A trilayer pellicle that consists of a high-index center layer that is symmetrically coated on both sides by a low-index film can be designed to produce differential reflection and transmission phase shifts of ∓90° at oblique incidence and equal throughput for the p and the s polarizations. Such a device splits a beam of incident linearly polarized light into two orthogonal circularly polarized components that travel in well-separated angular directions. Examples of infrared dual quarter-wave retarders that use a symmetrically coated Ge pellicle at 77° angle of incidence are presented. A 50–50% splitter requires a symmetric pellicle with at least five layers. Error analysis shows that the thicknesses of the high-index layers must be tightly controlled. These circular polarization beam splitters are intended for operation with a well-collimated light source and can be used as the basis of a novel circular polarization Michelson interferometer

Azzam, Rasheed M.A.

Mahmoud, Fadi A.

English

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University of New Orleans 
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Electrical Engineering Faculty Publications Department of Electrical Engineering 
1-1-2002 
Symmetrically coated pellicle beam splitters for dual quarter-wave 
retardation in reflection and transmission 
Rasheed M.A. Azzam 
University of New Orleans, razzam@uno.edu 
Fadi A. Mahmoud 
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 Part of the Electrical and Electronics Commons 
Recommended Citation 
Rasheed M. A. Azzam and Fadi A. Mahmoud, "Symmetrically Coated Pellicle Beam Splitters for Dual 
Quarter-Wave Retardation in Reflection and Transmission," Appl. Opt. 41, 235-238 (2002) 
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Symmetrically coated pellicle beam splitters for dual
quarter-wave retardation in reflection and transmission
Rasheed M. A. Azzam and Fadi A. Mahmoud
A trilayer pellicle that consists of a high-index center layer that is symmetrically coated on both sides by
a low-index film can be designed to produce differential reflection and transmission phase shifts of 90°
at oblique incidence and equal throughput for the p and the s polarizations. Such a device splits a beam
of incident linearly polarized light into two orthogonal circularly polarized components that travel in
well-separated angular directions. Examples of infrared dual quarter-wave retarders that use a sym-
metrically coated Ge pellicle at 77° angle of incidence are presented. A 50–50% splitter requires a
symmetric pellicle with at least five layers. Error analysis shows that the thicknesses of the high-index
layers must be tightly controlled. These circular polarization beam splitters are intended for operation
with a well-collimated light source and can be used as the basis of a novel circular polarization Michelson
interferometer. © 2002 Optical Society of America
OCIS codes: 120.2130, 260.5430, 230.5440.
1. Introduction
Quarter-wave retarders are widely used in optics for
polarization analysis and control and as linear-to-
circular or circular-to-linear polarization transform-
ers.1 They are usually designed by use of natural or
induced linear birefringence, total internal reflection,
or interference in thin films.
Coherent multiple-beam interference in a uniform,
tilted, thin dielectric slab pellicle produces a differ-
ential transmission phase shift between the p and s
linear polarizations parallel and perpendicular to
the plane of incidence, respectively or a net retar-
dance t 90°. Therefore a tilted pellicle of a homo-
geneous optically isotropic material acts as the
simplest possible fractional-wave less than quarter-
wave retardation plate.2 Light interference in a
suitably designed tilted bilayer membrane realizes a
quarter-wave retardation QWR in transmission
t  90° at a high incidence angle.3 In general,
these pellicle retarders have unequal throughput for
the p and s polarizations, hence they exhibit diat-
tenuation.4
In this paper we show that dual QWR in transmis-
sion and reflection t  90° and r  90° can be
produced, without diattenuation, by use of a trilayer
pellicle that consists of a high-index center layer that
is symmetrically coated on both sides with a low-
index thin film. Therefore a simple circular polar-
ization beam splitter is realized, for the first time to
our knowledge, by use of conventional nongyro-
tropic thin-film optics. Earlier circular polarization
