,z), k _p and k _d are k-vectors of the incident and diffracted ray, respectively. k _d is computed by the k–sphere formulation. The intensity I of the diffracted ray is η ₀×η, where η is computed by Eq. (10).

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Using the geometrical relationships of the k-sphere, the diffracted ray vector k _d and the Bragg mismatch vector δ k _d are identified to be

and

Continuing with the implementation of the UDS, we normalize the diffracted ray vector k _d and directly write the result into the ray data structure of diffracted rays.

The attenuation of diffraction efficiency as a result of Bragg mismatch is taken into account by δ k _d in (5). Since the normalized diffraction efficiency is equivalent to the relative transmittance of the surface for the given ray, the corresponding parameter of the data structure is rel surf tran, defined as

where η ₀(x,y, z) is the local diffraction efficiency. This feature allows changing absolute efficiency locally. Typically manufacturers provide hologram efficiency maps, hence the data can be substituted into η ₀(x,y, z). Note that k _p,K _g,k _d and rel_surf_tran are repeatedly computed for all rays being traced.

4. Examples

In this section, we present simulated results generated by Zemax ^®and our volume hologram model. First we reproduce the experimental data [1] by simulation, and then we address new directions of using Zemax ^®in VHI system design.

4.1. Depth selectivity

In the case of the VHI system with a plane wave reference hologram as shown in Figs. 2(a) and (b), maximum Bragg diffraction is achieved if the light from a point source is collimated and the resulting plane wave enters the volume hologram at the Bragg matched angle. If the point source leaves the front focal plane and moves along the optical axis, due to the defocus, then the wavefront entering the volume hologram is no longer a plane wave, causing Bragg mismatch except for a Bragg matched location on the optical axis. This example demonstrates how the Bragg condition can be exploited to achieve depth selectivity with volume holograms. We used a 4–f VHI system as shown in Fig. 2(b) (focal lengths f ₁=f ₂=50.2 mm) with a slanted volume hologram (radius a=3.5 mm, θ _f=0°, θ _s=12°, L=2 mm) at the Fourier plane, and the wavelength of recording and probe beam is 532 nm.

Figure 4(a) shows the ray–tracing simulated plot of the total diffracted intensity at the detector plane versus the defocus of the point source. The longitudinal PSF is defined[1] as the normalized diffraction efficiency with respect to defocus. Figure 4(b) shows an analytical simulation, overlayed with an experimental result [1] for the comparison.

 

Fig. 4. Longitudinal PSF of a 4–f VHI system (see text for parameters)

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4.2. Effect of geometrical aberrations in volumetric diffraction

An important advantage of the ray–tracing approach over analytical methods is the straightforward fashion in which aberrations can be taken into account. In the analytical model [1], the paraxial approximationwas used and rotational symmetrywas assumed for the wavefront inside the volume hologram. This prohibited aberrations other than spherical to be taken into account. Generally, we can assume that aberration reduces the Bragg diffraction efficiency, and consequently the depth and wavelength selectivity of VHI systems may be degraded. Therefore, the ability to model aberration effects is essential for making full use of analysis and optimization tools in the design process.

Figure 5 presents a simple example showing the effect of aberration to the longitudinal PSF. In this case, we used the ideal volume hologram represented by one K _g vector and two different objective lenses: first, an ideal thin lens (which, observing the paraxial approximation does not introduce any aberration in Zemax ^®); and second, a realistic bi–convex lens made of BK7. The effective focal lengths of the two lenses are identical at the wavelength of 532 nm.

 

Fig. 5. Longitudinal PSFs with paraxial and aberrated lens. (f ₁=50.2 mm, L=2 mm, θ _s=23.1°, λ _f=λ _p=532 nm)

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The aberrated lens yields slightly lower Bragg diffraction than the paraxial lens; however, unexpectedly the FWHM(Full Width Half Maximum) of the longitudinal PSF is improved by 10% after the normalization. This implies that the depth resolution (defined as FWHM of the longitudinal PSF) can be engineered by managing aberrations.

