ECE184
Computer Generated Holography for Optical Signal Processing
OBJECTIVE
To design, create and evaluate computer-generated holograms using Lee and
Burckhardt encoding techniques.
BACKGROUND
In this lab we will design a Fourier transform hologram. For this specific hologram the
computer generated hologram (CGH) design procedure can be better understood using the
setup shown in Fig.1. Imagine that a plane wave illuminates a two-dimensional object
with amplitude transmittance t, located in the front focal plane of a lens. The observation
plane is located in the back focal plane of the lens. If we ignore the vignetting effect of the lens
due to finite lens aperture, the complex amplitude of the optical field in the back focal plane
is described by Fourier transform of the input amplitude transmittance
(1)
where are space coordinates in the object plane, x, y are space coordinates in the Fourier
transform plane. Focal length of the lens is f, is the wavelength of the illuminating light. The
complex amplitude Uh (x,y) in the plane of hologram is calculated, encoded and recorded
using an optical storage medium (transparency, photographic film, spatial light modulator,
etc). The original object does not need to exist: given the amplitude transmittance
function tone can use a computer to calculate optical field distribution in the plane of
hologram and then encode and record the calculated field distribution.
Fig. 1: Fourier hologram recording setup
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Fig. 2: Fourier hologram reconstruction setup
Since the hologram contains information about the Fourier transform of the original
object, the reconstruction procedure should involve inverse Fourier transformation of the
recorded hologram. A lens, as shown in Fig.2, is usually used to do this transformation.
The process of creating a computer-generated hologram can be divided into the following major
steps:
1. Choosing the object (desired image).
2. Calculating the object wave front at the plane of hologram.
3. Encoding the calculated wave front into a hologram transmittance.
4. Optical reconstruction of the image using the hologram (optical decoding of
hologram transmittance)
Choosing the object
Since the hologram is calculated in the computer, all the data has to be in digital
(discrete) form. The input image is sampled in space and amplitude. For an input object with
amplitude transmittance t(x, y) one defines a spatial sampling function f(x, y)

f ( x, y)  comb(x/X)comb(y/Y)

where X, Y are sampling periods. The spatially sampled amplitude transmittance is then
represented by
(2)
ts ( x, y)  f(x, y) t(x, y) = comb(x/X)comb(y/Y) t(x, y)
(3)
This sampling process is shown in Fig. 3.
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Fig. 3. Spatial sampling of continuous amplitude transmittance function
It is intuitively clear, that if the samples are taken close enough, the sampled image
becomes a close representation of the original image. For a bandlimited continuous image
(with the spatial spectrum satisfying –BX < fX < BX, –BY < fY < BY) the sampling period has to
satisfy the Nyquist criterion:
1
1
X
, Y
2BX
2BY
to completely preserve the information about original image. Sampling rates lower than
Nyquist result in aliasing in the spatial frequency domain.
For a given two-dimensional object spatial sampling can be performed using an
optical scanner, digital camera, or a similar device. Alternatively, one can start with
creating a raster image directly in a computer. The sampled image, tS(x, y) can be represented
in the computer using a two-dimensional matrix. Each element of the matrix carries
amplitude information, while the matrix indices n, m of the element correspond to spatial
location, as shown in Fig.4.
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Fig. 4: Continuous image and matrix representation of the sampled image.
Calculation of the object wave front at the plane of a hologram
After the object is selected and sampled with appropriate sampling rate one can calculate
optical field distribution at the hologram plane using the discrete Fourier transform,
which in one dimension is given by:
where x[j] is a one dimensional sequence of samples, X[k] is its discrete spectrum, N is the
total number of samples in the sequence and j, k are the sample indices. One of the fastest
ways to calculate the digital Fourier transform is using the FFT algorithm. It requires
that the total number of samples be a power of 2. If the total number of samples is not a
power of 2, the data can be zero-padded before using FFT.
Encoding the calculated wave front into the hologram transmittance
The calculated optical field at the hologram plane contains amplitude and phase
information. For successful hologram reconstruction the amplitude and phase information has
to be recorded in the optical storage medium (photographic plate, transparency, etc.). Even
though it is possible to produce a plate with desired phase and amplitude transmittance, the
manufacturing process is quite involved. Several different techniques have been developed
for encoding the calculated optical field distribution in the hologram. These techniques
allow recording complex amplitude information using only amplitude modulation. In this
lab you will use encoding techniques proposed by Lee and Burckhardt, which are
modifications of the detour-phase encoding introduced by Lohmann.
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In the detour-phase technique the hologram plane is divided into a 2D array of cells
as shown in Fig. 5.
Fig.5. Detour-phase hologram encoding concept.
The cell represents an opaque screen with a small aperture (subcell) of variable size
and location inside the cell. The hologram is illuminated by a coherent light source.
The light diffracts at each aperture and arrives at the observation point from each cell
with a certain amplitude and phase. The amplitude of the arriving light is controlled
by the size of the aperture (the amount of light passing through), while the phase
depends on the relative location of the aperture inside the cell. Therefore amplitude
and phase information about the optical field is stored in the hologram using only
amplitude modulation.
The first detour-phase CGH developed by Lohmann was printed on paper and then
photo reduced by a photo camera. This technique implies that the size of the aperture as
well as its position inside the cell can be precisely controlled – continuous distribution
of aperture size and location was assumed. Printing devices, however, also process
discrete data and this was taken into account by Lee and Burckhardt.
Im(z)
Z
e2
e3
Im(z)
Z
e1
e2
e1
Re(z)
Re(z)
e4
e3
U[n,m]
Lee encoding scheme
U[n,m]
Burckhart encoding scheme
Fig. 6. Lee and Burckhardt hologram encoding techniques.
