Download DISPERSIVE BIREFRINGENT FILTERS Pochi YEH 1. Introduction

OPTICS COMMUNICATIONS
Volume 37, number 3
1 May 1981
DISPERSIVE BIREFRINGENT FILTERS
Pochi YEH
Rockwell International
Science Center, Thousand Oaks, CA 91360,
USA
Received.1 December 1980
Polarization interference filters that employ crystals with a strong dispersion of birefringence as the wave plates can
accommodate a very wide field-of-view while at the same time maintaining a very narrow bandwidth. It is noted, in particular, that CdS exhibits a strong dispersion in the spectral regime between 5200 A and 5400 A, with a rate of dispersion
that could provide a passband of only 1 A with a filter structure of several millimeters thick. This paper investigates the
properties of these filters. The results are presented and discussed.
1. Introduction
Birefringent filters play an important role in many
optical systems whenever filters of extremely narrow
,bandwidth with wide angular fields or with tuning
capability are required. For example, in the area of
solar physics the distribution of hydrogen may be
measured by photographing the solar corona in the
light of the H, (X = 6563 A) line. In view of the large
amount of light present in neighboring wavelengths,
a filter of extremely narrow bandwidth (“1 A) is required if reasonable discrimination is to be attained.
In the area of laser communication when the signal
is transmitted through a random medium such as an
atmosphere or sea water, the received signal is carried
by a scattered component of the laser radiation which
will appear to come from a wide field-of-view, often
up to several steradians. Optical communications under
these severe circumstances requires a filter with not
only an extremely narrow bandwidth to reject the
unwanted background light and, hence, increase the
signal to noise ratio, but also a large angular aperture
in order to receive as much signal as possible. These
filters are composed of birefringent crystal plates
(wave plates) and polarizers. Two basic versions of the
birefringent filters had been invented. They are the
Lyot-&man filters [l-4] and the sole filters [5-71.
These filters are based on the interference of polarized
light. This polarization interference requires a phase
retardation between the components of the light
polarized parallel to the fast and slow axes of the crystal when radiation passes through it. Since the phase
retardation introduced by a wave plate is proportional
to the birefringence of the crystal. It is desirable to
have crystals with large birefringence (n, -. no) for
filter construction. Currently, the most commonly used
materials are quartz, calcite and ADP. The field-of-view
of these basic birefringent filters is too small for many
applications. By using the wide field elements designed
by Lyot [2] or the split element developed by Evans
[4], the field-of-view can be improved enormously.
However, a subangstrom birefringent filter made of
these materials (quartz, calcite and ADP) requires very
thick crystal plates and yet only provides a moderate
field of view.
This paper investigates the properties of dispersive
birefringent filters that are made of materials with a
strong dispersion in birefringence. Many crystals are
found to exhibit anomalous dispersion of birefringence
in the spectral regimes near the absorption band edges.
These include cadmium sulfide (5 150-5400 A) [8],
magnesium fluoride (1150-l 500 A) [9] and sapphire
(1430 - 1530 A) [9]. In fact, there is even a change
of sign of the birefringence (zero-crossing) after passing
through zero, in the case of magnesium fluoride and
cadmium sulfide. The operation of birefringent filters
based on the zero-crossing of birefringence was first
suggested by Yeh [lo]. This filter can accommodate a
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very wide field-of-view (-2n) while at the same time
maintaining an extremely narrow passband. The widefield arises from the vanishing of the birefringence
while the narrow bandwidth is a result of the strong
dispersion of the birefringence. A birefringent filter
can also operate near the isotropic point of certain unaxial crystals when the birefringence is small and yet
exhibits strong dispersion. It was first suggested by
Chandrasekharan and Damany [9] that the operation
of Lyot filters using magnesium fluoride near its isotropic points (1194 A) can provide an extremely
narrow bandwidth because of the sharp dispersion of
its birefringence. We will show in this paper that the
sharp dispersion and the smallness of the birefringence
also give rise to an enormous field-of-view for the dispersive birefringent filter.
2. Principle of operation
2.1. Differential phase retard0 tion
The phase retardation r introduced by a uniaxial
crystal plate of thickness d when radiation passes
through it is given by
r = (2n/A) and,
(3)
where Q!is the rate of dispersion of the birefringence
and is defined as
154
(5)
The last equality is true provided CY
% b/h. We find
that for birefringent crystals with a strong dispersion
(large o) the bandwidth is inversely proportional to
the product of the dispersion rate cuand the plate thickness d. The separation between the.peaks is given by
(2)
Maximum transmission occurs at the wavelengths
when the phase retardation is an integral multiple of
2n, i.e., r = 2Mn. The transmission characteristics are
governed by the variation of the phase retardation r
with respect to the wavelength h. This variation is
described by the following differential
(Y= (ah/ax)
The variation of phase retardation I’ with respect to
wavelength h (see eq. (3)) consists of two parts. First,
the variation of r with respect to h can be due to the
factor l/X in (1). This variation is represented by the
second term -A@ in the parenthesis in (3). Second,
the variation of r with respect to X can also be resulted
from the dispersion of the birefringence. The latter
contribution to the differential phase retardation is
described by the first term OLin the parenthesis in (3).
