388

Research Article

Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America A

Wave description of geometric phase
Luis Garza-Soto,1 Nathan Hagen,1, *Dorilian Lopez-Mago,2

1

AND Yukitoshi Otani

1

Department of Optical Engineering, Utsunomiya University, 7-2-1 Yoto, Utsunomiya, Tochigi 321-8585, Japan
Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., 64849, Mexico*Corresponding author: nh@hagenlab.org
2

Received 14 November 2022; revised 3 January 2023; accepted 3 January 2023; posted 4 January 2023; published 1 February 2023

Since Pancharatnam’s 1956 discovery of optical geometric phase and Berry’s 1984 discovery of geometric phase inquantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches
using the Jones calculus, or on spherical trigonometry approaches using the Poincaré sphere. The abstracted
mathematics of the former and the abstracted geometry of the latter obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves andthe resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how
geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by aninterferogram, and provide the conditions under which the two match one another. © 2023 Optica Publishing Grouphttps://doi.org/10.1364/JOSAA.480814

1. INTRODUCTION
As we have come to learn more about geometric phase over the
past 38 years [1,2], we have acquired the ability to calculate it
for an increasing array of circumstances. Yet, even now we lack aphysical model for visualizing geometric phase. For polarization
optics, the Poincaré sphere has been a widely used tool to visualize the various calculations and to describe how the geometricphase relates to transformations of polarization states, but its
use opens up even more questions [3]. For example, when using
the Poincaré sphere to calculate the geometric phase, the typicaldefinition states that when a state of polarization (SOP) undergoes a series of transformations, the various transformations
induce a phase delay γ to the wave, equal to half the solid angle
subtended on the surface of the sphere: γ = /2. However,when drawing the curves used to delineate the spherical area,
researchers often ignore the fact that the actual physical pathtaken by the polarization state is not that of the curves they draw.
Thus, while the calculation proceeds correctly, the underlying
physics that drives the calculation methods remains unclear.
The following presents a list of some of the curious rules
under which the Poincaré sphere areas must be calculated toobtain correct estimates of the geometric phase.
1. Pancharatnam’s original work never considered a cycle ofstates, but rather considered only two input states A andB, and the state generated by their sum, C = A + B [1].
States A and B, together with C 0 (the antipode to C ), forma triangle of points on the Poincaré sphere from which one
can calculate the subtended solid angle,  [4].
2. The subtended angle formed by the actual physical path ofthe polarization state is generally not the correct solid angle
1084-7529/23/020388-09 Journal © 2023 Optica Publishing Group needed to get the correct geometric phase. Rather, for thecorrect , one must use the shortest geodesic arc connecting the pair of SOPs before and after each homogeneouspolarization element [3,5]. In addition, if the physical pathof the polarization state is a geodesic, such as when using
a half-wave plate (HWP), then one must use the geodesic
that coincides with the physical path.
3. One of the consequences of separating the physical path ofthe polarization state from the geodesic path of calculationis that it becomes difficult to determine the value of thegeometric phase inside optical elements such as a linear
retarder. Rather, analyses are generally limited to calculating γ before entering or after leaving an element, but
not the continuous changes that occur while propagating
through it.
4. The areas on the Poincaré sphere are actually signed areas—
negative if clockwise, positive if anticlockwise, and that if apath crosses itself, then one can have positive and negative
areas partially cancelling one another [6–8].
5. While the spherical angle is usually defined using a closed
cycle of polarization states, [9,10] a cycle of states can neverin practice be exactly closed, and it is necessary to have aprocedure for calculating the geometric phase that allows
for a set of states that are only approximately closed, or not
closed at all [11]. This procedure is as follows: the phaseof non-closed loop of states A → B → C is equal to thePancharatnam–Berry phase related to the closed loop
A → B → C → A, plus the phase of the Pancharatnam
connection related to the projection of A → C [7].
6. Under some conditions, the geometric phase can undergo a
π shift singularity [5,11–13].

