The diffraction efficiency of interferometrically formed holographic lenses is influenced by the recording geometry and properties of the recording material. Variations in efficiency increase when attempting to make high numerical aperture elements. The factors which influence the diffraction efficiency of high numerical aperture holographic lenses are examined

Kostuk, Raymond K.

English

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Appendix L -" _ 211
N9 2,'-1 1 4
Practical Design Considerations and Performance Charac
of High Numerical Aperture Holographic Lenses
Raymond K. !(ostuk
teristics
.J
2.:ectricai and Computer Engineering Department. and Optical
.'/ciences Center, University of Arizona Tacson, AZ. 85721. US._ , ;.'_.,
1(¢ .... " "'"
ABSTRACT i} ' "
The diffraction etficiency of interferome_rically formed holographic lenses is influenced
by the recording geometry and properties of the recording materiaL. Variations in efficiency
increase wi_en attempting to make high numerical aperture elements. In this presentation
the factors which influence the diffraction efficiency of high numerical aperture holographic
lenses are examined.
1. Introduction
Many factors influence the diffraction efficiency of holographic lenses. These include
changes in _he visibility of interfering fields across the hologram aperture due to intensity
and polarization variations, changes in the average refractive index and thickness of the
recording material between exposure and processing steps, and variation of the interbeam
angle over the aperture of the hologram. Considering high efficiency phase gratings, these
effects physically alter the refractive index modulation at different locations across tile
aperture.
in order to analyze the influence of these factors, high numerical aperture (0.54 N.A.)
kolographic objectives were formed in bleached silver halide emulsions using a reversal
bleach process. The total efficiency of this lens was 43% compared to a planar grating
etficiency of 47!2 formed with two collimated beams and similar geometrical conditions.
The e/ficiencies of higher diffraction orders for both s- and p- polarized reconstruction
beams are mapped across the aperture, and related to the interbeam angles of the con-
struction beams at different locations. Although several evaluations of high N.A. lenses
have previously been made _-a, these investigations only considered a section of the lens
containing the grating vector (K) which was illuminated with light polarized perpendicular
to this plane. Tb.is paper provides experimental results for the efficiency of the grating
with different K vector and reconstruction field polarization orientations.
'2. Characterization of Holograms formed in Bleached Silver Halide Emulsions
itolograms for t_his evaluation were formed in Agfa 8E75HD silver haIide emulsions ex-
posed with 632.8 nm illumination, and processed with an Itford reversal chemistry (SP678C
deveLoper/SP679C bleach). This material was chosen because its properties b.ave been welt
characterized, and because pracessing is relativeb' simple and provides consistent results.
OF PO0_ Qbbx"_
https://ntrs.nasa.gov/search.jsp?R=19920005696 2020-03-17T14:13:40+00:00Z
212 appendix L
Although the efficiency is iess than that obtained for dichromated gelatin and some pho-
topol3"mers, many of the same factors affecting silver haiide holograms wiiI also appear in
gratings formed in other materials.
The e_ciencv a.s a function of exposure for a series of unslanted and slanted planar
gratings processed with the reversal chemistry are simwn in Figures 1 and 2. IThe diffrac-
tion eificiency for this analysis is equal to the power in a particular order divided by the
incident power, t An interbeam angle of 40 ° was used for both cases, and the slantedgrat-
ing formed with one Ueam at normal incidence to the emulsion and the second at 40 ° to
the normal in air. (This geometry corresponds to the interfering rays at the center of the
focusing holographic lens.) Nla.ximum efficiency for the unslanted grating occurs when the
hologram is illuminated at the construction angle. However, for the slanted grating, the
maximum diffraction efficiency occurs at a different angle from that used during construc-
tion. Figure 2 shows that their is about a 10% change in absolute diffraction efficiency
,.0_0 relative) near the optimum exposure for this process. This results from changes inO Ow
the emuision _hickness which effectively rotates the grating plane, and a change in the av-
erage refractive index further detunes the grating from peak efficiency at the construction
angle. A reversal bleach removes the exposed silver halide crystals which were converted
to silver during developmen_ a. This mechanism reduces both :he emulsion thickness and
average refractive index producing a drop in efficiency. This same mechanism however,
also reduces the negative effects of scatter and noise gratings, and gives reasonably high
diffraction e_ciency. Since the thickness and average refractive index change could be
quantified, the reversal bte_ch process was used for this evaluation. Measurement of an
emulsion exposed with 115 _J/crn = showed an average refractive index change from 1.64
_,o 1.60, and a thickness reduction from 5.0 #m to 4.5 ;am.
