Direct fringe writing architecture for photorefractive
polymer-based holographic displays: analysis and
implementation
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Citation
Jolly, Sundeep. “Direct fringe writing architecture for
photorefractive polymer-based holographic displays: analysis
and implementation.” Optical Engineering 52, no. 5 (May 1,
2013): 055801. © 2013 Society of Photo-Optical Instrumentation
Engineers
As Published
http://dx.doi.org/10.1117/1.oe.52.5.055801
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SPIE
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Thu May 26 09:00:44 EDT 2016
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http://hdl.handle.net/1721.1/80734
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Detailed Terms
Direct fringe writing architecture for
photorefractive polymer-based
holographic displays: analysis and
implementation
Sundeep Jolly
Daniel E. Smalley
James Barabas
V. Michael Bove Jr.
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Optical Engineering 52(5), 055801 (May 2013)
Direct fringe writing architecture for photorefractive
polymer-based holographic displays: analysis and
implementation
Sundeep Jolly
Daniel E. Smalley
James Barabas
V. Michael Bove Jr.
Massachusetts Institute of Technology
Media Lab
77 Massachusetts Avenue, Rm. E15-443d
Cambridge, Massachusetts 02139
E-mail: sjolly@media.mit.edu
Abstract. An optical architecture for updatable photorefractive polymerbased holographic displays via the direct fringe writing of computergenerated holograms is presented. In contrast to interference-based
stereogram techniques for hologram exposure in photorefractive polymer
(PRP) materials, the direct fringe writing architecture simplifies system
design, reduces system footprint and cost, and offers greater affordances
over the types of holographic images that can be recorded. This paper
reviews motivations and goals for employing a direct fringe writing architecture for photorefractive holographic imagers, describes our implementation of direct fringe transfer, presents a phase-space analysis of
the coherent imaging of fringe patterns from spatial light modulator to
PRP, and presents resulting experimental holographic images on the
PRP resulting from direct fringe transfer. © 2013 Society of Photo-Optical
Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.52.5.055801]
Subject terms: dynamic holographic displays; photorefractive polymer; computergenerated holography; direct fringe writing; phase-space optics; Wigner distribution
function.
Paper 130447P received Mar. 21, 2013; revised manuscript received Apr. 9, 2013;
accepted for publication Apr. 12, 2013; published online May 7, 2013; corrected May
8, 2013.
1 Introduction
1.1 Background and Motivation
Updatable, dynamic holographic displays have a wide
variety of potential applications in consumer, medical, and
military imaging contexts. Such displays are typically built
around spatial light modulators (SLMs) that act to generate
three-dimensional imagery via diffraction from dynamically
updated computer-generated holograms. To date, achievement of the requisite space-bandwidth product needed for
large area and large viewing angle in such a dynamic holographic display has proven difficult.1
Photorefractive materials have been studied extensively,
in the last several decades for their application in holographic
recording, particularly in the domain of holographic data
storage.2,3 Because of constraints on their surface area,
most photorefractive materials have proven unsuitable for
application in holographic displays. However, recent
advances in photorefractive polymeric (PRP) materials have
resulted in the development of a PRP with high sensitivity,
fast write times, rapid erasure, and large area by researchers
at Nitto Denko Technical Corporation.4 This combination of
properties has enabled the development of an updatable,
PRP-based holographic display with large viewing area by
researchers at the University of Arizona.4,5 In contrast to
SLM-based dynamic holographic displays, PRP-based displays exploit the “optical memory” offered by PRPs
which allows large-area holograms to be written serially
over time and with persistence in the material for viewing
at the completion of the writing process. Although not
refreshable at video rates, these displays do allow for images
0091-3286/2013/$25.00 © 2013 SPIE
Optical Engineering
to be refreshed as desired (up to a rate of 1∕2 Hz) and can
mitigate some of the viewing angle and display area issues
that hinder SLM-based holographic displays.6
Current PRP-based holographic display systems employ
the conventional holographic stereogram recording technique, which involves the optical interference of a hogelmodulated object beam with a mutually coherent reference
beam.1 While such an approach can offer a convincing threedimensional image, there exist some drawbacks inherent to
the stereogram recording process, namely the lack of accommodation cues in the resulting 3-D images and the complexity of the supporting optical architecture needed to generate
diffraction fringes via coherent optical interference.
