566 Appendix 4
the (α,β) system in Chapter 8. It basically just counts the pixels on the SLM. In Chapter 8, we derived
I-to-S transformations, which dened the data S
αβ
at the SLM pixel (α,β) in terms of the raw camera or
image data. We can therefore imagine the (α,β) coordinate system corresponding to the (y
o
,z
o
) coordinate
system. Specically, we can write, following the results of Chapter 8
*
,
z
N
y
o
A
o
=
−
−
−
=−
−
21
1
1
2
21
()
()
tan
()
(
α
β
Ψ
PH
NN
B
−
−
1
1
2)
tan
Ψ
PV
(A4.24)
where Ψ
PH
and Ψ
PV
are the printer eld of views (horizontal and vertical) as apodised by the SLM.
If we write data, S
αβ
to the SLM at the pixel (α,β) corresponding to the coordinates (y
o
,z
o
) on the
projected SLM plane, ?′ at unit distance from the hologram then we know that on replay the ray will
effectively move to a new position (y
i
,z
i
) on ?′. We can now dene the system (αʹ,βʹ) to dene the effective
pixel position this image ray corresponds to
z
N
y
i
A
i
=
′
−
−
−
=−
′
−
21
1
1
2
21
()
()
tan
(
α
β
Ψ
PH
))
()
tan
N
B
−
−
1
1
2
Ψ
PV
(A4.25)
So what we actually want to do is not to write the data S
αβ
to the pixel (α,β). Rather, we want to write
the data S
αʹβʹ
to the pixel (α,β). This way, on replay, the data S
αʹβʹ
ends up in the right place. Another way
of saying this is that we should redene the SLM data so that
′
=
′′
SS
αβ αβ
(A4.26)
where
′
=+
−
+
αα
β1
1
2
1
2
N
z
A
i
(,)cot
Ψ
PH
′
=+
−
−
βα
β 1
1
2
1
2
N
y
B
i
(,)cot
Ψ
PV
(A4.27)
In this way, we end up writing an image data byte to a different pixel location on the SLM than we
would have done if there were no emulsion thickness change. However, this shift in position is just what
is required such that the distortion induced by the emulsion thickness change effectively moves the ray
back to the position where it should have been.
The index transformation (Equation A4.27) may be calculated just one time and then applied to
all hogels for a given primary colour channel. Typically, bilinear or bicubic interpolation is used (see
Appendix 7). Note that the index transformation depends on illumination wavelength, so there will be
different transformations necessary for each primary colour.
A4.5 Compensation for Chromatic Aberration
Our reection hologram is assumed to be illuminated by a broadband white-light source. As such, each
ray will be associated with an optimal replay wavelength that is, in general, different from λ
r
. This
*
Note that we are using a non-conjugate SLM position here as per Figure A4.1.
567Appendix 4
optimal wavelength is determined by the parameter, ε, in the ray equation (Equation A4.22). For the
axial ray of each hogel
ε
λ
λ
τ
ττ θ
==
++ −
cr
rc
r
n
n
2
11
22
()cos
(A4.28)
If we consider the case of a three colour reection hologram λ
r
will be replaced by the three laser
wavelengths used to record the hologram-namely λ
R
, λ
G
, and λ
B
.
With no emulsion change, the tristimulus values associated with a given object/image ray from a given
hogel are given by
Xk xxx
Yk yy
=++
{}
=++
SSS
SS
GG RR
BB
GG RR
() ()
()
() ()
λλλ
λλSS
SSS
BB
GG RR
BB
y
Zk zzz
()
() ()
()
λ
λλλ
{}
=++
{}
(A4.29)
Here,
x
,
y
and
z
are the colour-matching functions of the CIE Standard Colorimetric Observer (see, for
example, Giorgianni and Madden [1]) and k is a normalising factor. The parameters S
G
, S
R
and S
B
are,
respectively, the green, red and blue brightness data written to the three primary-colour SLMs for the
case of zero emulsion shrinkage.