3. Factors Affecting the Efficiency of Focusing HOEs
Consider the construction geometry for a holographic lens shown in Fig. 3. In this
arrangement an on-axis spherical beam interferes with an off-axis collimated reference
beam. The potarization of the reference field is along the y-axis. The spherical beam
is formed by focusing collimated beam with its field polarized along the y-axis. The
polarization of this beam ham a different orientation for each ray illuminating the aperture.
Therefore, as the NA of the element increases there will be a larger difference between
the polarization vectors of the spherical and reference beams. High numerical aperture
elements accentuate this difference and make it necessary to consider vector effects both
during construction and reconstrucion.
The configuration of Figure 3 provides a relatively large interbeam angle over much
of the grating aperture. This reduces the grating period, increases the overall efficiency,
and tends to equalize the efnciency of s- and p- diffracted light s . Another advantage of
this arrangement is that non-diffracted light in the zero order does not overlap with the
focusing beam during reconstruction which would reduce the signal-to-noise level in the
region of focus. _n addition, the off-axis geometry can also be used to help circularize the
reconstruction beam.
Appendix L 2 13
[n order to analyze the varying efficiencyof this hologram a localized planar grating
approximation is made at discrete points across the aperture '_. At the iens center the
corresr_onding planar grating has an interbeam angle of 40 _ with both fields poiarized in
the y _ii',ection. Plots of the experimental diffraction efficiency vs. reconstruction angle
!or tire ¢entrai region of a 0.54 NA focusing element and for a hologram, formed witi_
two coil!mated fields are illustrated in Figures 4 and 5 respectively. These holograms
were m.ade using :he same exposing and processing conditions and show reasonabiv good
correspondence in maximum efficiency, however :here is an angular displacement indicating
that tl_e probe beam diameter(lmm i may have exceeded the limit for the locai plane gra:ing
appromma_ioa.
The e_ciency of a volume hologram depends on the visibility of the interfering fie[&
and the exposure ievet at the film plane. "When the emulsion response is linear, this
dependence can be approximated: by the relation
77= SE,-._,"
where _ is a film sensitivity factor, Eo is _.he average film exposure, and V is the visibility
of the interfering fields within tb.e emulsion. The film sensitivity factor will depend on
the particular type of emulsion and processing chemistry. The exposure will vary across
the aperture of _he focusing element due to changes in the path lengths between the
center and edge of the aperture for the expanding spherical beam, and changes in the
fresnei coefficients. These differences affect the beam ratio (R) which in turn influence the
visibility since
_Rl,_-cos(fl)
(I,R)
with gl the relativeorientation of the polarizations of the interferingfields.
Combining tl_ese factors and calculating _he ratio of the efficiency at the edge of the
hologram aperture relative to the center shows (Fig.6) that the expected fall-off for a 0.55
NA clement, illuminated with an s-polarized reconstruction beam is about 15%, and 25,°,o
with a p-polarized beam.
As stated earlier, these calculations assume that rl varies linearly with exposure. This
is a good assumption for dichtomated gelatin and many photopolymers, however materials
such as silver halide tend to saturate after reaching a maximum value (Figure 1). This
can be used to advantage by exposing the emulsion beyond the linear range of the film.