For a conventional wave plate such as quartz at 5000
A, the variation of phase retardation with respect to
wavelength is dominated by the term &/A (-1.8
X 10m6 A-l) because of the small dispersion (o - 1.6
X 10m7 ,8-l). For the case of a cadmium sulfide plate
at 5300 A, Anh is only 1.l X 10m6 A-l whereas (Yis
0.7 X 10e4 A-l. Therefore, the variation of phase
retardation r of a cadmium sulfide plate with respect
to the wavelength X is dominated by the rapid variation
of An with respect to h, while the factor l/h plays
only a very minor role. Thus, the transmission of a
relatively thin plate of those anomalous dispersive
birefringent crystals will have a very narrow bandwidth.
From (l), (2) and (3) the full width at half maximum
Ah,,, (FWHM) for each peak (I’ = 2/l&) is given by
(1)
where An = n, - no is the birefringence of the crystal
at wavelength h n, and no are the refractive indices
of the crystal. The transmission of polarized light
through such a plate when sandwiched between
parallel polarizers is given by
T= cos=Y/2 .
1 May 1981
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(4)
x
AX=d(n-An/A)~d
h
’
This separation Ah is called the free spectral range.
2.2. Lyo t-&man filters [ 141
Consider the Lyot&nan
filter which consists of
a set of dispersive birefringent plates separated by
parallel polarizers. The plate thicknesses are in geometric
progression, i.e., d, 2d, 4d, ...2N-1d. The transmission
of this filter for polarized light is given by
2
N
T=,!
COS=
[2’-l r/2]=
,
GE;:,
(7)
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1 May 1981
OPTICS COMMUNICATIONS
where r is the phase retardation of the thinnest plate
given by eq. (1).
From (7) it is seen that the transmission spectrum
is similar to that of a grating with 2N lines. The bandwidth AXyZ (FWHM) of the system is governed by
that of the band from the thickest plate and is given
by *
(i.e., I’ = 2Mn). The transmission is therefore also described by (11) and (12) except that AI’ is given by
AQz. = 0.886 x
= 0.886 & ,
(8)
2Nod
where N is the number of stages and D is the thickness
of the thickest plate. The free spectral range (FSR)
of this filter is determined by the thickness of the
thinnest plate and is given by
AXV2= 0.80 X/Nod
Ah = X1o.d.
(9)
The finesse of this filter, defined as the ratio of the
free spectral range to the bandwidth, is.thus given by
F= 1.13 X y
(10)
Ar=r
--mff
(fan type).
According to (1 l), the half-maximum transmission occurs at AT’= 40.87r/N. The full bandwidth at halfmaximum (FWHM) for each peak is therefore given by
(11)
where
x2 = 1 + (NAr/n)2 ,
(12)
with
I)rf
(16)
The finesse of this filter is given by
Consider a folded type Sole filter which consists
of N identical plates of dispersive birefringent materials placed between a pair of crossed polarizers. The
azimuth angles of the plates are in a prescribed manner,
i.e., p, --p, p, -p, ... (-lyY_‘p with p = 45’/N.
Maximum transmission occurs when the retardation
F of each plate is an odd integral multiple of n (i.e.,
r = (2M t 1)~). The transmission of such a filter structure for polarized light in the spectral regime around
r=(2M+ 1)n is [ll]
Ar=r-(2Mt
(15)
where again we assume that Q 9 An/h.
Notice that the bandwidth is inversely proportional to the total thickness of the crystal plates (Nd)
and is also inversely proportional to cy.The free spectral range (FSR) of filter structures is determined by
the thickness of the plate and is obtained by finding
the wavelength at which the plate becomes a fullwave plate (I’ = Z&r) for the fan type and a half-wave
plate (I’ = (2iV t 1)n) for the folded type filters. The
result is
Ah= X/cud.
2.3. &Zc filters [S-7]
(14)
(folded type).
