Angular directivity of diffracted wave in Bragg-mismatched readout of volume holographic gratings

We investigated theoretically and experimentally angular directivity of a diffracted beam in volume holographic gratings. We measured the angular direction of the diffracted beam as a function of Bragg-angle deviation of the read beam and showed that the experimental result agrees well with the Ewald sphere vector model (ESVM). We also showed that the Kogelnik’s coupled-wave theory (CWT) is correct in predicting the diffraction efficiency, but is incomplete in its description of the direction of the diffracted wave. We show that the ESVM and the CWT theories taken together produce a self-consistent mathematical model of wave propagation inside the gratings that is confirmed with experimental results. The proper model for the direction of the output beam as presented here is important in developing theoretical models of image propagation through thick gratings for holographic imaging and correlation applications. 2007 Elsevier B.V. All rights reserved. Holographic optical correlators offer potential advantages in speed for image processing applications, because of the inherent parallelism in optics [1,2]. An efficient implementation of a holographic correlator requires that the device performance be invariant to translations of the target in the field of view [3–5]. The underlying physics of shift-invariant correlation is Bragg-mismatched diffraction from holographic gratings. In designing shift-invariant holographic correlators, it is important to predict the exact angular direction of the diffracted beam in Bragg-mismatched readout of gratings. The direction of the diffracted beam is also important in developing the generic imaging properties of thick gratings [6,7]. In this paper, we derive the angular directivity using the Ewald sphere vector model (ESVM) [8–10] that matches with the experimental data very well. We also show that the angular direction of the diffracted wave in off-Bragg incidence is not predicted correctly within the framework of Kogelnik’s coupled-wave theory (CWT) [11,12], because of the presence of phase factors that were not discussed in 0030-4018/$ see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.08.047 * Corresponding author. Tel.: +1 847 467 0395. E-mail address: [email protected] (A. Heifetz). the original derivation of the CWT. Nevertheless, the CWT is accurate in predicting the angular bandwidth of a volume holographic grating. We show that the ESVM and the CWT theories taken together produce a self-consistent mathematical model of wave propagation inside the gratings that is confirmed with experimental results. Fig. 1 shows the model of a volume holographic grating which is used for our analysis. For simplicity, we restrict our attention to lossless transmission gratings; however the results presented here should also remain valid in the presence of loss. The z-axis is chosen in the direction of the wave propagation, the x-axis is in the plane of incidence and parallel to the medium boundaries, and the y-axis is perpendicular to the plane of incidence. In the general case the fringe planes are slanted with respect to the medium boundaries and the grating vector K is oriented perpendicular to the fringe planes. The magnitude of the grating vector is K = 2p/K, whereK is the period of the grating, and the angle of the grating vector is /, measured with respect to the zaxis. The fringes of the grating are represented by a spatial modulation of the refractive index n = n0 + n1cos(K Æ r), where n1 is the amplitude of the spatial modulation, n0 is the average refractive index, and r is the position vector. Fig. 1. Model of thick holographic gratings readout. 312 A. Heifetz et al. / Optics Communications 280 (2007) 311–316 The read beam is denotedR and the diffracted signal beamS. The propagation vectors q and r contain the information about the propagation constants and the directions of propagation of R and S, respectively For Bragg-matched incidence, jqj = jrj = b, where b = 2pn0/k is the average propagation constant and k is the wavelength in free space. Vector diagrams illustrating Bragg diffraction are shown in Fig. 2 on the Ewald sphere, which is drawn on a plane as a circle of radius b. Bragg-matched readout is shown in Fig. 2 with dotted arrows. For Bragg-matched incidence, the propagation vector r is related to q and the grating vector by