7 Geometric phases in optics. The Poincaré sphere method
Tải bản đầy đủ - 0trang
173
Application of the Poincaré sphere method
(a)
(b)
N
A
N
C
A
B
O
2χ
2α
O
0°
0°
C'
S
(c)
N
S
(d)
(e)
N
N
A
B
2θ
O
2α A
0°
2χ
O
S
O
0°
2α
0°
S
S
A
Fig. 7.1: Representation of geometric phases of the Poincaré sphere.
due to different polarizations (PP1):
δ=π−
ΘABC
.
2
where ΘABC is the solid angle with vertex at the sphere center (point O ) subtended
by the spherical triangle ABC , with C being the point diametrically opposite to C .
Since δ is measured in radians and ΘABC in steradians, the formula implies a numerical equality. If the Poincaré sphere has a unit radius, the solid angle ΘABC is
numerically equal to the area of the triangle SABC . Note that δ can be evaluated using the Jones matrix method [794].
7.1.1.2 Type II Pancharatnam phase
The deﬁnition of the type II Pancharatnam phase (PP2) [665] is a lot more complicated. PP2 can only be uniquely deﬁned for cyclic evolution of polarization state; however, even in this case, PP2 does not always correspond to a real phase change of light
oscillations. Although this issue has been addressed in numerous theoretical and experimental studies (see the bibliography in [891]), no general conditions have been
174
Geometric phases in optics. The Poincaré sphere method
established under which there is a correspondence between PP2 and actual phase
changes of light oscillations. We will analyze this issue below for the general case.
Previously, PP2 was considered with regard to the so-called spiral (circular) photons [891]). Spiral photons correspond to a ﬁeld with a certain value of the spin operator (or to the so-called pure states); a photon (an elementary particle with zero rest
mass) can have only two values of the intrinsic angular momentum (spin) projection,
±1, onto the direction of its velocity [90]. However, in polarization optics, geometric
phases are calculated for an arbitrary polarization state.
In our paper [556] we considered three simple examples where a monochromatic light beam of arbitrary polarization state performs a cyclic evolution as it passes
different polarizers as well as circular and linear phase plates (see Figs. 7.1c–e, where
the arrows indicate the direction of change of the polarization state). We will not reproduce simple but cumbersome calculations for the cyclic evolution of polarization
state using Jones matrices. However, we note that the real phase always coincides
with PP2 in the only case of cyclic evolution of the polarization state as the light
beam passes a set of polarizers (see Fig. 7.1c, which shows how linearly polarized
light passes sequentially three polarizers, one circular and two linear). When an elliptically polarized light beam passes an optically active medium, a circular phase
plate with a phase shift of 2π , the real phase difference between the light traveling along the slow and fast axes of the plate equals π , regardless of the polarization state, while the magnitude of PP2 does depend on the ellipticity of the polarization state (see Fig. 7.1d, Θ = π2 − 2χ ). Quite similarly, as a linearly polarized light
beam passes a linear phase plate with a phase shift of 2π , the real phase difference between the light traveling along the slow and fast axes of the plate equals 2π ,
while PP2 depends on the linear polarization azimuth relative to the plate axes (see
Fig. 7.1e). The results of the study [556] show that if orthogonal polarizations are excited with different weights, the PP2 obtained by the Poincaré sphere method may
not correspond to the real phase obtained by the Jones matrix method. If the polarization state is such that only one of the orthogonal polarizations is excited in
the phase plate (such a polarization state is called a fundamental polarization state
of the phase plate), then the so-called dynamic phase arises (DP) [891]. A change in
the dynamic phase is not correlated with a change in the polarization state; when
a light beam in its fundamental polarization state propagates through an anisotropic
medium, its polarization state remains unchanged and the image point representing
this state on the Poincaré sphere does not move, whereas the dynamic phase changes
by ±π Δnz/λ (z is the distance that the light travels).
If both fundamental polarization modes are excited and have unequal weights,
it is generally impossible to separate PP2 and DP [891]. The attempt by Martinelli and Vavassori [592] to represent the total phase increment as the sum of PP2 and
DP proved to be unsatisfactory – by the deﬁnition of [592], DP is identically zero for
the case of cyclic evolution of the polarization state, which is not true.
Application of the Poincaré sphere method
175
To sum up, for cyclic evolution of polarization state, the type II Pancharatnam
phase is generally equal to the real phase change (which is measured from the interference between a test beam and a reference beam, whose phase and state of polarization are known) only if the light passes different sequences of polarizers of arbitrary
type. For phase plates, PP2 is generally not equal to the real phase change if both
orthogonal polarization modes are excited with different weights.