1.2 Related and Previous Work
In addition to their use in the generation of diffraction fringe
patterns for dynamic holographic displays, techniques for
computer-generated holography (CGH) have been utilized
for physically fabricated holographic optical elements and
display holograms. Techniques for fabricating CGHs include
electron-beam lithography and optical dot-matrix lithography.7,8 Additionally, researchers at Nihon University have
demonstrated direct optical writing of CGH patterns
displayed on spatial light modulators into permanent photosensitive media for display hologram fabrication.9,10
The authors’ research group has developed a direct optical
CGH fringe writing architecture for PRP-based updatable
holographic display.11,12 Relative to interference-based holographic stereogram recording techniques, direct writing of
holographic fringes offers several advantages to a PRP-based
holographic display, including simpler optical architectures,
smaller footprint and increased portability, lower cost, and
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Jolly et al.: Direct fringe writing architecture for photorefractive polymer-based holographic displays. . .
Fig. 1 System overview for direct optical fringe writing in PRP. The direct fringe writing architecture encompasses fringe computation, display on
spatial light modulator, demagnification and transfer to PRP, and PRP exposure with appropriate computer-control for spatial multiplexing for largeimage generation.
increased affordances in the types of holograms that can be
recorded.
2 Experimental Methods
Our architecture for direct writing of holographic fringes in
PRPs encompasses fringe computation, display on spatial
light modulators, optical projection and demagnification for
transfer to PRP, and spatial multiplexing for large-image
recording. An overview of the system process is depicted
in Fig. 1.
2.2 Fringe Transfer and Total Hologram Rastering
Direct writing of holographic fringes into the PRP material
involves demagnification and transfer of the fringe pattern
displayed on a spatial light modulator. The experimental
setup is depicted in Figs. 3 and 4.
A CW beam from a DPSS laser at λ ¼ 532 nm is beamexpanded and collimated. This beam illuminates a liquidcrystal-on-silicon (LCoS) modulator displaying the information corresponding to an elemental fringe pattern. The LCoS
module, used in the current study, has been furnished by
silicon micro display and has an 8.5 μm pixel pitch and
2.1 Fringe Computation
For the purposes of the current study, fringe computation is
performed using the Reconfigurable Image Projection (RIP)
hologram algorithm which was originally developed at the
MIT Media Lab for video-rate, horizontal parallax-only
(HPO) fringe pattern generation. This method produces
fringe patterns from computer graphics data and is implemented using shader processing on graphics processing units
for real-time computation. This technique has been described
in detail elsewhere.13
At the moment, the computational architecture employs
precomputed HPO fringe patterns. Because these fringe
patterns have a horizontal resolution that is much greater
than what can be displayed on typical spatial light modulators, the fringe patterns are horizontally segmented into a
series of elemental fringe patterns suitable for display on
such an SLM. This process is depicted in Fig. 2.
L
BEO
L1
Fully-Computed HPO Fringe Pattern
L2
S
L
M
Segmentation into Elemental Fringe Patterns
P
R
P
PBS
Elemental Fringe Patterns
z1
Fig. 2 Generation of elemental fringe patterns from fully computed
HPO holographic fringe patterns. Note that the segmentation creates
elemental fringe patterns suitable for display on the SLM used in the
fringe writing setup.
