COL 12(6), 060021(2014) CHINESE OPTICS LETTERS June 10, 2014
Fast calculation of wave front amplitude propagation: a tool
to analyze the 3D image on a hologram
(Invited Paper)
J. -S. Chen and D. P. Chu
∗
University of Cambridge, Photonics and Sensors Group, Department of Engineering,
9 JJ Thomson Avenue, Cambridge, UK
∗
Corresponding author: dpc31@cam.ac.uk
Received March 4, 2014; accepted April 23, 2014; posted online May 30, 2014
A simple approach to calculate the amplitude component of a wave front propagating in space from a
hologram is proposed. It is able to calculate the amplitude distribution on a plane at any distance rapidly
using a standard GPU. This is useful for analyzing and reconstructing the 3D image encoded on a hologram.
OCIS codes: 090.1995, 070.7345, 100.6890.
doi: 10.3788/COL201412.060021.
1. Introduction
Digital hologram calculation has been developed for
decades and the efficiency of algorithms has also been
improved at the same time. However, there wasn’t an
efficient method to calculate a single cross -section of the
3D sce ne encoded on a hologram. In principle one can
apply Huygens-Fresnel principle to compute the wave
propagation
[1]
. This method calculates all wave fronts
propagated from every point on the hologram plane to
every point on a targeted image plane, and then sums
them up at each pixel of the image plane. Apparently,
this method is extre mely time-consuming because a dig-
ital hologra m can have millions or even billions of pixels
and the targeted image plane as well. Therefore the total
calculation can be in the order of quadrillions, and makes
it difficult to be used in practice. Researchers from Ni-
hon University pro posed a method ba sed on fast Fourier
transform (FFT) to simulate a reconstructed 3D image
from a hologram
[2,3]
. They deal with small segments of
a hologram to decide the direction and a mplitude of the
light ray from each segment and a pply a ray model to
simulate the reconstructed image. This method tre ats
each se gment as an individual comp onent, which is a
simplification of the real situation. Lights from different
pixels on a hologra m do interfere with each other even
if they are not fr om the same segment. In addition, this
method is based on ray tracing which does not faithfully
reconstruct a wave front. These two drawbacks make
it difficult to ex tract the cross-section image at a given
distance from a hologram. As a res ult, to check the
quality of the 3D scene encoded on a hologram one has
to either perform a large number of c alculations or opti-
cally reconstruct the whole 3D image and then capture
it to analyze, which are both time consuming and labor
intensive.
To simplify the way to analyze the 3D scene encoded
on a hologram, a fast simulation method is pro posed here
based o n the layer-based method which was developed
for easy genera tion of 3D holograms
[4]
. It should be men-
tioned that this proposed method can only deal with the
amplitude comp onent of the wave front as propagated
from a 3D hologram, but this is sufficient for most ima ge
analysis applications.
2. Layer-based method
The layer-based method is a way to generate 3D holo-
grams using an optical Fourier transform (FT) which
links the image plane and hologra m plane in an f -f
system, as shown in Fig. 1(a) where I is the targeted
image at the image plane and H(f) the hologram at the
hologram plane. These two planes ar e located at two
sides of the lens by the distance of f , which is the focal
length of the lens. Computationally, the image plane and
the hologram plane are each other’s FT. Note that an
f-f sy stem is symmetric, so FT and FT
−1
have the same
optical meaning. This means if we place a hologram at
any s ide of a lens at the focal length distance and provide
a planar light wave to illuminate it along the lens’s axis
toward the lens, the wave fro nt on the image plane a t the
fo c al length distance on the other side of the lens will be
the FT of this hologra m. If we co uld move the hologram
physically forward to the lens plane without affecting
the lens itself, the phase component of the wave front
at the image plane would change but not the amplitude
component, which is of little concer n in analyzing the re-
constructed image since human eyes don’t detect phase.
This concept is illustrated in Fig. 1(b) where φ is the
change of phase after moving the hologram forward, L(f )
the phase representation of the lens with a focal length
of f and H(f) ⊗ L(f) the wave front right in front of the
lens. The “⊗” operator means the direct multiplication
between the pair of the corresponding matrix elements.
The fact that moving the hologram along the lens axis
doesn’t change the amplitude profile at the image plane
can be shown by Eq. (1)
[5]
, where I
f
(u, v) is the com-
plex image on the image plane, (u, v) the 2D coordinate
normal to the lens axis, h
FT
f
(u, v) the complex value of
the Fourier transform of H(f), d the distance between
hologram and the lens, λ the wavelength of the light (532
nm in our experiment) and k the wavenumber defined
as 2π/λ. It is clear that changing d will only affect the
phase part but not the amplitude part of the image.
1671-7694/2014/060021(4) 060021-1
c
2014 Chinese O ptics Letters
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