(17)
3. Spectral selectivity
An important feature of this dispersive birefringent
filter is its spectral selectivity. Unlike the conventional
birefringent filters, this type of filter has a bandwidth
AhV2 which is determined by the product of the total
thickness and the dispersion rate Q. Many materials
exhibit anomalous birefringence dispersion with a dispersion rate of the order of 10m4 A-l (e.g., CdS,
AgGaS,, MgF,, etc.). Thus, crystals like CdS or
AgGaS, with a total thickness of 2 mm would have
a bandwidth of 2 A at h - 5000 A according to (8)
and (15). A conventional wide field Lyot 1 filter with
the same bandwidth would require a thickest plate of
about 8 cm.
(13)
The transmission characteristics of a fan type Sole
filter is identical to that of the folded type except
that maximum transmission occurs when the phase
retardation of each plate is an integral multiple of 2n
* The factor 0.886 is due to the presence of other stages.
F= 1.25 N
4. Field-of-view
Another important feature of this new type of fiter is its large field-of-view. This wide field-of-view
arises from the operation of this filter in the spectral
regime near the zero-crossing point where the bire155
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fringence & is small and yet the dispersion rate a! is
large.
To study the angular dependence of the transmission characteristics, we assume that the c-axis of the
plates are parallel to the surfaces of the plates. Let 0
be the angle of incidence in air and 9 be the angle
between the plane of incidence and the c-axis of a
typical plate. The phase retardation of the plate for
light at this direction of incidence (0, $) is given by
F-21
r=(kez-k,)d,
with
(18)
koz= (2n/h)(nz - sin2fI)V2,
(19)
= (2n/X)[ni - sin20 sin24
k,
- (n,2/n,2)sin20 cos2#] lj2,
(20)
where d is the thickness of the plate and k,, , k, are
the normal components of the wave vectors of the
ordinary and extraordinary waves, respectively. For
light with non-normal angle of incidence, the phase
retardation r is no longer given by the simple expression (1). It is a function of both 0 yd 4. If the angle
of incidence is such that sin28 4 n , the phase retardation can be given approximately by
I’ = (2n/X)(n, - n,)d
X[l-$(l-‘+sin2#)].
0
In the spectral regimes around the zero-crossing point
where the birefringence is small (i.e., no w n,), the
phase retardation becomes
r=2&pn
x
1 -sin20
2
II
(1 - 2 sin2#)] ,
(22)
where n is the averaged value of no and n,. Since the
phase retardation I’ of each plate is a function of both
0 and 4, the transmission maximum of a filter is expected to be dependent upon the angle of incidence
(0, 4). For the case of Lyot-ohman birefringent
filters, the spectral shift in wavelength is given by
Ah=
an
2&X - Anjh)
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1 May 1981
OPTICS COMMUNICATIONS
sin2e(1
- 2 sin2@).
(23)
In a conventional Lyot-Ohman filter made of crystals
such as quartz, the dispersion Q!is small and the spectral shift is
Ax = (X/2.n2) sin20 (1 - 2 sin2$)
(conventional).
(24)
In a dispersive Lyot-Ohman filter made of crystals
such as CdS in the spectral regime where (Yis large
(i.e., a % Arr/X), the spectral shift (23) becomes
Ah = (&/2n20)
sin28 (1 - 2 sin2$)
(dispersive).
(25)
Notice that the spectral shift for the dispersive birefringent filter is small because of the small birefringence and the strong dispersion (large CY).As a result,
the angular field-of-view of a dispersive birefringent
filter is expected to be very large compared with the
conventional birefringent filter. If the field-of-view is
defined as the angle of incidence 0 when the spectral
shift is one half of the bandwidth, then according to
(25) the field-of-view of a dispersive Lyot-Ohman birefringent filter is
e = ti(c~A$,~/L\n)~~,
(26)
where Ahy2 is the bandwidth given by (8). A conventional Lyot-Ohman filter would have a field-of-view
of only 8 = ti(Ahl12/X) ‘I’. For a 1 .&dispersive LyotOhman filter made of CdS at X = 5300 A, the field-ofview is approximately * 16’. A conventional LyotOhman filter made of quartz crystal would have a field.
of-view of only +2’. Notice that, according to (26)
the field-of-view is arbitrarily large for the passband at
the isotropic point when An = 0.
A further increase in the field-of-view can be obtained by using the wide-field eleT;nts [l-4]. An increase by a factor of (2n/&)
in the field-of-view is
obtained in both the conventional and the dispersive
Lyot 1 filter. This field-of-view is thus given by
e = ~~(2nl~)u2(,AXy2/a”)112
(27)
for a Lyot 1 dispersive birefringent filter. Table 1 compares the field-of-view and plate thickness of a dispersive birefringent falter with those of a conventional
birefringent filter.