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z2=f1+f2
z3
Fig. 3 Optical setup for direct fringe writing. L ¼ DPSS CW laser
at λ ¼ 532 nm, BEO ¼ beam-expanding optics, SLM ¼ LCoS spatial
light modulator, PBS ¼ polarizing beam-splitter, L1 ¼ input objective
objective of the telecentric imaging system with focal length f 1 ¼
250 mm, L2 ¼ output objective of the telecentric imaging system with
focal length f 2 ¼ 50 mm, PRP ¼ photorefractive polymer. z 2 is dictated by the focal lengths of the lenses used, whereas z 1 and z 3
are linearly related.
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Table 1 Writing parameters for HPO teacup stereogram.
PR polymer bias voltage
Total number of elemental fringe
patterns
35
Exposure time per elemental
5s
Total writing time
Laser output power
Fig. 4 Photograph of the optical setup for direct fringe transfer. Left to
right: SLM, PBS, L1 , aperture stop at Fourier plane of L1 (completely
blocking diffracted orders higher than 1), L2 , PRP.
1920 × 1080 resolution. A polarizing beam-splitter (PBS) is
used to operate the LCoS module in an amplitude modulation configuration. The modulated light is then projected into
the PRP material by means of a bilaterally telecentric optical
system comprised of two cylindrical singlet lenses in a
Keplerian telescope configuration. The focal length of the
input objective L1 is f 1 ¼ 250 mm and the focal length
of the output objective L2 is f 2 ¼ 50 mm. This configuration
yields a demagnification of 5x. Given the 8.5 μm pixel
dimension of the SLM and the 5x demagnification specified
by the telescoping optics, the nominal smallest pixel dimension in the optical irradiance distribution arriving at the PRP
is 1.7 μm.
The PRP used in this experiment has been introduced by
researchers at the University of Arizona and Nitto Denko
Technical Corporation.4 The PRP composite has an active
layer thickness of 100 μm and is positioned between two
indium-tin-oxide-coated glass electrodes. A detailed characterization of the recording and decay dynamics for this PRP
has been reported previously.5 Of interest to the current study
is the exposure time that yields 50 percent diffraction efficiency, which has been reported to be 6 s, for an applied writing irradiance of 100 mW∕cm2 (i.e., exposure energy of
600 mJ∕cm2 ) for an empirically determined optimum
applied voltage. Writing parameters used in the current
experiment are listed in Table 1. The PRP sample is
obliquely slanted relative to the optical axis by a projection
angle of 35 deg. This oblique projection of the writing beam
onto the PRP sample is necessary to induce the space-charge
field that enables refractive index modulation via the photorefractive effect.14 The PRP sample is mounted on a motorized, computer-controlled translation stage that appropriately
translates the PRP in between exposures of neighboring
elemental fringe patterns.
Note that this optical transfer scheme can be considered to
be an afocal projection of the pattern displayed on the SLM.
However, because the fringe pattern displayed on the SLM
generates information-carrying diffraction orders upon
coherent illumination, a demagnified projection of the original fringe pattern is not obtained at all points along the optical axis after the output objective. Careful positioning of the
PRP sample at a distance z3 is required for coherent image
synthesis. This condition is elaborated on in Sec. 3.
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5.5 kV
Maximum irradiance per elemental fringe
pattern
Maximum exposure energy density per
elemental fringe pattern
245 s
200 mW
150 mW∕cm2
750 mJ∕cm2
3 Phase Space Coherent Imaging Analysis
Our method for direct fringe transfer is best modeled with
coherent imaging theory. Analysis of the optical propagation
of diffracted fields from fringe patterns through projection
optics is further aided by the use of a time-frequency distribution for localization of spatial frequencies as the complex
optical field evolves through propagation in freespace and modulation by lenses. In the context of Fourier
optics, the Wigner distribution function (WDF) is a useful
Fig. 5 Simulation geometry for direct optical fringe writing system.