Now consider the important issue of reciprocity of the Pancharatnam phases. Of
course, PP1 principally cannot be reciprocal or nonreciprocal, since it is just an additional phase difference between two light beams, which is due to the difference
in their polarizations. As far as PP2 is concerned, its reciprocity depends on which
type of birefringence causes the cyclic evolution of the optical medium. For circular
or linear birefringence, PP2 is reciprocal. However, if circular birefringence is caused
by the Faraday effect or if linear birefringence is caused by an electromagnetic optical nonreciprocal birefringence arising in crossed electric and magnetic ﬁelds [4,
494, 651] (this issue will be considered in Section 7.4), then PP2 is also nonreciprocal,
since the nonreciprocity term Δn is added to the refractive index with different signs
for counterpropagating modes. When the phase increment is calculated by the Jones
matrix method, this issue does not arise, since the phase sign of counterpropagating
waves is considered automatically.
The above suggests that both the phase difference between two interfering beams
with different polarization and the phase difference increment due to evolution of
the polarization state is generally more reasonable to evaluate using the Jones matrix method or, equivalently, using the method of ﬁeld decomposition into geometricoptical (normal) waves [280, 281, 951]. Note that normal waves are plane, while polarization modes in single-mode ﬁbers have a nearly Gaussian cross-sectional distribution of intensity. However, in most cases (e. g., in determining the polarization state
of light and its phase in single-mode ﬁbers by the Jones matrix method), the transverse distribution of light intensity is not taken into account. Therefore, as applied to
SMFs, normal waves are nothing but mutually orthogonal polarization modes, which
are generally elliptical.
7.1.2 Birefringence in SMFs due to mechanical deformations
7.1.2.1 Kinematic phase in SMFs
First we consider the phase increment in an isotropic ﬁber. As noted above, in polarization isotropic ﬁbers, there always exits a phase shift increment proportional to
the optical path length Ln, or a kinematic phase (KP), ϕ = 2π Ln/λ, where L is
the ﬁber length, n is the effective refractive index, and λ is the light wavelength in
vacuum. The kinematic phase is independent of the type of winding of the ﬁber, provided that the deformations due to winding do not affect n. Since counterpropagating
modes in the FRI circuit travel the same optical path, the kinematic phases of counter-
176
Geometric phases in optics. The Poincaré sphere method
propagating waves are the same even if winding deformations change n. Consequently, the kinematic phase cannot affect the result of interference of counterpropagating
waves.
7.1.2.2 Bending induced linear birefringence of SMFs
In isotropic SMFs, the orthogonal polarization modes are degenerate. However, actual
SMFs have one or another kind of polarization anisotropy. As a results, the effective
refractive indices for the orthogonal modes a slightly different, ranging from 10−3
for high-linear-birefringence SMFs to 10−9–10−8 for the so-called connected SMFs,
in which linear birefringence is largely suppressed. Non-plane winding always suggests some bending of the ﬁber, which produces some additional linear birefringence
β (rad/m) [710, 716, 872, 875]:
β=
2π ρ d
2π Δn
=
λ
λ
D
2
,
where Δn is the refractive index difference between the slow and fast polarization
modes of the ﬁber, d is the ﬁber diameter, D is the winding diameter, and ρ = 0.133
for quartz SMFs. Mathematically, in the plane wave approximation, an SMF is no different from a linear phase plate. The bending-induced linear birefringence disturbs
the intrinsic (unperturbed) linear birefringence (if any) so that the total birefringence
is the vector sum of the bending-induced and intrinsic components.
7.1.2.3 Twisting-induced circular birefringence of SMFs. The spiral polarization
modes
A ﬁber can be twisted in the process of winding and so can acquire torsion deformations producing some additional circular birefringence βc ; the degree of this birefringence is expressed as [73, 715, 876]
βc = 2 (1 − g) τ .
where g is the photoelastic coefﬁcient, which equals 0.065–0.08 for quartz ﬁbers [73,
715, 876], τ is the ﬁber twist per unit length (rad/m). If the ﬁber has an intrinsic linear birefringence, its fundamental polarization modes become elliptical [572, 951].
The description of a twisted SMF in the laboratory frame of reference is rather inconvenient – even with uniform twisting, the ellipticity and azimuth of the major axis of
the natural polarization mode ellipse, treated as integral characteristics of the entire
ﬁber, are periodic functions of the ﬁber length.