Since the efficiency does not change rapidly with exposure in this region, variations of 77
across the aperture can be reduced. (It is assumed that nonlinearities in the refractive
index modulation were small since measurements showed that very little power went into
higher diffraction orders.)
For a highly linear responding material the slope of the efficiency vs exposure can be
reduced by using a thinner emulsion. Exposing the emulsion at level slightly above the
first 77 maximum will then keep the total efficiency high over a relatively large exposure
range.
ORK_b,i._/..;' . :9
OF POO!l QUALi'Tv
2 14 Append/x L
Other factors which influence the diffraction efficiency of high numerical aperture HOEs
are differences in emulsion thickness and average refractive index between the exposure and
pos_ process phase of hologram recording. The effects of these variations on the diffraction
,:if.cienc,/for the slanted planar grating described earlier corresponding to the local grating
at the cea_er of :he focusing HOE were determined using coupled wave anaI.vsis s and are
iilus:rated in Figures 7 and 8. As indicated a change in emulsion thickness produces
significant displacement and reduction in the eHciency, however a change in the average
refractive ip,ciex of 0.06 results in only a small displacement of the curve (i.e. 2°). Values for
the average refractive index and thickness change for emulsions processed with the reversal
i_leach presented in the previous section indicate that the change in average refractive index
is not significant, however the thickness change will produce a major shift in the et_ciency
performance.
The numerical values in Figure 3 show the normalized coordinates in the x direction
and the corresponding interbeam angles _,nter for a 0.55 NA lens. These angles can
then be used to compute the appropriate grating vector and diffraction efficiency for local
planar gratings. For emulsions on the order of 5 _m thick, average refractive index of !.6,
and index modulation of 0.05, the volume grating conditions will not be satisfied across
the aperture. This will give rise to higher d =.faction orders which extract power from the
desired order.
4. Fabrication and Evaluation of High N.A. Holographic Lenses
Holographic lenses were formed with a spherical beam produced with a 0.55 N.A.
long working distance microscope objective and a collimated reference beam at 40 ° to
the surface normal of the emulsion. A beam ratio of 1 at the center of the exposed area
was obtained by placing a mask with a small diameter aperture in the film plane and
/ Othen measuring the power in each beam. The film was then exposed with 115 ,z J/crn" of
622.8nm illumination from a HeNe laser as mentioned in Section 1, and processed with
a standard l[ford reversal chemistry. This exposure level was beyond the linear range of
_he film/process combination (Figure 1), and was expected to improve uniformity in the
efficiency over the aperture of the HOg..
After processing the hologram was illuminated with the conjugate of the planar refer-
ence beam. A mask with a 1 mm aperture was mounted on an x-y translational stage to
probe the efficiency at different locations in the aperture. Both s- and p- polarized light
was used to illuminate discrete positions along the x- and y-axes of the hologram. The
results of these measurements are shown in Figures 9-12. The smallest interbeam angle
exists at the x = -t.0 positions on the x-axis efficiency plots. The +1 and -1 diffraction
orders are equal at this coordinate indicating that the grating acts like a thin sinusoidal
grating. There are several weaker diffraction orders (Ro, T(-2), and R(-1) ) not shown on
these figures for clarity which account for an additional 15°70 of the incident illumination at
the x = -t position. In each case the efficiency of the primary order also decreases near the
edge of the aperture with the largest interbeam angle. It is not exactly clear if this drop is
a result of a decrease in exposure or is due to an overcoupling effect. From Figures 1 and 2
it can be seen that a slanted grating does not saturate in the same manner as an unslanted
Appendix L 215
one. Tl,is e:_ect "._::d !;e accentuated near the edge of the hoiogram aperture where the
lnterber-nt angie in air is close to 73 ° Given tr, ese iimitations however, the totals- and p-
etficie;tcies over tire complete aperture were 43.1 and 4[.7c._ respectively. A corresponding
planar gratinzZ formed "with a normally inciden: beam and a second beam at 40: to the
emulsion normal itad an s-polarized reconstruction beam emciency of 47_. Using this az a
reference for the maximum erSciency obtainable with this film and process chemistry, the
focusing element efficiency is approximately 92c,_ of the possible planar grating erEciency.
