Wigner analysis of three dimensional pupil with finite lateral aperture.

A three dimensional (3D) pupil is an optical element, most commonly implemented on a volume hologram, that processes the incident optical field on a 3D fashion. Here we analyze the diffraction properties of a 3D pupil with finite lateral aperture in the 4-f imaging system configuration, using the Wigner Distribution Function (WDF) formulation. Since 3D imaging pupil is finite in both lateral and longitudinal directions, the WDF of the volume holographic 4-f imager theoretically predicts distinct Bragg diffraction patterns in phase space. These result in asymmetric profiles of diffracted coherent point spread function between degenerate diffraction and Bragg diffraction, elucidating the fundamental performance of volume holographic imaging. Experimental measurements are also presented, confirming the theoretical predictions.


Introduction
The use of volume holographic (VH) pupils in imaging systems includes applications such as hyper-spectral image acquisition [1], profilometric imagers [2,3], unhindered imaging capability under broadband illumination, and multi-focal microscopic imaging [4,5]. Compared to conventional 2D imaging pupils such as thin gratings and lenses, the VH pupil at a 4-f imagers' Fourier plane acts as a 3D optical element, providing unique opportunities to process the optical field in three spatial dimensions as well as wavelength [6,7].
Utilizing holographic recording techniques and advances in material [5], the 3D VH imaging pupil can be further functionally engineered to achieve wavelength-coded [8] and phase-coded [9] VH imaging gratings. Several mathematical models, including coupled wave theory [10], k-sphere formulation [11], and weak diffraction approximation [12], have been used for 3D pupil analysis. Here we are interested in phase space (i.e. space-spatial frequency) information transport [13] between the object and image planes through the 3D pupil. Recent, prior work [14] introduced the Wigner Distribution Function (WDF) analysis method for that purpose; however, infinite lateral aperture was assumed and, therefore, several important features of the resulting diffraction patterns and point spread function (PSF) were missed.
In this paper, we analyze a weakly diffracting VH pupil of arbitrary thickness and width. We derive basic volume holographic properties of angular selectivity in phase space as well as phase space information of diffracted beams at various locations along the VH pupil, we also analyze of the relationship between the response of 3D pupils in phase space and point spread function as it impacts imaging performance we compare; the simulation results with experiments under different aperture conditions validating the theory. Fig. 1 illustrates a 4-f imager consisting of a 3D pupil, recorded as a volume hologram by two mutually coherent plane waves with an inter-beam angle s θ . Figure 1(a) shows the recording geometry, whereas Fig. 1(b) is the corresponding k-sphere diagram [11] illustrating the Bragg-matching relationship between the diffracted and probe wavevectors d k  and p k  , respectively, and the grating vector K  . The Bragg-matching condition allows the 3D VH pupil to perform its unique spatial-spectral filtering and imaging functions [8,11,12,15].

Theory
To simplify the analysis of the 3D VH pupil onto Winger space, we rather consider the impulse response of a 4-f VH system based on Fig. 1(c). Although we are neglecting the undiffracted (i.e. 0-order) beam for convenience, the geometry is convenient since its 3D spatial transfer function of the 4-f VH system has been reported previously without missing the general Bragg properties of the 3D pupil [12]. In Fig. 1(c), the 3D pupil is located at the Fourier plane and has finite aperture width a , and thickness L. In the vicinity of the Fourier plane, the input and output WDF 3 W , 4 W respectively to the VH pupil can be related to input ( , ) ( , ; , ) ( , ) .
where 3D K is the kernel describing the action of the 3D pupil. Fig. 1 . Illustration of (a) volume holographic recording geometry, (b) k-sphere diagram probed and recorded using wavevectors kp and kd, respectively, for the recorded grating vector K  ; (c) 4-f imaging system geometry with 3D pupil, where f is the focal length, a and L are the lateral and axial apertures, respectively, and red dash-line denotes the Fourier plane. To obtain a clear pupil, the VH would be removed from the system and be replaced by a simple aperture of the same size, located at the Fourier plane.
For a volume hologram, and VHIS h is the coherent PSF of the volume holographic imaging system. Assuming the 1storder Born approximation [12], VHIS h is obtained as Note that for L = 0, Eq. (5) becomes identical to the diffraction-limited PSF of a conventional 4-f imaging system with clear pupil. Substituting Eq. (5) into Eq. (4), we obtain In the case of 2D clear pupil i.e. L = 0, the Wigner function 2D K can be simplified as where Λ denotes the triangle function.

