Angular Bandwidth in DCG

 Posts: 772
 Joined: Wed Jan 07, 2015 2:10 pm
Angular Bandwidth in DCG
I was reading a paper which suggested that "angular bandwidth" (which I assume is the number of viewing angles a hologram can be seen from, i.e., narrow angular bandwidth = restricted viewing angles) can be related to spectral bandwidth. The paper suggested that narrow spectral bandwidth is associated with narrow angular bandwidth in DCG. Can anyone shed some light on this? Thanks in advance, Din
Re: Angular Bandwidth in DCG
Yes, true.
Consider a set of Bragg planes. If the planes are all parallel from two perfectly collimated and perfectly monochromatic, sources (physically impossible), then all the Bragg planes would be at a specific angle and a specific spacing and reconstruction only happens when the reconstruction angle is highly specific; according to Kogelnik, such a set of planes would only cause diffraction for 2.5 degrees around the perfect reconstruction angle, and the "perfect" angular bandwidth would be +/ 2.5 deg.
However, if you recorded with a relatively narrow band, ie a low but significant range of wavelengths, and a divergent source such as you would in a display hologram, then the hologram would "see" a range of collimated wavefronts at different angles; any divergent source is the equivalent of a set of collimated beams over a range of angles. Each collimated beam would cause a set of parallel planes at an angle based on the angle of the collimated source, causing a set of planes at slightly different angles and slightly different spacing. If you were to 'reconstruct' the hologram at this point  see the latent image  the hologram would be visible over a narrow range of angles dependent on the bandwidth and divergence (angular bandwidth) of the source.
If you now develop the hologram by putting it in water, you'd get differential swelling  the water would initially penetrate the upper layer of the hologram, which would continue to swell as the water penetrated deeper. This differential swelling would exacerbate the variation of angles and spacing of the planes. hence creating a larger spectral bandwidth (greater variation of plane spacing), but also a greater angular bandwidth (greater variation in angles of the planes).
Therefore, the greater is the spacing of the planes  spectral bandwidth  the greater the range of angles of the planes. As you view the hologram, the reconstruction source "picks up" a different set of planes at different angles. The precise variation of angular bandwidth as a function of variation of spectral bandwidth can be derived from Kogelnik.
Consider a set of Bragg planes. If the planes are all parallel from two perfectly collimated and perfectly monochromatic, sources (physically impossible), then all the Bragg planes would be at a specific angle and a specific spacing and reconstruction only happens when the reconstruction angle is highly specific; according to Kogelnik, such a set of planes would only cause diffraction for 2.5 degrees around the perfect reconstruction angle, and the "perfect" angular bandwidth would be +/ 2.5 deg.
However, if you recorded with a relatively narrow band, ie a low but significant range of wavelengths, and a divergent source such as you would in a display hologram, then the hologram would "see" a range of collimated wavefronts at different angles; any divergent source is the equivalent of a set of collimated beams over a range of angles. Each collimated beam would cause a set of parallel planes at an angle based on the angle of the collimated source, causing a set of planes at slightly different angles and slightly different spacing. If you were to 'reconstruct' the hologram at this point  see the latent image  the hologram would be visible over a narrow range of angles dependent on the bandwidth and divergence (angular bandwidth) of the source.
If you now develop the hologram by putting it in water, you'd get differential swelling  the water would initially penetrate the upper layer of the hologram, which would continue to swell as the water penetrated deeper. This differential swelling would exacerbate the variation of angles and spacing of the planes. hence creating a larger spectral bandwidth (greater variation of plane spacing), but also a greater angular bandwidth (greater variation in angles of the planes).
Therefore, the greater is the spacing of the planes  spectral bandwidth  the greater the range of angles of the planes. As you view the hologram, the reconstruction source "picks up" a different set of planes at different angles. The precise variation of angular bandwidth as a function of variation of spectral bandwidth can be derived from Kogelnik.

 Posts: 772
 Joined: Wed Jan 07, 2015 2:10 pm
Re: Angular Bandwidth in DCG
That's fascinating. Thanks for the explanation.