We analyze the bulk elastic transformation of volume holograms as a general approach for three-dimensional pupil engineering. The physical relationship between transformation and the resulting point spread function is discussed by deriving the corresponding analytical expressions. For affine transformations, an analytical solution is directly possible. However, for nonaffine transformations, the analytical solution is not straightforward and we must turn to quasi-analytical solutions using the approximation of the stationary phase. Transformational volume holography offers richer design flexibility and real-time adjustment capabilities for imaging systems.
© 2014 Optical Society of America
Volume holograms (VHs), given their three-dimensional (3D) nature, are inherently shift-variant and are able to provide richer opportunities for point spread function (PSF) design compared to conventional two-dimensional pupils. VHs have been widely utilized in various imaging applications as 3D pupils at the pupil planes of imaging systems [1–5]. However, since generally a VH is recorded by interference of two mutually coherent beams, the degrees of freedom in specifying the PSF are limited by those of the spatial-light modulators (SLMs) used to shape the two interfering wavefronts. To achieve more flexibility, we propose “transformational volume holography,” an analysis and design procedure for engineering the PSF by deforming the exterior of a hologram.
Previously, we have investigated the exterior deformations of the VH pupils using a superposition of point indenters [6–9]. However, not all PSFs are achievable by multiple point indenters. Here, we propose bulk elastic deformations besides point load, including compression, shearing, bending, twisting, etc. Through transformation, the system performance can in principle be tuned to fit more design criteria such as spectral composition of the PSF, anisotropic behavior, and so forth. This general approach is called transformational volume holography. It allows for real-time adjustment of an optical system, e.g., a microscope, for different imaging purposes. One can simply change the elastic deformations applied on the VHs to achieve various PSFs without having to replace the pupils. Here, we address only the forward problem of computing the PSF given the elastic deformation. However, the inverse problem, where given a desired PSF, we seek the deformation that produces it, is much more interesting but much more difficult to solve; hence, we have postponed it for future work.
Numerically, it is straightforward to compute the final PSF given a certain type of transformation. However, these computations require 3D integrations of highly oscillatory integrands. Hence, excessive sampling is required, especially in 3D. This poses unattainable demands on the CPU and extensive memory costs, while still allowing for the danger of numerical instabilities. Furthermore, a physically intuitive relationship between the transformation of the VH pupils and the resulting PSFs is lost. In this paper, we show how to find quasi-analytical expressions for the PSFs. In Section 2, we present the general architecture of volume holographic imaging systems. In Section 3, analysis of coordinate transformations as a result of elastic deformations is performed. In Section 4, we focus on the analytical solution of the PSFs under affine transformations. In Section 5, nonaffine transformations are investigated and the approximation of the stationary phase is utilized.
2. VOLUME HOLOGRAPHIC IMAGING SYSTEMS
In this paper, without loss of generality, we use the transmissive geometry of the volume holographic imaging system shown in Fig. 1. The hologram is recorded by two plane waves originating from point sources located at and . After recording, the hologram is probed by another plane wave, which is exactly the same as the reference beam used for recording (i.e., ). In this case, the hologram is perfectly Bragg-matched and the final optical field collected by the detector can be calculated as [2,3]5.C. We are able to analytically perform the integral of Eq. (1), which gives
More generally, Eq. (1) can be written as2].
3. TRANSFORMATION ANALYSIS
To include this coordinate transformation in the VH analysis, we use the coordinates after the transformation for integration, as shown in Fig. 2. That is to say, the coordinates used for the probe beam and the Fourier transform should be transformed. In addition, the area (or volume) of each integration grid changes. Therefore, as a compensation, the Jacobian of the transformation matrix should be applied on , since mathematically, the Jacobian denotes the scaling of the grid area (or volume). This approach yields
Alternatively, the same coordinates without transformation can be used for integration. The coordinates of permittivity distributions and hologram support functions would then be mapped back to the original coordinates; and due to the change of the hologram support function, the integration boundaries would be modified accordingly. Numerically, these two methods do not make much difference. However, the second method makes it difficult to find an analytical solution.
Note that we assume all transformations are small enough so that only material (permittivity or refractive index) redistribution is considered. For large transformations, anisotropy  becomes non-negligible and should be included. However, in transformational volume holography, a small transformation assumption suffices for most cases of interest because most large transformations result in significant Bragg-mismatch with the incoming probe beam so that the diffracted field becomes minimal.