This issue was resolved by Ginzburg [280] in 1944 (see also [281]:2 he introduced
the notion of a spiral polarization mode (SPM), a polarization mode in the frame of
2 The study [281] considered a linear phase plate with torsion stresses.
Application of the Poincaré sphere method
177
reference following the twisting. In this case, the polarization mode ellipticity is independent of the ﬁber length, the elliptical birefringence is βe = β2 + β2c , and the azimuth of the major axis of the polarization mode ellipse rotates with the spiral frame
of reference. In calculations, the rotation of the reference frame must be taken into
account at the output of a ﬁber segment. Ginzburg also formulated an applicability
condition for geometric optics in the case of uniform twisting [280, 281]:
2τ
β.
(7.2)
When this conditions holds, the principal states of polarization of the optical medium with unperturbed linear birefringence and torsion are weakly elliptical (close to
linear) and, hence, can be treated as spiral linear polarization modes [280, 281]. Conβ, which suggests that the twisting-induced
dition (7.2) can be rewritten as βc
circular birefringence must be much less than the intrinsic birefringence of the medium. If there is no initial linear birefringence, twisting does not produce any circular
birefringence at all [280, 281]. The study [951] formulated an adiabaticity condition
for the case of varying twisting, which states: the ellipticity of normal waves must
not change signiﬁcantly on the polarization beat length: Lb = λ/Δn. Whenever this
condition holds, one can neglect the linear interaction in describing the propagation
of normal waves. The energy exchange between two normal waves is conventionally
characterized by the polarization holding parameter h (see Chapters 3 and 4).
Using spiral polarization modes is essentially equivalent to the standard Jones
matrix method in a linear basis (reference frame) and, in some cases, can be more
advantageous. To use this approach, one should change to the elliptical basis corresponding to the mutually orthogonal SPMs [68]. In the elliptical basis, the Jones matrix of a ﬁber segment with natural SPMs will be diagonal, resulting in a considerable
simpliﬁcation of the analysis.
7.1.3 Rytov effect and the Rytov–Vladimirskii phase in SMFs and FRIs in the case of
noncoplanar winding
7.1.3.1 Rytov effect in the FRI circuit ﬁber
We now return to consider the ﬁber ring interferometer (FRI). Since the FRI circuit
consists of numerous coils of a single-mode ﬁber wound on a reel without intersections, the end and beginning of each coil are at least one cross-sectional ﬁber diameter apart. This kind of winding in noncoplanar (nonplane) and the ﬁber exhibits
the Rytov effect [737, 738], rotation of the light polarization plane. Apart from that,
the Rytov effect also involves a rotation of the transverse structure of the light beam if
any [361].
The angle of rotation of the polarization plane due to the Rytov effect will be referred to as the Rytov angle (RA). For one ﬁber coil on a cylindrical reel, the Rytov
178
Geometric phases in optics. The Poincaré sphere method
angle is expressed as [891]
γ = 2π 1 −
H
(π D)2 + H 2
= 2π (1 − cos θ) ,
(7.3)
where D is the reel diameter, H is the pitch of the winding, and θ is the angle between
the ﬁber axis and reel axis. The total Rytov angle for all coils is given by
αRyt = Nγ ,
(7.4)
where N is the number of coils.
Consider two variants of winding an SMF on the FRI circuit frame (reel).
1. Suppose an optical cable is wound on a cylindrical reel in such a way that
the principal linear birefringence axes of the single-mode ﬁber preserve their
orientation from coil to coil as shown in Fig. 7.2a. Note that this condition is automatically satisﬁed for the so-called ribbon-like single-mode ﬁber [59, 157, 802];
it is this kind of ﬁber that is shown in Fig. 7.2a. The cross-section of a ribbon-like
ﬁber is close to a rectangle and the birefringence axes are perpendicular to its
sides. In this case, the ﬁber experiences torsional twisting. The ﬁber twist per
unit length, τ , and rotation angle of the linear birefringence axes in one coil, Γ ,
are expressed as. [891]
τ=
Γ =
2π H
,
(π D)2 + H 2
2π H
(π D)2 + H 2
(7.5)
= 2π − γ .
Although the linear birefringence axes preserve their orientation from coil to coil,
the natural polarization modes of the ﬁber become elliptical due to twisting. If
the ﬁber birefringence was elliptical, this kind of winding would cause a change
in the ellipticity of the natural modes.