beam splitters are based on reflection by chiral me-
dia5 or diffraction by liquid-crystal gratings.6
2. Design Procedure
Changes in the state of polarization of light upon
transmission reflection by a trilayer pellicle, Fig. 1,
are determined by the ratios of complex-amplitude
transmission reflection coefficients for the p and the
s polarizations:
	  TpTs tan 
t exp jt, (1)
  RpRs tan 
r exp jr. (2)
The transmission and reflection coefficients of a mul-
tilayer system are determined by use of the scatter-
ing matrix method.7 For a monochromatic light
beam of a given wavelength , which is incident on
R. M. A. Azzam razzam@uno.edu is with the University of New
Orleans, Department of Electrical Engineering, Lakefront, New
Orleans, Louisiana 70148. F. A. Mahmoud is with Adaptec, In-
corporated, 691 South Milpitas Boulevard, Milpitas, California
95035.
Received 21 May 2001.
0003-693502010235-04$15.000
© 2002 Optical Society of America
1 January 2002  Vol. 41, No. 1  APPLIED OPTICS 235
the pellicle at an angle of incidence , the ellipsomet-
ric functions7 of Eqs. 1 and 2 can be put in the form
	  f , n1, n2, 1, 2, (3)
  g, n1, n2, 1, 2. (4)
We assume a symmetrically coated pellicle, Fig. 1,
with identical outer layers 1 and 3 of the same low
refractive index n3  n1 and the same normalized
thickness 3  1 and a center layer 2 with a differ-
ent high refractive index n2 and normalized thick-
ness 2. The immersion medium 0 on both sides of
the pellicle is assumed to be air or vacuum with re-
fractive index 1. All media are assumed to be ho-
mogeneous, optically isotropic, and separated by
parallel-plane boundaries. The normalized thick-
ness i of the ith film is defined by
i  di/Di, i  1, 2, 3, (5)
where
Di  2ni2 sin2 12, i  1, 2, 3 (6)
is the angle-of-incidence-dependent film thickness
period. To achieve pure QWR in transmission, the
following two conditions must be satisfied simulta-
neously:
	  TpTs  1, (7)
arg	  t  90°. (8)
For an all-transparent pellicle,
  RpRs  1 (9)
is satisfied simultaneously with Eq. 7. Equations
7 and 9 indicate the absence of diattenuation in
transmission and reflection. Finally, to obtain QWR
in reflection, we should also have
arg  r  90°. (10)
For given film refractive indices n1, n2 and a given
angle of incidence , the normalized film thicknesses
1, 2 are iterated on to satisfy Eqs. 7–10 individ-
ually or simultaneously. To achieve QWR at the
lowest possible angle of incidence, by use of a sym-
metric trilayer pellicle in air, the refractive-index
contrast between the center and the outer layers
should be sufficiently high. In Section 3 we consider
a pellicle that consists of a high-index center layer of
Ge n2  4 that is symmetrically coated by a low-
index fluoride8 thin film n1  1.35 for incident
infrared radiation.
3. Symmetric Trilayer Pellicle as Dual Transmission
and Reflection Quarter-Wave Retardation at 75° Angle
of Incidence
As a specific example, we take n1, n2, n3  1.35, 4,
1.35, which corresponds to a symmetrically coated
pellicle, with a Ge center layer and fluoride outer
coating at 10.6-m CO2-laser wavelength and  
75° angle of incidence. This operating angle is 7°
lower than that at which the bilayer membrane of
Ref. 2 functions as a QWR and leads to an angular
separation of 30° between the reflected and the trans-
mitted beams.
Figure 2 shows the loci of multiple solutions 1, 2
of Eq. 8 for transmission quarter-wave retarders for
both t  90° and t  90° as represented by the
two small closed contours.
Superimposed on Fig. 2 are the corresponding so-
lution loci of Eq. 7, 	  1, for the same trilayer.
The two solution loci do not intersect, hence Eqs. 7
Fig. 1. Reflection and transmission of light by a tilted pellicle that
consists of three optically isotropic layers 1, 2, and 3with uniform
thicknesses d1, d2, and d3. The pellicle is immersed in a trans-
parent ambient medium 0. The linear polarizations p and s are
parallel and perpendicular to the plane of incidence, respectively,
and  is the angle of incidence.
Fig. 2. Loci of multiple solutions 1, 2 of Eq. 8 for symmetric
trilayer transmission quarter-wave retarders for both t  90°
and t  90° are presented by the closed contours. Superim-
posed are the corresponding solution loci for Eq. 7, 	  1, for a
coated Ge trilayer with indices 1.35, 4, 1.35 at   75° angle of
incidence. Note that the two solution loci do not intersect.
236 APPLIED OPTICS  Vol. 41, No. 1  1 January 2002