4. AFFINE TRANSFORMATIONS
We first discuss some common affine transformations where exact analytical solutions are possible. Note that in this section, all transformations, as well as the VH itself, are assumed to be invariant along the axis; thus, we only calculate along the plane. Without loss of generality, for all the numerical examples and discussions below, the following parameters are used for the volume holographic imaging system: wavelength , angle of the signal beam , angle of the reference beam , and size of the hologram , .
A. Hologram Shrinkage6) and we can derive an analytical equation for the final field as 6] using a perturbation theory approach. Note that, interestingly, the resulting PSF is simply a reduction of the intensity when compared with the undeformed case. This makes sense since shrinkage along does not change the grating vector of the hologram, which is along the direction. The hologram is still perfectly Bragg-matched. The lowered intensity is due to the reduced hologram area as a result of shrinkage. An example of the final PSF is demonstrated in Fig. 3(b), which is a scaled sinc function. The two sinc-function curves are shown separately as well in Figs. 3(c) and 3(d). Note that we are plotting the intensity, which is proportional to .
We then consider hologram shrinkage only along the axis [see Fig. 3(c)]. Similar to the previous case, now the transformation matrix becomes3(d). The mainlobe is shifted because the shrinkage increases the grating vector of the hologram along so that in order to make the best Bragg match, the diffracted beam should be at a different angle. The two sinc-function curves are shown separately as well in Figs. 3(g) and 3(h).
B. Elastic Affine Deformations
1. Axial Loading
For axial loading (compression) along the axis, which tends to extend the hologram along the axis,
For a hologram that is rotated counterclockwise by ,
For shearing of the hologram along the direction,
5. NON-AFFINE TRANSFORMATIONS
For the first example of nonaffine deformations, we consider bending as shown in Fig. 4(a). Bending curves the hologram with a radius of curvature equal to the bending radius . Note that by the convention used here, a bending curved to the left [see Fig. 4(a)] results in a negative since the radius extends to the left of the hologram. Radius is positive when the bending curves to the right. We define the “bending ratio” as a convenient measure of the amount of bending exerted on the hologram.
The transformation matrix of bending deformation can be expressed as4(b). It can be seen that for larger bending ratios, the PSF widens and flattens, and its peak also decreases.
The integral along can be analytically performed without any approximation, and this reduces Eq. (19) into a single integral along so that20) can no longer be reduced analytically. Next, we show how the stationary phase method can be used as an approximation to find an analytical solution.
B. Stationary Phase Method22) [17,18]:
The stationary phase has been widely used in electromagnetic scattering, diffraction, and radiation problems as a standard approach to solve the diffraction integrals [20–22]. In this section, we propose that a similar approximation can be used to facilitate the integral of calculating PSFs of VH imaging systems under nonaffine deformations and help reach a quasi-analytical solution.
1. Integration Boundaries and Poles
We first note that for the integral of Eq. (20), the corresponding is a summation of three terms:20) as a sum of three integrals. We refer to these three integrals as Part 1, Part 2, and Part 3. The following discussion focuses on Part 1, while Part 2 and Part 3 can be analyzed similarly. The comparison of contributions from all three components will also be illustrated later. Furthermore, the term is not slowly varying but it can be written as . By comparisons with the standard expression of the stationary phase method shown in Eq. (22), Part 1 can be rewritten as
Now, we are able to solve Part 1. We first realize that the term expressed in Eq. (21) can be zero within the integral range resulting in a pole (one type of critical point) denoted as . Besides, introduces two integration boundaries at and . At these boundaries, oscillations from exponential terms stop abruptly; thus, the contributions from these points should be considered since the fast-varying expression does not cancel out with these sharp changes. Considering these two types of critical points (poles and integration boundaries) together, the final result is 
2. Close Stationary Points
We plotted and for the integral at different detector positions in Fig. 5. At [Figs. 5(c) and 5(d)], it can be observed that two stationary points appear within the integral range where . Furthermore, these two stationary points are close to each other (within a few cycles of exponential oscillations), and they are also both close enough to the pole of . That is to say, the contribution from each critical point should not be added independently; instead, all critical points need to be combined in order to find the total contribution. For detailed quantitative analysis on how close two critical points should be so that their individual contributions become dependent, see .