._cknowledgements
The author wishes to acknowledge support for this work from IBNI, the University of
Arizona Optical Data Storage Center, and NSF contract =- ECS-8910067.
5. References
1. C. Kojima, K. Miyahara, K. Hazegawa, T. Otobe, and H. Ooki, "In-Line Holographic
Lenses of I{igil Numerical Aperture," Japan. J. Appl. Phys., 20, Supplement 20-1, 199-
°-04 (1980).
2. R.II..t. Svms and L. Solymar, "Higher diffraction orders in on-axis holographic lenses,"
Appl. Opt., 21, 3263-3268 (1982).
3. K.P_..-k. Syms and L. Solymar,"ExperimentaI and theoretical evaluation or" tile efficiency
o[" alloil'-axis volume holographic lens," App[. Phys. B 32, 165-173 (1983).
4. G. Saxby, Practical Holography, Prentice Hall, p.73, (1988).
5. ,%.I3. O'ColLnor and R..K. Kostuk, "Polarization properties of high numerical aperture
holographic objectives," OSA Topical Meeting on Optical Data Storage, Los Angeles, CA.
17-19 Jan. 1989, Technical Digest p.95-97.
6. R..R.A. Syms and L. Solymar,"Localized one-dimensional theory for volume holograms,"
Opt. Quant. Electron., Vo[. 13,415-419 (1981).
7. L.H. Lin, "hlethod of characterizing hologram-recording materials," J.Opt. Soc. Am.,
Vol. 6t, 203-208 (1971).
8. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J.,
"/oI. ,iS, 2909-2947 (1969).
216 Appendix L
o.6-! ,
0.5
i
0.4
0.3
0,2 i
0.1
0 20 40 60 80 1O0 120 140
Exposure (uJ/cm2)
Figure i. Diffraction efficiency vs. exposure for Agfa 8E75HD emulsions processed with
Ilford SP678C developer and SP679C reversal bleach. Grating planes are normal to the
emulsion surface, and was formed with two coiiimzted beams having a 40 ° interbeam angle
in air.
0.6
0.58(
0.56-_
+
O.54
0.52
uJ 0.5
mr
0.46-
0.42:0"44"_, { • On-Sragg + Max DE t
0.4
50 60 70 80 90 100 110 120
Eo(uJ/cm2)
Figure z. Diffraction efficiency vs. exposure for a planar grating formed in Agf_ 8E75HD
emulsions processed with Iiford SP678C developer and SP679C bleach. The slanted grating
was formed with one beam normal to the emulsion surface and the second at 40 ° to the
normal in air. Tile upper curve shows maximum efficiency obtained by rotating the grating,
and the second is reconstructed at the formation angle.
Append_ L 217
A
7
- . - !
< ,";/
j,,t.
/
- ':,o0 -T'
-0.5 !_
-0.5 ....
-0,25 70.9°
0.00 ..:.0.0_
0.25 "9.I °
0.50 ., ._
0.75 65._
1.00 2 "
Figure :_. Construction geometry for a high N.
and an off-axis collimated reference wave. Also
air) which exist along the x-axis.
06
A. lens formed with an on-axis spherical
shown are different n:erbeam angles (in
0.5-
04
0.2-
I
0.11
0 1
0
Figure t.
/'N
/ _ -- ÷1
i ..... •
10 20
Diffraction efficiency vs.
focusin_ HOE.
......... " 1
7
/
f
/
/
/'
/
¢
30 40 50 60
Reconstruction Angle
reconstruction angle at the center of _lle 0.5.t N.A.
