We will introduce a new simple analytic formula of the Fourier coefficient of the 3D field distribution of a point light source to generate a cylindrical angular spectrum which captures the object wave in 360° in the 3D Fourier space. Conceptually, the cylindrical angular spectrum can be understood as a cylindrical version of the omnidirectional spectral approach of Sando et al. Our Fourier coefficient formula is based on an intuitive observation that a point light radiates uniformly in all directions. Our formula is defined over all frequency vectors lying on the entire sphere in the 3D Fourier space and is more natural and computationally more efficient for all around recording of the object wave than that of the previous omnidirectional spectral method. A generalized frequency-based occlusion culling method for an arbitrary complex object is also proposed to enhance the 3D quality of a hologram. As a practical application of the cylindrical angular spectrum, an interactive hologram example is presented together with implementation details.

© 2015 Optical Society of America

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2014 (1)

2013 (2)

2012 (1)

2011 (3)

2010 (1)

2008 (2)

1982 (1)

1967 (1)

Barada, D.

Distante, C.

Fernandes, J.

Ferraro, P.

Finizio, A.

Fujii, T.

Hahn, J.

Hwang, C.-Y.

Jackin, B.

Javidi, B.

Jeong, I. K.

Jeong, T.

Kang, H.

Kim, H.

Lee, B.

Lee, S.-K.

Lim, Y.

Memmolo, P.

Morley, R.

P. Shirley and R. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).

Oh, S.

Onural, L.

Park, G.

Park, J.-H.

Paturzo, M.

Sando, Y.

Shirley, P.

P. Shirley and R. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).

Soares, O.

Yamaguchi, T.

Yaras, F.

Yatagai, T.

Yoshikawa, H.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (5)

Opt. Lett. (2)

Other (1)

P. Shirley and R. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).

Supplementary Material (2)

» Visualization 1: MOV (7908 KB)      360 degree reconstruction result
» Visualization 2: MOV (27950 KB)      realtime capture video

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Figures (6)

Fig. 1
Fig. 1 Cylindrical grid structure in the 3D Fourier space for cylindrical angular spectrum.
Fig. 2
Fig. 2 CGH synthesis at view angle θ and distance d(a) and rotation of cylindrical angular spectrum patch(b). P colored in gray is an angular spectrum patch determined by the given CGH and P′ is the rotated angular spectrum patch whose center is aligned to the +γ-axis. k is a frequency vector in P and k′ is the rotated frequency vector in P′.
Fig. 3
Fig. 3 Bipolar intensity pattern at z = 0(a) and focus analysis(b) for the field generated by a point source at (0, 0, 1). In the graph of (b), the peak point represents the focus distance.
Fig. 4
Fig. 4 A result of frequency-based occlusion culling. Before occlusion culling(a) and after occlusion culling(b). The front object(left lower) is focused for both cases.
Fig. 5
Fig. 5 Numerical reconstructions for view angle −20°, 70°, 160°, 250° from cylindrical angular spectrum(see Visualization 1 and Visualization 2 for 360° and interactive results).
Fig. 6
Fig. 6 Numerical reconstructions with different focus distance for view angle −20°. The chest(a) and the nose(b) are focused, respectively.

Tables (1)

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Table 1 Computation time to synthesize a CGH with 2,048 × 2048, pixels from a cylindrical angular spectrum with 162,628 × 2,048 grid points for a bunny model with 34,834 points.

Equations (10)

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U ˜ ( x ) = U ˜ ( x , y , z ) = k 3 δ S 1 / λ ( k ) A ( k ) exp ( j 2 π k x ) d k
= k S 1 / λ A ( k ) exp ( j 2 π k x ) d S ,
U p ( x ) = k S 1 / λ A p ( k ) exp ( j 2 π k x ) d S ,
A p ( k ) = exp ( j 2 π k p ) .
A ( k ) = i U ¯ p i A p i ( k ) = i U ¯ p i exp ( j 2 π k p i ) .
k n p > 0 ,
A p ( k ) = 0 .
A ( k ) = 1 γ A ( k ) ,
A z = d ( α , β ) = A z = 0 ( α , β ) exp ( j 2 π d 1 λ 2 α 2 β 2 ) .
[ a b = [ v v v w v w w w 1 [ ( x p 0 ) v ( x p 0 ) w .