Simulation and analysis
Fig. 2(i-iv) shows the 3 3 x u − planar cross sections of the 4-dimensional 3D K at various coordinates 4 x at the back plane of 3D pupil. The width of the visible slit in the 3 3 x u − space is proportional to the pupil thickness L indicating the Fourier-conjugate relationship with the width of the visible slit in the input 1 x plane due to Bragg selectivity. On the contrary, this thickness-induced visible slit does not appear in the case of a clear pupil, as shown in Fig. 3, owing to 3 4 ( K . Moreover, there exists an offset between the center of probing location 3 x and the maximum diffracted location 4 x in VHIS; this is because of the Braggmatching condition. For example, in case (i) of Fig. 2 its diffracted beam is from 4 0 x = with its corresponding 3 0.25 x = − , while in case (iv) its diffracted beam originates from 4 0.8 x = (outside the recorded area) with its corresponding 3 0.375 x = . This is true independent of the probe beam position, i.e. near the center or edge of the VH pupil. Due to the lateral extent along the 3 x plane as well as finite lateral pupil width, while the probe beam moves towards the pupil boundary, peak values of the 3D K kernel gradually reduce (right-hand side color-bar in Fig. 2). This indicates that Bragg diffraction effects become weaker at the vicinity of the pupil edge, which may be thought of as the grating having reduced effective thickness near the edges. Furthermore, Fig. 4 shows the cross section between 3 x and 4 x under Bragg match condition with 3 u and 4 s u θ λ = at various coordinates. The projection along 4 x in Fig. 4(c) provides Gaussian-like shape, whose width is wider than that along 3 x , and there is an offset between the peak values. These show agreement with previous finding in Fig. 2.  Rather than analyzing the Bragg selectivity of the 3D pupil at 1 2 x x − spatial coordinates [12], 3 4 u u − phase coordinates reveal a more detailed transformation relationship at various locations on the 3D pupil. , depending on the Bragg-matched angle. In addition, the bandwidth of 3D K near the pupil center (in Fig. 5(b)) is narrower than that near the pupil edge (in Fig. 5 (c)), while maximum value on Fig. 5(b) (i.e. Bragg diffraction efficiency) of the kernel in the vicinity of the pupil center is much higher than that in Fig. 5(c). In other words, Bragg diffraction is weaker and angular selectivity is less sensitive near the edge, which show agreement with the previous findings in Fig. 2.  Next, we investigate how the 3D pupil with thickness and finite width affects VHIS performance. Since WDF is a bilinear transform [19], its intensity projected along spatial frequency domain and the point spread function of a 4-f imager can be expressed as ( ) ( , ) , ( ) ( , ) , 2 where 4 W is calculated at the condition of 3 3 3 3 ( , ) ( ) W x u u δ = with coherent on-axis plane wave. 4 W under different recording angles s θ is plotted in Fig. 6 and Fig. 7, respectively.   Compared to a conventional 4-f imager with 2D pupil in Figs. 6(b) and 7(b), 4 I of the 3D pupil in VHIS has wider cut-off range and decays gradually towards the pupil boundary, while 4 I with 2D pupil, as expected, has rectangular profile with narrower cut-off range. Therefore, Figs. 6(c) and 7(c) show that PSF in VHIS exhibits much fewer side-lobes than that in a conventional 4-f imager with 2D pupil. These PSF simulation results also indicate that in VHIS there exist fewer side-lobes along the degeneracy axis (i.e. red line in Figs. 6(c) and 7(c)), and more side-lobes along the non-degeneracy axis (i.e. blue line in Figs. 6(c) and 7(c)), which acts as a conventional 4-f imager with 2D pupil. In addition, an experimental setup was built to further verify our theoretical analysis with PSF performance measurements. The 3D pupil implemented by VH was located in the Fourier plane based on Fig. 1(c). Thickness of the 3D pupil in the PSF measurements was ~1.2mm with 68 s θ =  , and Bragg-matched operation wavelength was 532nm (Millennia, Spectra-Physics). The 3D pupil was illuminated by a collimated beam, and an iris was placed in front of the VH to adjust the pupil width at will. The measured PSF performance at two different values of 3D pupil width is shown in Figs. 8(a) and 8(c). In the measurement, we deliberately saturated the intensity of main lobe in order to identify the clearer side lobe, and the simulated PSF was also saturated for fair comparison. Simulation results are shown respectively in Figs. 8(b) and 8(d), and show good agreement of the lobes' distribution with the measurements. Indeed, both cases result in fewer side-lobes along the degeneracy axis ( 2 x axis). The correlation coefficients between simulated and measured 1-D PSFs passing through peak of main lobe along x and y axis were also calculated according to

PSF measurements and conclusion
where M and S denotes the measurements and simulations of 1-D PSFs, respectively. The i and n is the index and total pixel number of M and S, respectively, and is the average operator. For 1-D PSF along 2 x axis, the correlation coefficients are 0.95 and 0.92, for 0.57 a mm = and 1 a mm = respectively. In case of 1-D PSF along 2 y axis, the correlation coefficients are 0.93 and 0.97, for 0.57 a mm = and 1 a mm = respectively. The corresponding intensity profiles passing through the peak of the main lobe along 2 x and 2 y axis in both simulations and measurements are shown in Figs. 8(i-iv). The possible error could be occurred from neglecting aberration and absorption in our model. This proposed phase space derivation and simulation can be also potentially applied to analyze 3D pupils recorded by spherical reference wave. Future work will include a study of aberration in the imager, and absorption caused by the VH in VHIS.