Diffraction of primary orders from a generalized holographic grating
displayed on the SLM is depicted; higher-order modes of diffraction
from the SLM are neglected in this diagram. z 1 is the distance
from SLM to the input objective lens, z 2 ¼ f 1 þ f 2 is the distance
from the first objective to the second objective, and z 3 is the distance
from the second objective to the recording medium at the point of diffracted order re-convergence. W 0 is the WDF at the plane of the SLM,
W −1 is the WDF at the plane of the first objective lens prior to phase
transformation, W þ
1 is the WDF immediately after phase transformation by the first objective lens, W −2 is the WDF at the plane of the second objective lens, W þ
2 is the WDF immediately after phase
transformation by the second objective lens, and W 3 is the WDF
at the plane for PRP exposure (diffractive image plane).
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space-frequency representation of a signal that enables
examination of the spatial distribution of spatial frequencies
and allows for phase-space operations corresponding to freespace Fresnel propagation, fractional Fourier transformation,
or lens-induced phase modulation to be applied via simple
transformations.15,16 The WDF of a one-dimensional complex optical field distribution UðxÞ is given as
Z ∞ x0
x 0 −j2πx 0 s 0
Wðx; sÞ ¼
U xþ
dx ;
(1)
U x−
e
2
2
−∞
where Wðx; sÞ is the WDF, s is the spatial frequency variable,
x 0 is a spatial variable used as an integration reference, and
denotes complex conjugation.
The goal of the projection optics, described in Sec. 2, is to
transfer the fringe pattern displayed on the SLM to the PRP
with a demagnification factor given by the ratio of focal
lengths of the lenses used. A mathematical analysis indicates
that this demagnified irradiance pattern is synthesized via the
re-convergence of diffracted orders from the SLM; this process is depicted in Fig. 5. This notion is consistent with
Abbe’s theory of coherent image synthesis.17,18 It is possible
to analytically derive the distance z3 after the exit objective at
which this convergence occurs (see Appendix A for this derivation). The distance is given as
2
f 22
f2
z1 ;
(2)
z3 ¼ f 2 þ −
f1
f1
where z1 , f 1 , and f 2 are all system distances following the
depictions in Figs. 3 and 5. Note that this condition holds for
any given input pattern regardless of what or how many spatial frequency components may be present. Also note that
this distance corresponds to the imaging condition dictated
by geometrical optics (see Appendix B for this equivalence).
Figure 5 also depicts the geometry used here for simulating the evolution of the WDF of a typical diffracted optical
field. This evolution is depicted in Fig. 6 for an input fringe
pattern corresponding to a simple sinusoidal diffraction grating with Λ ¼ 15 μm, an SLM-L1 distance of z1 ¼ 5 cm,
and lens parameters identical to those used in the experimental setup. Figure 6(a) depicts the input WDF W 0 ðx; sÞ corresponding to the sinusoidal grating displayed on the SLM.
Note that the spatial frequency corresponding to the grating
period is depicted by the top-most and bottom-most energycarrying regions in the WDF at s ¼ 6.66 × 104 m−1 . Freespace propagation and phase transformations by lenses are
modeled via Fresnel and thin-lens transformers in phase
space and mathematical details are provided in Appendix A.
Through a series of these transformations, the output WDF
W 3 ðx; sÞ is computed [Figure 6(f)] and depicts a spatial scaling of 1∕5x and a spatial frequency scaling of 5x (Λ ¼ 3 μm,
s ¼ 3.33 × 105 m−1 ) which is exactly that expected for projection through a Keplerian telescope with angular magnification M ¼ −f 1 ∕f 2 ¼ −5.