and 8 cannot be satisfied simultaneously. Conse-
quently, it is not possible to achieve QWR without
diattenuation at   75°. The azimuth of incident
linearly polarized light can be adjusted to circularly
polarize one of the two split beams at a time, but not
both at the same time.
To quote a specific design, a trilayer with 1, 2 
0.2430, 0.7066 and least metric thicknesses d1, d2
 1365.5, 964.7 nm functions as QWR in both re-
flection and transmission with opposite signs of re-
tardance at   75°. The intensity transmittances
for the p and s polarizations are 39.13% and 18.01%,
respectively. Incident linearly polarized light with
an azimuth of 55.84°, measured from the p direction,
produces a circularly polarized transmitted beam.
In general, we find that, for symmetric pellicles, if
QWR is achieved in transmission, QWR is also ob-
tained simultaneously in reflection, with 90° retar-
dance in the two split beams. However, the reverse
of this statement is not true, i.e., QWR in reflection
does not guarantee QWR in transmission. This im-
portant conclusion for which an analytical proof may
be possible extends to symmetric multilayer pellicles
with any odd number of layers.
To satisfy Eqs. 7 and 8 simultaneously, and to
achieve orthogonal circular polarizations in the two
split beams at the same time, the angle of incidence
 must be increased to just above 76°. In Section 4
we present a design that operates at   77°.
4. Symmetric Trilayer Pellicle as Dual Quarter-Wave
Retardation without Diattenuation at 77° Angle of
Incidence
We assume the same material system that we con-
sidered in Section 3. The solution loci of Eqs. 7 and
8 for the normalized film thickness 1, 2 appear in
Fig. 3 at   77°. Figure 4 shows the corresponding
solution loci for Eqs. 9 and 10.
Figures 3 and 4 confirm that Eqs. 7 and 9 have
the same solution, as one expects for a transparent
system. Figure 4 shows that the solution loci for Eq.
10 consist of two closed contours that exactly coin-
cide with the corresponding closed contours in Fig. 3,
except that when t  90°, r  90°, and vice
versa. Figure 4 indicates two additional solution
branches for Eq. 10 for which QWR is achieved only
in reflection but not in transmission.
The intersection points x, y, u, and v in Figs. 3 and
4 represent solutions for which QWR is achieved si-
multaneously in transmission and reflection without
diattenuation.
We further consider the design marked x in Figs. 3
and 4. The normalized and least metric film thick-
nesses for this symmetric trilayer are 0.225855,
0.803674 and 1281.0, 1097.9 nm, respectively.
The transmission and reflection differential phase
shifts are t  90° and r  90°, respectively.
The polarization-independent same for p and s
transmittance and reflectance are 37.78% and
62.22%, respectively. It is not possible to achieve
dual QWR with equal split fractions by use of a
trilayer pellicle in air. In Section 5 we briefly dis-
cuss a 50–50% design using a symmetric five-layer
pellicle.
It is important to specify the sensitivity of this
design to small deviations of incidence angle, wave-
length, and film thicknesses from their design values.
To maintain the phase error in reflection and trans-
mission to within1°,  and must be kept to within
0.2° and 40 nm of their design values, respec-
tively. This indicates that this circular polarization
beam splitter is suited mainly for well-collimated la-
ser radiation. The associated tolerances for the
thicknesses of layers 1, 2, and 3, for the same 1°
phase error, are given by 4%, 0.5%, and 4%, respec-
tively. Thus stringent thickness control is required
mainly for the middle high-index layer.
5. Symmetric, Five-Layer, 50–50% Pellicle Beam
Splitter with Dual Quarter-Wave Retardation without
Diattenuation at 78° Angle of Incidence
Assume that we wish to design a circular polarization
pellicle beam splitter immersed in air that satisfies
all the following requirements simultaneously:
1. dual quarter-wave retardations with opposite
signs in reflection and transmission,
2. equal throughput for the p and the s polariza-
tions i.e., no diattenuation, and
3. 50–50% split ratio.
Fig. 3. Loci of multiple solutions 1, 2 of Eq. 8 for symmetric
trilayer transmission quarter-wave retarders for both t  90°
and t  90° are presented by the closed contours. Superim-
posed are the corresponding solution loci for Eq. 7, 	  1, for a
coated Ge trilayer with indices 1.35, 4, 1.35 at   77° angle of