(a)
(b)
2
1
3
2
1
Fig. 7.2: Longitudinal section of a reel with a ribbon-like ﬁber wound on it: a, optical ﬁber (2) would
directly on a reel (1) and b, optical ﬁber (2) inside a Teﬂon tube wound on a reel (1). The arrow
indicate the directions of the linear birefringence axes.
Application of the Poincaré sphere method
2.
179
In order to detect the Rytov effect, one uses a different kind of winding. To avoid
torsion deformations in the course of a non-plane winding, the ﬁber is ﬁrst placed
inside a Teﬂon tube, which has a low friction coefﬁcient, and then wound on
a cylindrical reel [259, 260, 772, 773, 854, 886, 891]. The Teﬂon tube experiences
torsion deformations, while the ﬁber can untwist freely inside the tube so as to
remove the torsion stresses. This results in a noncoplanar coil structure without
ﬁber twists. Figure 7.2b displays the positions of the linear birefringence axes in
the cross-section of ribbon SMF enclosed in a Teﬂon tube wound on a reel. The ﬁgure graphically illustrates the physical meaning of the Rytov effect in SMFs with
no torsional twists, showing that the Rytov effect is the rotation of the ﬁber crosssection between coils (and hence, rotation of the linear birefringence axes) due to
a noncoplanar coil structure, or due to the geometric properties of the winding.
Accordingly, the Rytov effect is cross-sectional rotation of the ﬁber and, hence,
ﬁber axes in case there is no actual twisting of the light beam or the ﬁber.
It is noteworthy that the Rytov angle is independent of whether or not the ﬁber has
torsional twists and so in unrelated to the Rytov effect. Since the Rytov effect causes rotation of the polarization plane, it can formally reduced to the manifestation of
an additional optical activity, the effective circular birefringence, which equals (in
the laboratory reference frame)
βc = 2τef =
2γ
(π D)2 + H 2
,
(7.6)
where τef is the effective ﬁber twist per unit length due to the geometric characteristics of the winding. In the spiral frame of reference following the twisting [280, 572,
951], the natural (spiral) polarization modes preserve the ellipticity the ﬁber had before being wound on the cylinder. In particular, is the intrinsic (unperturbed) birefringence of the ﬁber are linear, then the spiral polarization modes will remain linear in
the spiral frame of reference. There is nothing surprising about this, since the optical
activity due to the Rytov effect is associated with the kinematic properties of parallel
transport of the Darboux trihedron along the light beam path or, which is the same,
along the ﬁber. Note that the Rytov effect is sometimes called pseudo-gyrotropic.
7.1.3.2 Rytov–Vladimirskii phase and PP2 in SMFs with noncoplanar winding
As shown above, the Rytov effect results in the appearance of an optical activity (or
a change in the optical activity) in SMFs. Therefore, as circularly polarized light propagates through a noncoplanar ﬁber, it acquires a dynamic phase, which Vinitskii et
al. [891] called the Rytov–Vladimirskii phase (RVP). If the light is linearly polarized,
it experiences an additional (with respect to the laboratory frame) change in the polarization state; in the spiral frame that follows the spatial twisting, the birefringence
remains unchanged and so the spiral polarization modes preserve their ellipticity. In
180
Geometric phases in optics. The Poincaré sphere method
turn, this change in the state of polarization produces an additional phase change,
which is actually the type II Pancharatnam phase for the case of an optically active
medium. Recall that RVP and PP2 cannot always be separated from each other.
As noted above (see also [556, 891]), for optically active media, PP2 represents
an actual phase change of light only if both circular polarized modes have equal
weights, which is when the incident light is linearly polarized. Consequently, the PP2
due to the Rytov effect corresponds to an actual phase change in light oscillations in
this case alone.
Whether RVP should be classiﬁed as a geometric of dynamic phase is the matter of deﬁnition. RVP is a consequence of a purely geometric phenomenon, the Rytov
effect, which causes an optical activity of geometric origin. In this sense, RVP is a geometric phase. However, if RVP is treated as the result of an optical activity, without
looking into what causes it, the RVP can be classiﬁed as a dynamic phase.
If the ﬁber does not possess a magnetic activity or a natural optical activity [893]
(circular birefringence unrelated to torsional twisting is not an optical activity) and
linear birefringence simultaneously, RVP is reciprocal, since an optical activity corresponding to the Rytov effect is also reciprocal. Otherwise, as shown in [951], the RVP
due to the Rytov effect is nonreciprocal. It is clear that the PP2 due to the Rytov effect
is also nonreciprocal in this case.