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Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America A

While the above rules are useful guides to obtaining thecorrect result, the abstraction of the spherical geometry leaves
little insight into the physical behavior guiding these rules,
and can even hinder understanding. Definitions found in theexisting literature, such as “Geometric phase is a consequence ofparallel transport in a curved topology” [14] or similar abstract
treatments [15], are not exactly wrong, but they can be considered misleading, in that much simpler processes are actually atwork. In the discussion below, we show that simply analyzing
the locations of wave maxima, without the abstraction of the
Poincaré sphere or of matrix algebra, generates the geometricphase properties while also allowing one to visualize the underlying physics. Moreover, in contrast to existing methods, our wave
model of the geometric phase provides a clear means of defining
the geometric phase at any point in its propagation through an
optical system—not only at points before and after traversing ahomogeneous optical element, but inside the element as well.
Section 2 starts with the simplest possible case—the superposition of waves in 1D—and introduces a reference planefrom which we can define geometric phase. When two waves
are superposed, the location of the wave amplitude peak of theirsum depends on both the relative phases of the input waves
and on their relative amplitudes. This change in position fromthe input reference plane to output state peak location is thegeometric phase.
Section 3 generalizes this superposition analysis to 2D, andshows how we can define the phase of an arbitrarily polarizedwave with respect to the two orthogonally polarized waves thatcompose it. This section also introduces a visualization aid forlocating the polarized wave peaks.
In Section 4, we provide an example of propagating polarizedwaves through a linear retarder. Although previous authors haveclaimed that single linear retarders cannot introduce geometricphase, [6] more recent work has argued that indeed they can,[16] and our work as well shows that such retarders do in generalcause a global phase shift.
Finally, we note that the above definitions of geometric phase
involve propagating waves, and so these are theoretical constructs that are not directly measurable. Section 6 introduces
a model for interfering a sample wave with a reference wave,
generating a stationary interferogram that is measurable. The
resulting interferogram includes the geometric phase γ previously obtained for propagating waves. However, one finds that
if the two arms of the interferometer deliver waves of different
polarization states, then the interferogram phase will differ fromthe wave geometric phase. This is similar to the requirement
sometimes stated in the historical literature that the geometricphase is defined only for “closed loop” transformations of thepolarization state.
2. WAVE COMPOSITION IN 1D
It is a well-known property of sinusoidal waves that the sumof any two waves is also a sinusoid, with an amplitude andphase that depends on the states of the input waves. For two
co-propagating electromagnetic waves E 1 and E 2 of arbitrary amplitude and phase E 1 = A 1 cos(kz − ωt − φ1 )and E 2 = A 2 cos(kz − ωt − φ2 ), their sum is given by the
harmonic addition theorem as [17,18]

E 1 + E 2 = A 3 cos(kz − ωt − γ ),

389

(1)

where k = 2π/λ is the wavenumber and ω the angularfrequency, and where
A 23 = A 21 + A 22 + 2A 1 A 2 cos(φ2 − φ1 ),tan γ =

A 1 sin φ1 + A 2 sin φ2
A 1 cos φ1 + A 2 cos φ2

(2)

(3)

give the resultant amplitude A 3 and phase γ . This can also be
generalized to the case of adding N arbitrary waves, as shownin Appendix A. Note that by using cosines to represent our two
waves, we have implicitly chosen the wave peaks to indicate
the phase origin (φ = 0) position. A phase advance is given by
φ > 0, and a phase delay by φ < 0.
If we choose to locate our reference plane halfway between theE 1 and E 2 peaks, the phase difference δ = φ2 − φ1 between thetwo waves is split in half on each side of the reference plane, suchthat φ1 = δ/2 and φ2 = −δ/2. In this case, the phase γ of thesum wave simplifies totan γ = tan(δ/2)

A1 − A2.
A1 + A2(4)

As indicated by our choice of variable, γ is the geometric phasefor this superposition [19]. If the amplitudes of the two input
waves are equal (A 1 = A 2 = A), using the product identity forcosines with Eq. (2) obtains
E 3 = 2A cos(δ/2) cos(kz − ωt).

(5)

Thus, when the two input wave amplitudes are equal, the sum
wave amplitude becomes A 3 = 2A cos(δ/2), and the sum wave
phase is exactly at the midpoint between the phases of the input
waves (γ = 0). The value of γ indicates the location of the sum
wave peak relative to the reference plane. If the two input wave
amplitudes are not equal (A 1 6 = A 2 ), then Eq. (4) indicatesthat the phase of the sum wave shifts towards the phase of the
input wave with greater amplitude. This situation is illustrated
in Fig. 1, where we see that the phase of the green sum wave
moves closer to the blue wave phase due to the latter’s higher
amplitude.
In Eq. (4), one might be concerned about the case whenA 1 = −A 2 , in which case the denominator in the equation
becomes zero. However, our choice of the phase delay between
the two component waves allows the negative sign to be replacedwith an extra phase delay of δ → δ + π , while maintaining
positive amplitudes. This avoids possible division by zero.
If the phase separation δ between the two input waves isgreater than π , then an ambiguity arises. Since we cannot generally measure the absolute phase of a wave, but can measure only
phase differences between waves, when δ > π, the nearest E 2
peak (red wave) with respect to the E 1 (blue wave) peak will nolonger be to the left of the origin, but rather to the right. In ourconvention, we always choose the nearest pair of peaks, so that δ
for E 3 (green wave) can never exceed π . As we will see below, thischange from using the left peak to the right peak corresponds
to behavior discussed in the existing literature on the geometricphase, in which the geometric phase measured in interferograms
is wrapped within a range of −π to +π .