incidence. The intersection points x, y, u, and v represent trilayer
pellicles that function as dual QWR in transmission and reflection
without diattenuation.
1 January 2002  Vol. 41, No. 1  APPLIED OPTICS 237
To satisfy all these conditions by numerical experi-
mentation, we find that a symmetric pellicle with at
least five layers is required. The parameters of one
such design are as follows:
• incidence angle   78°, wavelength   10.6
m;
• film refractive indices n1  n3  n5  1.35 and
n2  n4  4;
• normalized film thicknesses 1  5  0.022556,
3  0.36082, and 2  4  0.932137;
• metric film thicknesses d1  d5  128.5 nm,
d3  2055.3 nm, and d2  d4  1273.7 nm;
• differential reflection and transmission phase
shifts t  90.0002° and r  89.9998°;
• amplitude transmittance and reflectance ratios
	  0.999998 and   1.000002; and
• intensity reflectance and transmittances of
50.24% and 49.76%, respectively.
Of the three conditions cited above, the first two are
almost exactly satisfied, and the third is met to within
0.24%. However, the main problem with this design
is that it is sensitive to small thickness errors of the
even-numbered high-index layers. For example, a
1% error of d2 causes a phase error of approximately
10°. This makes this design impractical. To over-
come this remaining problem, it appears that the mul-
tilayer pellicle design would probably have to be
abandoned.
6. Conclusion
Pellicles are traditionally used as light, yet quite
sturdy, beam splitters.9 In this paper we have
shown that, by appropriate design, a trilayer pellicle
that consists of a high-index Ge center layer that is
symmetrically coated with a low-index fluoride thin
film can function as dual QWR in reflection and
transmission simultaneously without introducing
any diattenuation. To achieve a 50–50% split ratio,
a symmetric alternating high-index and low-index
stack of at least five layers is required. Such a de-
vice functions as a circular polarization beam splitter
for incident linearly polarized light, which can be
used as the key element of a circular polarization
Michelson interferometer.10
References
1. See, for example, W. A. Schucliff, Polarized Light Harvard
University, Cambridge, Mass., 1962.
2. D. A. Holmes, “Wave optics theory of rotatory compensators,”
J. Opt. Soc. Am. 54, 1340–1347 1964.
3. R. M. A. Azzam and F. A. Mahmoud, “Tilted bilayer mem-
branes as simple transmission quarter-wave plates,” J. Opt.
Soc. Am. A 18, 421–425 2001.
4. R. A. Chipman, “Polarization analysis of optical systems,” Opt.
Eng. 28, 90–99 1989.
5. S. D. Jacobs, K. A. Cerqua, K. L. Marshall, A. Schmid, M. J.
Guardalben, and K. J. Skerrett, “Liquid-crystal laser optics:
design, fabrication, and performance,” J. Opt. Soc. Am. B 5,
1962–1975 1988.
6. J. A. Davis, J. Adachi, C. R. Fernandez-Pousa, and I. Moreno,
“Polarization beam splitters using diffraction gratings,” Opt.
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7. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polar-
ized Light North-Holland, Amsterdam, 1987, Chap. 4.
8. E. Ritter, “Optical film materials and their applications,” Appl.
Opt. 15, 2318–2327 1976.
9. J. A. Dobrowolski, “Optical properties of films and coatings,”
Handbook of Optics, M. Bass, ed. McGraw-Hill, New York,
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10. R. M. A. Azzam and F. A. Mahmoud, “Circular polarization
Michelson interferometer,” presented at the 1999 Annual
Meeting of the Optical Society of America, Santa Clara, Calif.,
26–30 Sept. 1999, paper TuXX66.
Fig. 4. Loci of multiple solutions 1, 2 of Eq. 10 for symmetric
trilayer reflection quarter-wave retarders for both r  90° and
r  90°. There are four solution branches; the two branches
represented by the closed contours coincide with the closed con-
tours for transmission QWR in Fig. 3. Superimposed are the
corresponding solution loci for Eq. 9,   1, for the same coated
Ge trilayer with indices 1.35, 4, 1.35 at  77° angle of incidence.
The intersection points x, y, u, and v represent trilayer pellicles
that function as dual QWR in transmission and reflection without
diattenuation.
238 APPLIED OPTICS  Vol. 41, No. 1  1 January 2002


Symmetrically coated pellicle beam splitters for dual quarter-wave retardation in reflection and transmission

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Symmetrically coated pellicle beam splitters for dual quarter-wave retardation in reflection and transmission

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