Note that Kocharovskii [393] was the ﬁrst to consider the speciﬁc features of
the polarization state due to the Rytov effect as light propagates along a spiral ray.
The inﬂuence of twisting on the appearance of a geometric phase in microwave ﬁbers
was considered by Rivlin [724].
7.1.3.3 Rytov phase detection in FRIs
Frins and Dultz [259] considered an FRI with a weakly anisotropic SMF circuit, did not
have a polarizer, and used a monochromatic light source with a circular state of polarization. The ﬁber had a helicoidal winding and the detected the shift of the interference fringes from counterpropagating waves at the FRI output as the pitch of the helix
was changed. After the input splitter, one of the counterpropagating modes propagated along the right-hand circular polarization axis of birefringence and the other,
along the left-hand circular polarization axis. At the output, the two waves became
equally polarized and interfered with each other. Since the circular birefringence arising in the ﬁber was due to the Rytov effect and the input light was circularly polarized, the phase shift of the counterpropagating waves was equal to double the Rytov–
Vladimirskii phase, because one wave acquired RVP with the plus sign and the other,
with the minus sign. As a result, the FRI conﬁguration described in [259] made it possible to observe the polarization nonreciprocity due to the Rytov effect. In this case,
the polarization nonreciprocity was caused by dual linking and, hence, was PN2.
Senthilkumaran et al. [772, 773] considered FRIs with a weakly anisotropic ﬁber
circuit and a monochromatic or nonmonochromatic light source. The total light inten-
Application of the Poincaré sphere method
181
sity, Itot , was detected at the FRI output for different pitches of the helicoidal winding
of the ﬁber. Changes in the Rytov angle arising as the pitch was changed resulted in
the appearance of both PN1 and PN2 and the contributions of the two effects were
difﬁcult to separate. Since, as shown above, nonreciprocal effects caused by dual
linking (PN2) do not occur in FRIs with a nonmonochromatic light source, the experimental results of [773] are much easier to interpret as compared with [772], because they allow one to exclude the changes in PN2 and RVP, as the pitch of the winding was changed, the from consideration. The analysis performed in [556] shows that
the study [773] experimentally observed the Rytov effect, which resulted in changes
in the azimuth of the ﬁber axes at the circuit output as the pitch of the helicoidal
winding was changed.
To sum up, the study [259] observed RVP in FRIs, while [772, 773] observed the Rytov effect in FRIs. Note that Frins and Dultz [259] explain their experimental results
correctly, but call RVP the Berry phase.
The main conclusion that can be drawn from the section is that in experiments
with FRIs where the pitch of the helical winding of the ﬁber is changed, one essentially detects changes in the Rytov effect if a nonmonochromatic source is used and,
in addition, changes in the Rytov–Vladimirskii phase if a monochromatic source is
used.
The results of Section 7.2 can be summarized as follows.
1. It has been shown that the Rytov effect (rotation of the polarization plane due to
free noncoplanar winding of the ﬁber) causes the natural polarization modes of
the SMF to become polarization modes but does not lead to additional elliptical
birefringence. The Rytov effect in SMFs is the manifestation of an optical activity
that does not change birefringence in the spiral frame of reference.
2. The Rytov–Vladimirskii phase has an opto-geometric origin and so is a geometric phase; at the same time, it can be treated as a manifestation of the dynamic
phase, since it is deﬁned for a circular state of polarization in the case that the optical activity is due to the Rytov effect.
3. In optics, geometric phases, which are represented by points on the Poincaré
sphere, do not generally allow the calculation of actual phase changes corresponding to changes in the polarization state in a ﬁber segment of FRI. For example, the type II Pancharatnam phase does not correspond to the actual phase
even in the case of cyclic evolution of the polarization state in a medium with
arbitrary type of birefringence; in particular, this is true if the two natural polarization modes are excited with unequal weights.
4. The reason why geometric phases in polarization optics do not always correspond
to actual changes in the light phase is that these phases arise when light propagates through an anisotropic optical medium and cannot always be separated
from the usual kinematic an dynamic phases. This is in contrast to geometric
phases in classical mechanics, where translation of a rigid body can always be
separated from its rotation and conical motion.
182
5.
Geometric phases in optics. The Poincaré sphere method
PP2 and RVP in media that do not have a magnetic activity or natural optical activity and linear birefringence simultaneously are reciprocal. These phases can
be observed in ﬁber segments as well as Michelson and Mach–Zehnder ﬁber interferometers. In FRIs with a monochromatic light source, these phases can be
observed only if there is dual linking and so are manifestations of PN2.