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Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America Aq

A 2x cos2 (kz − ωt − φx ) + A 2y cos2 (kz − ωt − φ y ).
(8)
To locate the z position at which the electric field magnitude ismaximum, we take the derivative with respect to z (at constantt) and search for the location where the derivative is zero. Aftersome algebraic work, the derivative becomes|E | =

d |E (z)|= −k A 2x sin[2(kz − ωt − φx )]dz− k A 2y sin[2(kz − ωt − φ y )].
Fig. 1. Superposition of two waves with amplitudes A 1 = 1 (blue)
and A 2 = 0.5 (red) and a phase difference δ = 3π/4 between them.
The peak location γ of the resultant wave (green) shifts towards
the wave with greater amplitude (blue). The phase of each wave
(φi = ±3π/8) is defined with respect to a reference position (z = 0here) given by the midpoint between the two component wave peaks.
The phase γ = 38.8◦ of the sum wave’s peak is given by Eq. (4). (See
Visualization 1 for an illustration of wave composition for a range ofphase delay values..)

3. WAVE COMPOSITION IN 2D
Unlike the 1D case, we cannot simply add orthogonally vibrating waves and look for the resulting peak. Instead, we form thepolarization ellipse traced by the electric vector produced byadding the two orthogonal waves. The point at which the electric vector aligns with the major axis of the ellipse is designatedto be the location of the 2D wave peak—its phase. Generalizing
our approach to plane waves in 2D involves the same basic
procedure as in 1D, but we will see that we must work with thesquares of the electric fields rather than the fields themselves.
To start, we use the midpoint between the two orthogonalcomponent waves’ peaks to define a reference plane, and define
the geometric phase as the phase difference of the ellipticalwave’s sum peak from the reference plane. While this is easy to
do when working with linear polarization states, waves in 2D
are elliptically polarized in general—their instantaneous electricfield vectors have an orientation that varies with time. Somepolarization components are more conveniently represented
using a basis of two elliptical states, but if we want to define a“phase” for representing an elliptically polarized wave, we needa convention. Pancharatnam proposed the natural choice ofdefining φ = 0 at the position where the electromagnetic vector
is at its maximum positive displacement from the axis. Thisis also known as Pancharatnam’s connection, and can also be
stated as defining the relative phase φ between two interferingwaves such that φ = 0 (the two waves are “in phase”) when theirinterference is maximally constructive [20].
For a general elliptical SOP propagating along the z axis,
the real-valued electric fields of the E x and E y components are
given byE x = A x cos(kz − ωt − φx ),(6)

E y = A y cos(kz − ωt − φ y ).

(7)

At any given position z and time t, the vector magnitude of thesum of these two components is(9)

We can see that this has a form similar to Eq. (1), and so it comesas little surprise that after some algebra, we obtain a similar
expression for γ :tan(2γ ) =A 2x sin(2φx ) + A 2y sin(2φ y )A 2x cos(2φx ) + A 2y cos(2φ y )

.

(10)

This expression for γ for 2D waves closely resembles that for 1D
waves, Eq. (3), but the amplitudes are now squared, and the φ’s
are all doubled. The phase doubling is expected, since the squaring of the electric fields causes the wavelength to halve.
Thus, we can say that a geometric phase of, say, γ = 90◦ in
the case of 2D waves corresponds to exactly the same z location
as a 45◦ phase in the 1D case. These two cases indicate the same
distance, but in terms of different waves, and thus the factor of
two difference in phase value. Keeping track of different wavelengths in expressions becomes confusing, and so one should
always keep in mind that the “oscillation wave” (what we call thesinusoidal modulation in the 2D electric field vector length) has
half the wavelength, and that we always use φ to represent thephase expressed in terms of electric field wavelength λ and not
the “oscillation wavelength” λ/2.
As a result of this 2D composition of waves, we can say thatany 2D polarized wave possesses a geometric phase value.
And, as we show in Appendix B, Eq. (10) agrees with existing
expressions in the geometric phase literature obtained via Jones
calculus [21].
As with wave superposition in 1D, if there is no fixed reference plane, then it is convenient to choose the reference plane
to be such that φx and φ y are symmetrically displaced with thereference halfway between them: 2φx = −2φ y = δ. In such acase, Eq. (10) simplifies totan γ = tan(δ/2)

A 2x − A 2y
A 2x + A 2y

.

