Previous studies of the diffractive patterns have shown somewhat ambiguous results in estimating the shape of the crescent. For example, the shape has been attributed to the curvature of the hologram fringes [

1,

3], or it has been assumed exclusively circular or elliptical [

1,

11]. In some cases this ambiguity can be attributed to the utilization of the paraxial approximation, the effect of other optical components in the imaging system, and the utilization of a reduced FOV.

In order to minimize errors, our initial modeling and experiments use a simple setup indicated in . This setup minimizes the effects of undesired aberrations caused by other optical elements other than the hologram. In this setup, the beams are co-axially aligned before passing through a microscope objective (NA=0.55) which then focuses the beams onto a hologram. This setup permits to fill completely the objective aperture producing a cone of light incident on the hologram larger than the one produced in the imaging system shown in . This allows for a greater portion of the angular bandwidth of the volume grating to be illuminated, therefore more of the resulting diffraction pattern can be observed. The expansion of the FOV minimizes the error in the estimation of the diffractive shape.

shows bright crescents produced by two beams incident on a hologram using this setup. The light sources producing these images are the 488nm Argon laser and a 632.8 nm HeNe.

The best fit for the measured traces follows a curve given by hyperbolic equation given by

For the HeNe wavelength the curve parameters were [A,B,C]=[ 1, 1.08,8900], whereas for the 488 nm beam the curve parameters were [A,B,C]=[1, 0.445, 3900] .

The curve parameters were obtained by minimizing the error between fitting and measured curves (both shown in ). In order to simplify comparison, the fitting parameters are normalized by A. More details of this computation are shown in

Appendix I.

To model the crescent of this VHOE, the ACWA (described in Section 2) was implemented as a custom dynamic link library (dll) in Zemax™. Results of our general numerical model are shown in . In that figure the dotted lines represent the curve fitted to the hyperbolas obtained from previous measurements (). Comparison of modeled and measured traces shows very good accuracy (error < 3%).

However, the numerical model alone does not explain the causes for the obtained hyperbolic shapes. Neither, does it provide a good insight of the dependence of this diffractive pattern on the reconstruction wavelength and the hologram construction parameters. To pre-determine the image field and apply fast image processing techniques to VHIS it is important to understand and predict the diffractive image properties.

The analytical model described in Section 3.2 provides a criterion to determine the type of conic produced by the diffraction based on the hologram construction parameters and reconstruction wavelength.

In this model, the degree of tilt of the grating fringes is given by γ as shown in

(12). For the gratings utilized in S

^{2}-VHIS, it is desirable to kept γ>80°. The use of smaller angles increases the tilt of the fringes and affects negatively the resolution and FOV of the imaging systems.

Replacing γ=80° in

(13) it is found that hyperbolic shapes are produced when,

Also, the diffracted angle in the medium has to be lower than the total internal reflection angle of the hologram medium q < sin 90 / n

_{2}. These limits indicate that for almost all of the operational spectral range of VHIS, the x

^{2}-term coefficient in

(18) is positive. Therefore, using

(19) it can be found that in the majority of the cases using S

^{2}-VHIS diffracted shapes are hyperbolas. The asymptotes slopes of the hyperbolas obtained from

(18) are given by,

For the gratings diffraction patterns shown in and , γ=90° the asymptotes slopes are 1.49 for 488 nm and 0.96 for 633 nm. Theoretical and experimental values agree with error< 3% .

In cases where the HOE has large tilt in the K vector, such as cases close to the 90-degree geometry VHIS, γ is very small and it is possible to obtain other shapes such as a circle, ellipse or parabola. However, those grating construction geometries are not very useful for low aberration and high resolution VHIS.

For simplicity, the equations derived in this section were obtained assuming a camera aligned normal to the HOE (). However, we have derived the equation for other cases such as the one shown in [

11] in which the camera is aligned normal to the in-plane diffracted vector (zero components in the y-axis). In general, this case involves a rotation along the y axis to match the normal of the camera surface to the in-plane diffracted vector. Although this rotation affects the way the curvature of the diffracted pattern is projected in the camera, the discriminant classification shown in

(19) still holds for the hyperbolic shape produced by holograms used in S

^{2}-VHIS . In general, for a given curve defined by f(x,y), the different orientation of the camera will produce a curve given by f(ax+b,y), where

*a* and

*b* are parameters related with the coordinate system rotation [

19] . This means that an ellipse may be transformed in a circle or vice versa. However, there is not a transformation from the hyperbola to an ellipse or circle by simply rotating the camera around the y-axis.

The importance of pre-determining the crescent shape information in S

^{2}-VHIS can be appreciated in . This figure shows the image of a tissue sample at two different depths separated by ~ 50 μm. The images were obtained using an LED source with center wavelength of 633 nm and a two-grating multiplexed hologram as described in [

7]. In part a) of this figure two depth images are shown without segmentation or any other image processing technique. Part b) shows the segmented images using the hyperbolic shape pattern and spatial filtering optimized for each depth image. The segmentation is necessary to determine the boundaries of each depth section, and to select the best filter for each image. A prior-knowledge of the segmentation area and shape significantly reduce the image processing time. This also improves the SNR since spurious noise out of the segmented regions is avoided.

For 4D imaging, each hyperbolic line can be related to a specific wavelength of the illumination source. Therefore, an accurate knowledge of the diffractive shape at each position of the FOV allows partial recovery of spectral information.

It can be seen in this figure, that both planar and curved HOE produced hyperbolic traces. The hyperbolic shape is related to the construction parameters of the grating as described in the preceding sections of this paper. An example showing the derivation of the hyperbolic traces fitting parameters from the analytical model is shown in

Appendix I.