7.2 Polarization nonreciprocity in FRIs. Nonreciprocal geometric
phase of counterpropagating waves
In FRIs with a nonmonochromatic light source, the condition L
ldep is met practically always. Therefore, in the absence of random inhomogeneities in the SMF,
the counterpropagating waves that have traveled along different birefringence axes
of the SMF can only interfere pairwise. Accordingly, one can consider two output
independent interference patterns and, hence, two independent and generally different nonreciprocal phase shifts between counterpropagating modes due to PN1 (see
Chapter 5):
+
−∗
Im Ex,y
Ex,y
(x,y)
ϕnon = arctan
(7.7)
+
−∗ .
Re Ex,y
Ex,y
Let us show that the Poincaré sphere method allows one to determine the geometric phase corresponding to the nonreciprocal phase difference between counterpropagating modes for different FRI conﬁgurations with different orientation of
the birefringence axes of the ﬁber at the input end of the circuit.
Figure 7.3 displays four conﬁgurations: (i) an FRI without a polarizer (diagram 1),
(ii) the most common minimum conﬁguration with one polarizer between the beam
splitters (diagram 2), which was suggested by Ulrich [871], (iii) an FRI with two polarizers at the ends of the circuit (diagram 3), which was used in [289, 470, 761], and
(iv) an FRI with two polarizers where one is located after the light source and the other
placed before the photodetector (diagram 4), which was used in [873].
First we consider the FRI conﬁguration without a polarizer (Fig. 7.3, diagram 1).
Here and henceforth, we assume the ﬁber segment between the splitters to be short
enough so that the phase delays caused by its birefringence can be neglected. Then, if
the natural modes of the ﬁber are linearly polarized, then, as follows from the results
of Section 5.1 (see also [492]), the nonreciprocal phase difference between counterpropagating waves is given by (5.8). We will conﬁne ourselves to the case of adiabatic
evolution of the polarization state in the circuit. The adiabaticity condition can be
formulated as follows: the change of the orientation of the principal birefringence axes in the circuit ﬁber on the polarization beat length is much less than 90◦ . In other
words, this condition suggests that the rotation of the birefringence axes practically
does not affect the degree of birefringence and ellipticity of the ﬁber’s natural modes.
If the adiabaticity condition holds, then, as shown in [891, 951], it sufﬁces to know only the initial and ﬁnal orientations of the birefringence axes to calculate the geometric
183
Polarization nonreciprocity in FRIs. Nonreciprocal geometric phase
diagram 1
diagram 2
3
1
2
3
1
2
4
diagram 4
5
2
2
4
diagram 3
1
5
2
3
3
1
2
5
2
2
5
5
4
4
Fig. 7.3: FRI conﬁgurations with one or two polarizers placed at different locations (1, light source,
2, beam splitters, 3, circuit, 4, photodetector, and 5, polarizer).
phase, with the intermediate orientations being irrelevant. Adiabaticity conditions for
single-mode ﬁbers in the most general case were formulated in [951]. In what follows,
we assume the adiabaticity condition to hold.
As shown above, the geometric phase can be conveniently represented as the evolution of the polarization state on the Poincaré sphere – numerically, the geometric
phase equals the solid angle subtended by the closed curve corresponding to the evolution of the polarization state. If the Poincaré sphere has unit radius, the solid angle
is numerically equal to the spherical area enclosed by the curve. Of course, this only
applies to problems where the initial and ﬁnal polarization states coincide. In FRIs,
this condition is generally not satisﬁed; furthermore, since there are many polarization beats ﬁtting in the circuit length, the image point runs around the closed curve,
representing polarization beats, many times. If, in addition, we take into account that
the polarization beat length of nonmonochromatic light, Lb = 2π Δn/λ, depends of
the wavelength, we realize that the problem of determining the geometric phase corresponding to the nonreciprocal phase shift between counterpropagating modes is
practically unsolvable.
Nevertheless, the solution of the problem can be simpliﬁed by taking into account the fact, noted above, that in FRIs with a nonmonochromatic source, the waves
traveling along the slow and fast birefringence axes are incoherent if the condition
L
ldep holds. Moreover, the polarization state evolution on the Poincaré sphere
can be treated individually for either natural polarization mode of the circuit ﬁber,
or separately for the fast and slow birefringence axes. It is noteworthy that the orien-