Direct examination of the initial and final signals in the
spatial and Fourier domains can further elucidate the
Fig. 6 Simulated evolution of the Wigner distribution function of the optical field resulting via diffraction from a sinusoidal grating with Λ ¼ 15 μm as
it propagates through the optical system depicted in Fig. 5, having f 1 ¼ 250 mm and f 2 ¼ 50 mm. (a) W 0 ðx ; sÞ at the plane of the spatial light
modulator. Note that the top-most and bottom-most energy-carrying regions in the WDF correspond to the spatial frequency content of the grating
(s ¼ 6.66 × 104 m−1 ). (b) W −1 ðx ; sÞ at the plane of the entrance objective L1 , solved via a Fresnel transformation of W 0 ðx ; sÞ. (c) W þ
1 ðx ; sÞ at the
plane of the entrance objective L1 immediately after phase transformation, solved via a thin-lens transformation of W −1 ðx ; sÞ. (d) W −2 ðx ; sÞ at the
þ
plane of the exit objective L2 , solved via a Fresnel transformation of W þ
1 ðx; sÞ. (e) W 2 ðx ; sÞ at the plane of the exit objective L2 immediately after
phase transformation, solved via a thin-lens transformation of W þ
2 ðx ; sÞ. (f) W 3 ðx; sÞ at the plane of the PRP, solved via a Fresnel transformation of
5
−1
Wþ
2 ðx ; sÞ. Note that this final WDF indicates a spatial scaling of 1∕5x and spatial frequency scaling of 5x (Λ ¼ 3 μm, s ¼ 3.33 × 10 m ).
Optical Engineering
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Fig. 7 Comparison of initial and final irradiance and energy spectral density distributions resulting via diffraction from a sinusoidal grating with
Λ ¼ 15 μm in propagation through the optical system depicted in Fig. 5, having f 1 ¼ 250 mm and f 2 ¼ 50 mm. (a) Irradiance distribution, backcalculated from W 0 ðx ; sÞ, at the plane of the SLM depicting a sinusoidal grating with Λ ¼ 15 μm. (b) Irradiance distribution, backcalculated from
W 3 ðx ; sÞ, at the plane of the PRP depicting a sinusoidal grating with Λ ¼ 3 μm – a scaling of 1∕5x compared to the original period. (c) Energy
spectral density distribution, backcalculated from W 0 ðx; sÞ, at the plane of the SLM depicting the spatial frequency content of the grating at
s ¼ 6.66 × 104 m−1 . d) Energy spectral density distribution, backcalculated from W 3 ðx ; sÞ, at the plane of the PRP depicting the spatial frequency
content of the grating at s ¼ 3.33 × 105 m−1 – a scaling of 5x compared to the original spatial frequency.
behavior of the fringe transfer optics. The optical irradiance
distribution in the spatial domain and the energy spectral
density distribution in the Fourier domain are given as
projections of the Wigner distribution function as follows:
Z ∞
jUðxÞj2 ¼
Wðx; sÞds
(3)
−∞
Z
jFðsÞj2 ¼
∞
−∞
Wðx; sÞdx;
(4)
where FðsÞ is the Fourier transform of the optical field UðxÞ.
Figure 7 depicts these projections from the initial and final
WDFs W 0 ðx; sÞ and W 3 ðx; sÞ and affirms the observations
gathered via direct examination of the evolving WDF above.
To validate the notion that coherent image synthesis
occurs as expected for an input fringe pattern consisting of
an arbitrary summation of several spatial frequencies, the
evolution of the WDF of the diffracted optical field from an
input linearly spatially chirped grating is depicted in Fig. 8.
All system parameters are identical to those used for the simulation of the grating WDF. Note that the input WDF
W 0 ðx; sÞ depicts a spread of energy uniformly across the
spectral domain. Through the same series of phase-space
transformations corresponding to Fresnel diffraction and
thin-lens modulation, the output WDF W 3 ðx; sÞ is computed
Optical Engineering
[Fig. 8(f)] and depicts a spatial scaling of 1∕5x and a spatial
frequency scaling of 5x relative to the input WDF W 0 ðx; sÞ
[Fig. 8(a)] for all spatial frequencies present in the original
chirped grating.
4 Experimental Results and Discussion
In order to validate the direct fringe writing architecture and
to measure diffraction efficiency, a sinusoidal grating was
displayed on the SLM and transferred to the PRP sample
using an irradiance of 150 mW∕cm2 , exposure time of 5 s,
and PRP applied voltage of 5.5 kV. Upon readout with a
beam from a helium-neon laser at λ ¼ 632.8 nm, a measured
first-order diffraction efficiency of 12 percent was observed.
