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7 Geometric phases in optics. The Poincaré sphere method

7 Geometric phases in optics. The Poincaré sphere method

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Application of the Poincaré sphere method























2α A







Fig. 7.1: Representation of geometric phases of the Poincaré sphere.

due to different polarizations (PP1):





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]. 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


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


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 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-


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. 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]:


2π ρ d

2π Δn







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. Twisting-induced circular birefringence of SMFs. The spiral polarization


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


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]:



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 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


Geometric phases in optics. The Poincaré sphere method

angle is expressed as [891]

γ = 2π 1 −


(π D)2 + H 2

= 2π (1 − cos θ) ,


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 ,


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]


Γ =

2π H


(π D)2 + H 2

2π H

(π D)2 + H 2


= 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.








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



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 =

(π D)2 + H 2



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. 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


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]. 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


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


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.



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



ϕnon = arctan



−∗ .

Re 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


Polarization nonreciprocity in FRIs. Nonreciprocal geometric phase

diagram 1

diagram 2








diagram 4





diagram 3















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


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-

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7 Geometric phases in optics. The Poincaré sphere method

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