We now present another simple example to demonstrate the capabilities of our ray tracing model in conjunction with the built-in analysis features of Zemax ^®. In the paraxial 4–f system, we replace the first lens with an aberrated lens of constant focal length and variable q factor defined by

where R ₁,R ₂ are the radii of curvature of the front and back surfaces of a lens, respectively [11]. We conducted a simulation where we changed the both radii of curvature of the surfaces while maintaining its focal length. As is well known, the severity of the spherical aberration is a function of q in this case. The following movie shows the transformation of the first lens through different stages of shape factor, the resulting diffraction pattern and a ray–fan diagram to quantify the severity of aberration.

4.3. Optimization example

As in any optical system, VHI requires careful optimization of the overall optical elements, including the objective lens used to transform the object wavefront before it is diffracted by the hologram. In this section, we demonstrate a simple optimization of a VHI system with respect to aberration. We first run the optimization for both a regular 4–f system and VHI system with the same merit function. Intuitively, the optimization would yield different result because the characteristics of both systems are somewhat different. Then we try to obtain similar result with different merit function, which additional constraints are required in VHI optimization.

We first used the default merit function, which minimizes the overall optical path length for the evaluated rays from 18 predefined points. As the parameters to be optimized, we choose both radii of curvature of the aberrated lens. Column “DFM” (Default Merit Function) of Table 1 shows the result after 1,000 iterations.

In this particular example, the point source is on-axis so that spherical aberration is dominant. With a singlet, the minimum spherical aberration is achieved when the shape factor is

 

Fig. 6. Diffraction image, lens shape and ray fan. (1.4MB) [Media 1]

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Table 1. Optimization of 4–f and VHI system using the default merit function. The last column is computed by the IMAE operand in the merit function.

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where n is the refractive index [12]. In the example, we use BK7 glass (n=1.5195@532.8 nm), q_min.SA is 0.7437. Note that in the regular 4–f system, the optimized shape factor (0.7240) is almost same as q_min.SA, while the shape factor of VHI is only 0.0464. This is because of the merit function we chose, all rays diffracted by the hologram have the same weighting values with respect to the optimization goal. It is obvious that the optimization results in different shape factor for the VHI system.

Instead of using the default merit function, a more practical choice is to increase the intensity of the diffracted beams. Hence, we modified the merit function to give less weight to the rays attenuated due to Bragg mismatch. In Zemax ^®, this can be done by defining the merit function to maximize the IMAE operand, which measures the integrated intensity at the detector plane. The rightmost column labeled “DFM+IMAE” of Table 1 shows how the modified merit function leads to an improved overall diffraction efficiency of the VHI system. Note the optimized shape factor (0.5170) with IMAE operand is much closer to q_min.SA than the value (0.0464) computed with the default merit function only.

5. Discussion and conclusions

We described a ray–tracing approach for the simulation and optimization of VHI systems. Our model for volumetric diffraction is based on the k–sphere formulation, whose applicability is verified by comparison with results of our previous model [9]. We implemented a Zemax ^®User Defined Surface to demonstrate how the model can be easily integrated into current commercial ray–tracing software. Our method can simulate various volume holograms including reflection/transmission and slanted/unslated ones. Moreover, our model computes the local grating vector analytically or numerically from given data, which allows simulating not only plane/spherical wave reference holograms but also holograms with arbitrary grating vectors. Using the local efficiency parameter η ₀, we also account for absolute diffraction efficiency changes resulting from material properties or index variation,etc. Through several examples, we demonstrated how ray–tracing based analysis and optimization tools are used in conjunction with our model to assist in the design process of VHI systems. We showed that our model is ready to simulate aberrations in VHI systems and provided examples of how optimization tools can be applied to mitigate the effects of aberration or even benefit from them. We also want to mention that our approach can be integrated within other ray tracing software packages with proper modification. However, there is one limitation: the k–sphere formulation assumes that the lateral extent of the volume hologram is infinite. Currently, we have verified the model for spatially uniform volumetric gratings, e.g. an ideal plane wave reference volume hologram. It is left for future research to extend the model for non–uniform gratings, such as spherical reference wave holograms, holograms recorded with aberrated beams, etc.