HPO holographic fringes of a teacup model (see Fig. 9)
were computed using the RIP algorithm and transferred to
the PRP sample. Using the writing parameters listed in
Table 1 and after total hologram rastering, the resulting
holographic image upon readout with beam-expanded and
collimated light from a helium-neon laser at λ ¼ 632.8 nm
is depicted in Fig. 10. Note that this image exhibits much
higher brightness (diffraction efficiency), longer persistence,
better discriminability, and better apparent depth upon
reconstruction relative to previous images generated through
our direct fringe transfer architecture. However, discrimination of fine features is difficult due to the presence of noise
and other artifacts. Because the optical system used for
imaging the CGH pattern from the SLM to the PRP is
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Fig. 8 Simulated evolution of the Wigner distribution function of the optical field resulting via diffraction from a linearly chirped grating with spatial
frequencies ranging from s ¼ 0 m−1 through s ¼ 2.75 × 104 m−1 as it propagates through the optical system depicted in Fig. 5, having
f 1 ¼ 250 mm and f 2 ¼ 50 mm. (a) W 0 ðx ; sÞ at the plane of the spatial light modulator. Note a spread of energy over spatial frequencies, with
a cutoff in the spectral domain at the maximal frequency contained in the chirp (s ¼ 2.75 × 105 m−1 ). (b) W −1 ðx; sÞ at the plane of the entrance
objective L1 , solved via a Fresnel transformation of W 0 ðx ; sÞ. (c) W þ
1 ðx ; sÞ at the plane of the entrance objective L1 immediately after phase
transformation, solved via a thin-lens transformation of W −1 ðx ; sÞ. (d) W −2 ðx ; sÞ at the plane of the exit objective L2 , solved via a Fresnel transþ
formation of W þ
1 ðx ; sÞ. (e) W 2 ðx; sÞ at the plane of the exit objective L2 immediately after phase transformation, solved via a thin-lens transforðx
;
sÞ.
(f)
W
ðx;
sÞ
at the plane of the PRP, solved via a Fresnel transformation of W þ
mation of W þ
3
2
2 ðx; sÞ. Note that this final WDF indicates a spatial
scaling of 1∕5x and spatial frequency scaling of 5x and therefore the maximal spatial frequency present in the imaged chirp is
s ¼ 1.375 × 105 m−1 – 5x that of the original maximal spatial frequency.
not optimized for high imaged contrast and minimal
aberrations, anomalous diffraction due to imperfect imaging
is likely a contributing factor in the low contrast in the
three-dimensional reconstruction and the large amounts of
noise.
Fig. 9 Computer-generated model of a teacup.
Fig. 10 Holographic image of the computer-generated teacup model.
Note that the dark spot in the left side of the image is a region of
material degradation and not an artifact of the imaging process.
Optical Engineering
5 Conclusions
Our work on direct fringe writing architectures for PRPbased holographic displays has detailed the benefits,
affirmed the feasibility, and examined the challenges of
such an approach. While the current work represents a
feasibility study, a practical implementation of a direct fringe
writing architecture for high-resolution, noise-free, threedimensional image reconstruction will necessitate the use
of more sophisticated imaging optics than we have employed
here. Additionally, the nominal imaged feature size of 1.7 μm
presented here is not sufficiently small for direct viewing
with a large viewing angle, therefore, higher demagnification
factors are necessary. Higher-quality reconstructed images
with wider viewing angles are likely to be obtained with
the use of more sophisticated projection optics that can provide higher modulation transfer (i.e., imaged fringe
visibility) for the feature sizes of interest (e.g., <1 μm)
than the current system can provide. Although we have
not demonstrated dynamic updating of holographic images
in the current experiment, it should be noted that our architecture for direct fringe writing can readily be employed in a
dynamically updated imaging scheme (in lieu of, e.g., the
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hogel-based approach demonstrated in the University of
Arizona’s photorefractive holographic display system6).