Now, we have in total five critical points that include one pole, two stationary points, and two integration boundaries. The following identical relationship can be used here to eliminate one integration boundary:
According to , the combined contributions from these four critical points can be expressed as
3. Close Quasi-Stationary Points
In other positions, e.g., [see Figs. 5(e) and 5(f)], the first derivative of does not go to zero, which means that there are no stationary points. However, in this case, another type of critical point, the quasi-stationary point, becomes important when these two points are close to each other. Quasi-stationary points are virtual stationary points at “imaginary” positions and where . For example, if , it has quasi-stationary points at and .
In this case, we have four critical points, one pole, two quasi-stationary points, and one integration boundary. The total contributions can be expressed as 
After applying Eqs. (29) and (32), the resulting PSF is shown in Fig. 6(b). It can be clearly observed that the analytical result matches that of the full numerical solution. It is interesting to see that the contributions only from the poles [Eq. (27)] outline the envelope of the PSF [see Fig. 6(a)].
Similar analyses can be applied to Part 2 and Part 3 of Eq. (24). The total PSF is plotted in Fig. 7. We notice that the major contribution to the final PSF is from Part 1. This is because we assume a small bending so that and [see Eq. (20)]. Calculation of Part 1 suffices if the required accuracy is not high.
As another example of nonaffine deformation, we consider twisting. Twisting, which is different from all cases discussed above, does not have an invariant axis therefore all three coordinate axes should be considered. In this section, we consider twisting deformation along the optical axis ( axis). Instead of a rectangular-shaped VH, it is more convenient for the analysis to use a VH shaped as a cylinder (with the cylinder axis coincident with both the optical axis of propagation and the direction of the torque vector). A cylindrical hologram also allows for easy application of uniform twisting torque on the hologram.
For twisting, the rotating angles at different planes are 8. It can be observed that with increasing , the PSF extends along the axis with its peak reduced. This can be intuitively explained as shown in Fig. 9. The probe beam was diffracted by platelets located at different positions with grating vectors rotated at different angles, each contributing to a spot at different positions of the detector. Adding coherently the contributions from all platelets will result in the extended PSF.
In order to derive an analytical solution, we start with the transformation matrix of twisting,6), the resulting PSF is 23].
Without deformation, i.e., , the PSF can be calculated as8(a). However, with twisting, because the integral variable exists inside the jinc function, an exact analytical solution for Eq. (36) is not possible. A similar procedure for incorporating the stationary phase approximation like the one used in the bending transformation above should be used.
First of all, can be rewritten as24]
With Eqs. (41) and (42), the original 1D integral [Eq. (36)] has been reduced and expressed in the standard form of the stationary phase method [Eq. (22)]. Using the same analysis discussed in Section 5.B, a quasi-analytical solution can be derived. The resulting PSF at the detector is illustrated in Fig. 10, together with the difference for the result from the full numerical approach. The analytical result is in agreement with the numerical result.
In conclusion, we proposed a quasi-analytical method for calculating the PSF in transformational volume holography under bulk elastic deformations. For affine transformations, analytical equations of the final PSFs can be directly derived. For nonaffine transformations, we showed how to use the stationary phase to derive quasi-analytical expressions for the PSFs. Transformational volume holography provides more design flexibility for imaging systems. Additionally, the analytical solution not only reduces the computing costs, but it also enhances the physical intuition between the deformation and resulting PSF. The approach proposed here is general and can be applied to other types of bulk transformations or their superpositions.
We are grateful to Zhengyun Zhang for useful discussions. Funding support was from Singapore’s National Research Foundation through the Singapore–MIT Alliance for Research and Technology (SMART) Centre.
1. H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).
2. G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999). [CrossRef]
3. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004). [CrossRef]
4. Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011). [CrossRef]
5. H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013). [CrossRef]
6. K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005). [CrossRef]
7. H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.
8. H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.
9. H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).
10. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef]
11. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]
12. G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent 5,978,112 (November 2, 1999).
13. J. T. Gallo and C. M. Verber, “Model for the effects of material shrinkage on volume holograms,” Appl. Opt. 33, 6797–6804 (1994). [CrossRef]
14. P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).
15. D. H. R. Vilkomerson and D. Bostwick, “Some effects of emulsion shrinkage on a holograms image space,” Appl. Opt. 6, 1270–1272 (1967). [CrossRef]
16. R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).
17. V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).
18. H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).
19. J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982). [CrossRef]
20. V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).
21. O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001). [CrossRef]
22. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).
23. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).