For the case of Sole filters, the c-axis of each plate
is oriented at a prescribed angle. Therefore the phase
retardation I’ (2 1) for off-axis light is different for
Volume 37, number 3
1 May 1981
OPTICS COMMUNICATIONS
Table 1
Field-of-view and plate thickness of birefringent filters made of CdS and quartz at 5300 A
Quartz
CdS
AAll
= 1
A
simple
Lyot-Ohman
filter
wide-angle
Lyot 1
filter
simple
Lyot-Ohman
filter
wide-angle
Lyot 1
filter
0.34
0.34
13.8
13.8
e
16”
90”
1.2”
22.7”
D
3.4
3.4
138
138
5.2”
90”
0.4”
7.2”
D
(cm)
Ahq2
=
0.1 A
(cm)
8
D = Thickness of thickest plate. B = Field-of-view.
each plate in a Sole filter. The net result is that the
spectral shift in wavelength is expected to be a little
bit smaller than that of the Lyot-Ohman filter (eq.
(23)), especially the fan type Sole fnter. The freld-ofview of a dispersive Sole birefringent filter is therefore
also approximately given by (26) for a simple Sole
filter structure and by (27) for a Sole filter with wideangle elements.
5. Conclusions and discussions
T = f [exp(-a&/2) exp(-iI’/2)
In conclusion, we have proposed and analyzed a
new type of birefringent filter which consists of
crystals with a strong dispersion in birefringence. The
result shows that such a filter can accommodate a
very wide field-of-view while at the same time maintaining an extremely narrow bandwidth. The large
field-of-view arises from the operation of this filter
near the isotropic point where the birefringence is
small and yet the dispersion is strong. Many materials
exhibit anomalous dispersion (large o) near the absorption band edges. This offers a possibility of constructing narrowband filters (Axyi < 1 A) using a
relatively thin filter structure. The maximum tolerable
variation in thickness is approximately given by
Ad = M cuAQ2/2&
tolerable variation in thickness turns out to be of the
order of few microns both in the case of the thicker
quartz and the relatively thinner CdS filter. Therefore,
the dispersive birefringent filters (e.g., CdS) are expected to be less sensitive to temperature variations.
Anisotropic absorptions of light are found in these
anomalous dispersive materials. This will degrade the
polarization interference in these filters. As a result,
the transmission minima will no longer be zero. In fact,
eq. (2) is replaced by
(2%
for the dispersive birefringent filters, while M =
ti(AXu2/M) for the conventional birefringent fnters.
For the bandwidth of (AXu2 < 1 A), the maximum
+ exp(-cw,Z/z) exp(iIYz)] 2,
(29)
where 1 is the thickness of a plate and o,, , al are the
absorption coefficients (cm-‘) for light with electric
field parallel and perpendicular to the c-axis respectively. The fringe visibility of such a plate defined as the
ratio of minimum transmission to maximum transmission is given by
Tmh
-=
Tmax
exp(-a$/2)
- exp(-ollZ/2) 2
( exp(-a,,Z/2) + exp(-cu,l/2) 1 ’
(30)
For a 2 mm CdS plate with (Y,,= 0.5 cm-’ and Ok=
1.63 cm-’ [ 131, this ratio is 4 X 10m3. In a LyotOhman filter a factor of 0.4 is multiplied with this
ratio due to the presence of other stages. It turns out
that the anisotropic absorption is not a serious problem.
The problem of optical inhomogeneity is also expected to be small because the spectral shift due to the
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Volume 37, number 3
variation of the birefringence
of the sharp birefringence.
OPTICS COMMUNICATIONS
is very small as a result
References
[ 1]
[2]
[3]
[4]
[S]
158
B. Lyot, Compt. Rend. Acad. Sci. 197 (1933) 1593.
B. Lyot, Ann. Astrophys. 7 (1944) 31.
Y. Ohman, Nature 141 (1938) 157.
J. Evans, J. Opt. Sot. Am. 39 (1949) 229.
I. Sole, Cesk. Casopis 3 (1953) 366; 9 (1959) 237; 10
(1960) 16.
1 May 1981
[6] I. Sole, J. Opt. Sot. Am. 55 (1965) 261.
[7] J. Evans, J. Opt. Sot. Am. 48 (1958) 142.
[S] H. Cobrecht and A. Bartschat, Z. fur Physik 156 (1969)
131.
[9] V. Chandrasekharan and H. Damany, Appl. Optics 8
(1969) 671.
[lo] P. Yeh, Optics Comm. 35 (1980) 15.
[ 111 P. Yeh, Optics Comm. 29 (1979) 1.
[12] P. Yeh, J. Opt. Sot. Am. 69 (1979) 742.
[13] D. Dutton, Phys. Rev. 112 (1958) 785.