Because the algorithm employed here for fringe pattern generation is capable of driving electro-holographic displays at
video rates, the direct fringe writing approach is well-suited
for video-rate and PRP-based holographic display
systems. However, high update rates will require further
improvements in the PRP sensitivity and response time.14
The derivation of the coherent image synthesis condition
using a phase-space representation is aided by well-known
operations on Wigner distribution functions that correspond
to transformations due to Fresnel diffraction (i.e., free-space
propagation in the paraxial approximation) and modulation
by lenses.15,16 For a given input WDF W i ðx; sÞ, the transformation due to Fresnel diffraction through a distance z is
given by
(5)
where W o ðx; sÞ is the output WDF after propagation and λ is
the wavelength of the light used. Likewise, the transformation due to a thin lens with focal length f acting on an input
wavefront is given by
x
W o ðx; sÞ ¼ W i x; s þ
;
λf
(6)
where symbols have the same meaning as above. For the
simulation geometry depicted in Fig. 5, the relationship
between the input WDF W 0 ðx; sÞ and output WDF W 3 ðx; sÞ
can be derived via application of these transformations as
follows:
1. Free-space propagation over a distance z1 from the
plane of the spatial light modulator to the plane of
the input objective L1 over a distance z1 as W −1 ðx; sÞ ¼
W 0 ðx − λsz1 ; sÞ.
2. Thin lens transformation by L1 having focal length f 1
x
−
as W þ
1 ðx; sÞ ¼ W 1 ðx; s þ λf1 Þ.
3. Free-space propagation over a distance z2 from the
plane of the input objective L1 to the plane of the output objective L2 over a distance z2 as W −2 ðx; sÞ ¼
Wþ
1 ðx − λsz2 ; sÞ.
4. Thin lens transformation by L2 having focal length f 2
x
−
as W þ
2 ðx; sÞ ¼ W 2 ðx; s þ λf2 Þ.
5. Free-space propagation over a distance z3 from
the plane of the output objective L2 to the plane
of a screen over a distance z3 as W 3 ðx; sÞ ¼
Wþ
2 ðx − λsz3 ; sÞ.
A straightforward application of these transformations
yields the following relationship between W 0 ðx; sÞ and
W 3 ðx; sÞ:
Optical Engineering
x − λsz3
z2
W 3 ðx; sÞ ¼ W 0 x − λsz3 − λ s þ
λf 2
#
"
x−λsz3
x
−
λsz
−
λ
s
þ
z2
3
λf 2
x − λsz3
z1 ;
−λ sþ
þ
λf 2
λf 1
)
x−λsz3
x
−
λsz
−
λ
s
þ
z2
3
λf 2
x − λsz3
þ
sþ
.
λf 2
λf 1
(7)
Appendix A: Derivation of Abbe Coherent
Imaging Condition in Phase-Space
W o ðx; sÞ ¼ W i ðx − λsz; sÞ;
(
Noting that z2 ¼ f 1 þ f 2 , and after algebraic manipulations,
the relationship becomes
W 3 ðx; sÞ ¼
f1
f2
f1
f2
z þ z − f1 − f2 ; − s .
W 0 − x þ λs
f2
f1 1 f2 3
f1
(8)
When the spatial frequency dependence of the spatial component of W 0 on W 3 is set to zero as
λs
f2
f
z þ 1 z − f1 − f2
f1 1 f2 3
¼ 0;
(9)
the input-output relationship in Eq. (8) collapses to
f1
f2
W 3 ðx; sÞ ¼ W 0 − x; − s ;
f2
f1
(10)
which is the expected input-output relationship for an angular magnification operation with M ¼ −f 1 ∕f 2 . The condition specified in Eq. (9) is satisfied for nonzero spatial
frequency components when
z3 ¼ f 2 þ
2
f 22
f2
−
z1 .
f1
f1
(11)
Appendix B: Equivalance of Geometrical
Optics Imaging Condition and Abbe Diffractive
Imaging Condition
For a single thin lens, the imaging condition in geometrical
optics is given by the thin-lens formula,
1
1 1
þ ¼ ;
z o zi f
(12)
where zo is the distance from the plane of the object to be
imaged to the lens, zi is the distance from the lens to the
plane of the imaged object, and f is the focal length of
the lens.