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After the hologram matrix U[n,m] is calculated each matrix element contains complex
data, that has to be encoded. The element U[n,m] can be represented as a vector in the
complex plane, as shown in Fig. 6. Normally, only two base vectors – along the real and
imaginary axis, represent the value of the
element. However real and imaginary parts
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can have negative values, which cannot be
represented by amplitude modulation
ej+3
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techniques. Lee introduced two additional
base vectors. In his scheme the four base
ej+
vectors have a relative phase shift of /2 and
2
ej+
an arbitrary complex value can be
represented as a sum:
1
ej
U[n, m] A1e1 A2e 2 A3e 3 A4e 4 ,
Fig.7. Relative phase delay between four
apertures (base vectors) in Lee's encoding
technique.
Fig. 8: Binary phase and amplitude encoding using Lee’s technique.
a) Calculated hologram matrix
b) Expanded matrix with phase encoded using four projections
c) Expanded matrix with compensated cell dimension mismatch
d) Expanded matrix with binary amplitude and phase encoding
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where A1, A2, A3, A4 are the projections of U[n,m] into base vectors and only positive
values of the projections are taken. Two of the coefficients Ai are always equal to 0 for
any cell U[n,m].
One can imagine that each base vector corresponds to a separate location of an
aperture inside detour-phase hologram cell. Then relative phase delay between the base
vectors corresponds to the phase delay due to the optical path difference for the light
diffracting off the different apertures as shown in Fig. 7. To implement this encoding
scheme the original hologram matrix U with dimensions NxM is represented by 4NxM
matrix U1 by substitution each matrix element of U with 4 projections A1, A2, A3, A4. The
hologram cell now occupies 4 elements of matrix U1 (See Fig. 8b). When this matrix is
printed, this fact has to be taken into account to match the vertical and horizontal physical
cell dimensions. One can match the cell dimensions by repeating each row of matrix
U1 four times, as shown in Fig. 8c. After this procedure, the matrix can be printed with
the same resolution in the vertical and horizontal directions. Since most of the printing
devices are binary (only black or white), the second dimension is used for binary
amplitude encoding, as shown in Fig. 8d. All the amplitude values are divided into 5
discrete levels for Lee encoding. The matrix from Fig. 8d can be easily printed with
some printing device. It has amplitude and phase information from the original
calculated optical field distribution and can be used in the optical reconstruction setup
from Fig. 2.
In order to represent a complex number with only positive numbers it is sufficient to use
only three base vectors, as shown in Fig. 6. Burckhardt’s encoding technique is similar
to the one developed by Lee but it uses only 3 base vectors.
Optimization of CGH dynamic range and diffraction efficiency
The Diffraction efficiency of a CGH is the fraction of the incident optical power that
appears in a single diffraction order of the CGH. For better object reconstruction and
signal-to-noise ratio in the reconstruction plane one has to maximize diffraction
efficiency of the CGH. There are two ways to increase the diffraction efficiency of a
CGH: top clipping and object phase randomization.
During phase randomization the object amplitude transmittance function is multiplied by
a random phase distribution. The energy is distributed more evenly in the spatial
spectrum domain. As a result, better dynamic range and smaller peak-to-average ratio for
the CGH are achieved, while random phase in the reconstruction plane is not observed by
human eye.
To perform top clipping, the spatial spectrum of the object is multiplied by a positive
constant to make the maximum modulus greater than 1. Then all modulus values greater
than 1 are set equal to 1. This procedure decreases peak-to-average ratio, but affects the
reconstructed image: edges are enlarged.
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EXPERIMENTAL PROCEDURE
Because of the nature of this lab all the procedures, except image reconstruction are
performed in the computer. You will use Matlab to perform all the calculations.
1. Computer generated hologram design using the Lee encoding
a) Choose the object with the size 128x128 samples. Start with simple binary
objects first (for example use a square or a cross).
b) Calculate the optical field distribution at the hologram plane.
c) Apply the Lee 4-aperture phase encoding.
d) Apply 5-level amplitude discretization and perform binary amplitude encoding.
e) Perform dynamic range and diffraction efficiency optimization. At this point you
will need to look back. Random phase mask is usually imposed on the object
before the hologram is calculated. The result of this part of the lab is a bitmap file
(“.bmp”) containing a binary hologram with Lee phase encoding and binary 5level amplitude encoding (a 512x512 matrix).
f) Calculate the hologram of a “resolution target” (set of horizontal black/white bars
with decreasing width).
GRAY level?
2. Evaluation of the CGH
a) Set up hologram reconstruction using reflective Spatial Light Modulator connected
to computer as the second monitor.
b) Open hologram file to see it on the computer monitor.
c) Look for reconstructed image between the bright diffraction orders on an
observation screen.
d) Capture the reconstructed image using a CCD camera and save it as a bmp-file.
e) Use Matlab to analyze the signal-to-noise ratio (SNR) of the reconstructed image.
f) Estimate the spatial resolution of the reconstructed image using a resolution target
as the original object.
g) ???Estimate the dynamic range of the reconstructed image using a variable gray
level bar from the resolution target.
3. Computer generated hologram design and evaluation using the Burckhardt encoding
a) Design Burckhardt-encoded CGHs using a square and resolution target as
objects.
b) Evaluate the performance of the designed hologram using the procedure as in step
2.
c) ???Compare the SNR, spatial resolution and dynamic range provided by the Lee
and Burckhardt hologram encoding.
4. Extra credit.
Create CGH of some simple object (such as logo or else) and reconstruct the image.
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