When the primary wavelengths change according to Equation A4.22, the tristimulus values will also
change:
′
=
′
+
′
+
′
{}
′
=
′
+
Xk xxx
Yk y
SSS
SS
GG RR
BB
GG R
() ()
()
()
λλλ
λ y
yy
Zk zzz
()
()
() () (
′
+
′
{}
′
=
′
+
′
+
′
λλ
λλλ
RBB
GG RR
BB
S
SSS ))
{}
(A4.30)
This then describes the chromatic aberration of the ray in question. To ensure that there is zero chromatic
aberration, we must change the parameters S
G
, S
R
and S
B
and ensure that the tristimulus values are equal
to their primed values. Or in other words,
kx xxX
ky
′′
+
′′
+
′′
{}
=
′′
+
′
SSS
SS
GG RR BB
GG
() () ()
()
λλλ
λ
RRR BB
GG RR B
yyY
kz zz
() ()
() ()
′
+
′′
{}
=
′′
+
′′
+
′
λλ
λλ
S
SSS (()
′
{}
=λ
B
Z
(A4.31)
where X, Y and Z are given by Equation 4.29.
Equation 4.31 can be written in matrix form:
′
′
′
=
′′′
′′
S
S
S
G
R
B
GRB
G
xxx
yy
() () ()
() (
λλλ
λ
λλλ
λλλ
λ
RB
GRB
G
)()
() () ()
()
y
zzz
xx
′
′′′
×
−1
(() ()
() () ()
() () ()
λλ
λλλ
λλλ
RB
GRB
GRB
x
yyy
zzz
=
S
S
S
G
R
B
aaa
aaa
aaa
11 12 13
21 22 23
31 32 333
S
S
S
G
R
B
(A4.32)
568 Appendix 4
The matrix coefcients a
ij
need only be calculated for each ray one time, as they are identical (for a par-
ticular ray) for all hogels. Application of the transformation (Equation A4.32) to all the SLM brightness
data will effectively rebalance the chromatic equation and ensure that each ray from each hogel has the
correct tristimulus values.
The above analysis assumes that the hologram is rather thick and that the colour-matching functions
of the CIE Standard Colorimetric Observer are only sampled at one exact wavelength. In the case that
the hologram is thinner, we can calculate, the exact form of the spectral function for each ray using
the results of the PSM theory presented in Chapter 12. In addition, one can include the spectral power
distribution of the illumination source. In this case, each of the matrix elements in Equation A4.32 are
transformed in a similar fashion to
xF
xd
c
() () (, ,,)()λληλθψϕλλ
GG
→
∫
(A4.33)
Here, F(λ) is the spectral power distribution of the illumination source and η
G
is the diffractive efciency
in the green channel as given, for example, by Equation 12.90, with the coefcients in Equation 12.101,
in the case of illumination by σ-polarised light.
A4.6 Other Corrections
We have considered here only the case of a simple change in emulsion thickness with different refractive
indices on recording and processing. This is really the simplest case and serves to illustrate in the most
simple and straightforward manner how geometric and chromatic predistortion works. In practice, how-
ever, rather more complicated scenarios arise. In particular, often one wants to write a hologram with
a collimated reference beam and then replay it with a spot lamp at a certain distance. The mathematics
used above can be generalised rather easily to this situation—all that needs to be done is for the vector
k
c
in Equation A4.15 to be written in Cartesian form, thus describing a point source at the desired dis-
tance—this simply changes b
x
and b
y
and introduces a new b
z
parameter in Equation A4.18. Note, how-
ever, that both the geometric and chromatic predistortions are now different for all rays and all hogels.
It should also be underlined that predistortion has its limits. If too great a change in emulsion thick-
ness or index occurs or if too great a disparity in the illumination/recording geometry exists, then it may
just not be possible to compensate for the induced aberrations. We should also state that it is possible to
compensate for slightly larger chromatic aberration if one relaxes somewhat the condition of an axially
propagating image ray bundle. In this case one trades chromatic correction for geometric correction.
However this scheme can very quickly reduce the vertical eld of view.
REFERENCE
1. E. J. Giorgianni and T. E. Madden, Digital Colour Management—Encoding Solutions, Addison-Wesley,
Reading, MA (1998).
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