In a Keplerian telescope (and assuming that a given object
to be imaged is finitely distant), the input objective acts to
form an intermediate image of an object which is, thereafter,
imaged through the exit objective. Assuming the optical
geometry depicted in Fig. 5, the intermediate image is
formed at a distance
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Jolly et al.: Direct fringe writing architecture for photorefractive polymer-based holographic displays. . .
z2 1 ¼
z1 f 1
z1 − f 1
(13)
from the input objective. Noting the distance from this intermediate image to the output objective is given by
z22 ¼ f 1 þ f 2 − z21 ;
(14)
the distance z3 that satisfies the geometrical optics imaging
condition can be found as
z3 ¼
z2 2 f 2
;
z22 − f 2
which, after algebraic manipulations, reduces to
2
f2
f2
z3 ¼ f 2 þ 2 −
z1 .
f1
f1
(15)
13.
14.
15.
16.
17.
18.
Sundeep Jolly holds a BS degree in electrical engineering, a BS degree in physics, and
an MS degree in electrical and computer
engineering, all from the Georgia Institute
of Technology. He also holds an MS degree
from the Media Laboratory at the Massachusetts Institute of Technology, where he is currently a PhD student. His research interests
include electroholographic 3-D displays, photorefractive materials for holography, and signal processing methods in digital holography.
(16)
Note that this result is equivalent to that predicted by phasespace analysis.
Acknowledgments
This work has been supported in part by consortium funding
at the MIT Media Laboratory. This research was also funded
in part by the Office of the Director of National Intelligence
(ODNI), Intelligence Advanced Research Projects Activity
(IARPA), through the AFRL contract FA8650-10-C-7034.
All statements of fact, opinion, or conclusions contained
herein are those of the author and should not be construed
as representing the official views or policies of IARPA, the
ODNI, or the U.S. government. The authors gratefully
acknowledge the support of Silicon Micro Display in furnishing the LCoS spatial light modulators used in this
work. Furthermore, the authors would like to thank researchers at Nitto Denko Technical Corporation and at the
University of Arizona’s College of Optical Sciences for their
initial research efforts toward developing photorefractive
holographic imaging and for providing the PRP samples
used in this work.
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Daniel E. Smalley is a PhD candidate at the
MIT Media Laboratory, where he does work
on integrated optics for next-generation holographic video displays.
James Barabas received his BS degree in
computer science from Cornell University
and an MS in media arts and sciences
from MIT. As a research associate at the
Schepens Eye Research Institute, he developed computer graphics, motion tracking,
and haptic interfaces and conducted experiments to evaluate mobility in people with visual impairments. He is currently a PhD
candidate at the MIT Media Laboratory. His
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processing, visual psychophysics, and holographic displays.
V. Michael Bove Jr. holds an SBEE, an SM
in visual studies, and a PhD in media technology, all from the Massachusetts Institute of
Technology, where he is currently head of
the Object-Based Media Group at the
Media Laboratory and co-directs the Center
for Future Storytelling and the consumer
electronics working group CE2.0. He is the
author or co-author of over 60 journal or
conference papers on digital television systems, video processing hardware/software
design, multimedia, scene modeling, visual display technologies,
and optics. He holds patents on inventions relating to video recording,
hardcopy, interactive television, and medical imaging and has been a
member of several professional and government committees. He is
co-author with the late Stephen A. Benton of the book Holographic
Imaging (Wiley, 2008). He is on the board of editors of the Journal
of the Society of Motion Picture and Television Engineers and served
as associate editor of Optical Engineering. Bove is a fellow of SPIE.
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