Oi',.,_._i ,._._ PAGE IS
OF POOR QUALITY
218 Appendix L
0,6,
t.tJ
0,5_
0.4-_
0,1!
./"
iX_ lw+l .........
i .... -1, i
_° _ .,.% •,
10 20 30
Reconstruction Angle
t
0
0 40 50 60
Figure 5. Diffraction efficiency vs. reconstruction angle for a planar grating corresponding
to the local planar grating at the center of the 0.54 N.A. focusing HOE.
1 i
0.80"9 _- .... ..._
0,7-t
o.6-i
LU
rm 0.5
0.4-
0.3
0;
0.1
0
0.1
Figure 6.
_ s-Pol. _ p-Pol. I1
0.2 013 0.4 os ols 0
NA
Drop in diffraction efficiency with N.A. resulting from exposure variation, po-
lariza_ion mismatch, and different Fresnel coefficients across the aperture. A linear depen-
dence on exposure and visibility is assumed.
Appendix L 219
0.6_
uA
0.5-_
0.4-_
i
i
0.3-_
i
i
0.2-_
0.II
O_
15 _0 25 30 35 40 45 50 55 6[
Reconstruction Angle
Figure ":. Diffraction efficiency vs. reconstruction angle for a planar grating formed and
reconstructed with the same average refractive index, and with a change of 0.06 in index.
0.6
tJJ
C_
0.5 _,
0.4H
0.31
0.2-_
[
h
0.1_
OI
15 20 25 30 35 40 45 50 55 60
Reconstruction Angle
Figure 8. Diffraction efficiency vs. reconstruction angle for a planar grating formed and
reconstructed with the same average refractive index, and with a change of 0.06 in index.
ORi,_INAL PAGE IS
OF POOR QUALITY
220 Appendix L
0.5 ;; +________
0.45-_ /_/
0.4",
/
0.35. /
/
/ ..I=
/0.3-_ /
/
ILl
0.25 JL--.--_.
0,2 r__ r
0.15
/"I \
0.1 1 ..- \ l
, ./ \ I " TO + T(-I) _ T(+I) ,ii
0.05-4 • ,,
-1.0
II
-0,8 -0.6 -0.4 -0.2 -0.0 0.2 0.4. 0.6 0.8 .0
Normalized Aperture Coorclinate
Figure 9. Measured diffraction efficiency across the x-axis of a 0.54 N.A. HOE recon-
structed with s-polarized light.
LIJ
,51 L
°"1 / /
0"11 A \ _ TO * t(-1) "* T(+I) '
oo:y ,...7 ,
-1.0 -0,8 -0.6
_: K ; :$7 , ;i( L"
-0.4 -0.2 -0.0 012 0.4 0.6 0.8 1.0
Normalized Aperture Coordinate
Figure 10. Measured diffraction efficiency across the x-axis of a 0.54 N.A. HOE recon-
structed with p-polarized light.
Append/x L 22 l
0.5
0.45-_
0.4"_
,/
/
•'_ -4- 4-
\,,.
\
i
i
W 0.35-_
0.3]
0.25-_
0.2_
• To '" T(-1) :
t
-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
Normalizea Aperture Coorainate
Figure 11. Nfeasured diffraction efEciency across the y-axis of a 0.54 N.A. HOE recon-
structed with s-polarized light.
0.5,
0.48-_
0.46-_
0.444
b
0.42 _,
I
w 0.4-_
0.38-1
0.36.
0.34i
+
/
0.32-1
0.31
-1.0 -0.8 -0.6
4-,
P
I ,_, TO ÷ T(-1) I[
-0.4 -0'.2 -0.0 0.2 014
Normalized Aperture Coordinate
F
0.6 0.8 1.0
Figure 12. Measured diffraction efficiency across the y-axis of a 0.54 N.A. HOE recon-
structed with s-polarized light.
222 Appendix L


Practical design considerations and performance characteristics of high numerical aperture holographic lenses

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