(11)

Figure 2(a) illustrates two co-propagating linearly polarizedwaves, with electric field amplitudes A x = 2 and A y = 1. At
the location where each wave reaches its maximum, we draw aplane to indicate the wave position—the “wavefront.” For phasedifference δ 6 = 0, the superposition of these two waves will be
an elliptically polarized wave (not shown in the figure) that has
its maximum displacement at γ with respect to the midpointreference plane (dashed line). The location of the sum-wave’s
peak is indicated by γ .
Figure 2(b) shows a simplified drawing where the wavefronts
are represented by vertical lines. The length of each line indicatesthe amplitude of the wave, and the horizontal distance between

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Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America A

V H
y

1.0

(a) 0

2

x

V H

δ

2π

1

.5

1.0

(c)

−1
−2

y

0.5

x
1

0

δ
−1

0

−0.
5

π
2π
3π

−1.
0

4π

π

−0.
5

5π z

2π

−1.

0

3π
4π

z

H
(b)

391

V

(d)

E

V

γ

γ

δ

δ

Fig. 2. (a) Two waves oriented along the x and y axes, respectively, with A x = 2, A y = 1, and φx = π, φ y = π/2. The H and V “wavefronts” aredrawn at the wave peaks. (b) Wavefronts from (a). (c) Vertically polarized and elliptically polarized waves, with amplitudes A 1 = 1, A 2 = 1.5 andphases φ1 = π/2, φ2 = 5π/8. (d) Wavefronts from (c).

the two lines represents the phase difference δ between thetwo components. The reference plane is indicated by a verticaldashed line, given by the midpoint between the peaks of the twocomponent waves.
Figure 2(c) shows the case of superposing a vertically polarized and an elliptically polarized wave. Once again, the wave
positions are defined with respect to the field vector maxima.
(For the elliptical wave, this maximum is equal to half the lengthof the polarization ellipse’s major axis.) The corresponding
wavefront representation is shown in Fig. 2(d).
Figure 3 shows three examples of how this wavefront representation can be used to help visualize wave composition andthe resulting geometric phase γ . Figure 3(a) shows a horizontal
polarized wave and a vertical polarized wave that are in phaseand have the same amplitude. Their sum produces diagonallypolarized light (azimuth angle 45◦ ) with respect to the x axis.
From Eq. (10), we find that the position of the wavefront of thissum wave is given by γ = 0 since the relative phase between thecomponents is δ = 0. This means that the diagonally polarized
sum wave is in phase with both of the input component waves.
Figure 3(b) shows a horizontally polarized wave and a vertically polarized wave with equal amplitudes, and a phasedifference δ = π/2. The addition of the two waves produces
right-circularly polarized light. This time Eq. (10) gives γ = 0
because the amplitudes of the two component waves are equal.
Once again, the sum wave is in phase with the inputs.
Figure 3(c) shows the general case for summing two waves
using an x -y basis. The two input waves have different amplitudes A H 6 = A V , and a phase difference δ. The sum wave will
have a phase γ lying between the two input wavefront positions.
While Eqs. (3) and (10) give the expression for the geometricphase, this phase does not yet represent a physically measurable
quantity, because an interferogram is needed to detect a global
phase shift. Section 6 discusses how this expression can be generalized for the addition of 2D waves to produce a measurable
quantity. However, we can note that the expression given in

Fig. 3. Using the wavefront representation to aid wave composition
calculations. (a) Input horizontally polarized (H) and vertically polarized (V) waves sum to create a diagonally polarized (D) wave oriented
at 45◦ , in phase with the two input waves. (b) Same input waves as in(a) but now with a phase difference δ = π/2 between them. The resultant sum wave is elliptically polarized in general, but right-circularly
polarized (R) if A H = A V . Since the two input waves are in phase with
one another, the sum wave is in phase with them. (c) Input waves ofarbitrary amplitudes A H and A V and phase difference δ. The resultant
elliptical sum wave (E) has position γ relative to the midpoint between
the wavefront positions of the two input waves.

Eq. (11) contains much of the behavior of geometric phase that
excites curiosity. For example, if the waves were to pass througha retarder R, and we want to compare the input wave phase to
the output wave phase, we cannot simply subtract their phases
because the retardance induced by R will be present inside thetangent function of Eq. (11). This induces a nonlinear behavior
in the phase, such that the path by which one reaches the output
has an effect on the calculation.
4. EFFECT OF POLARIZATION COMPONENTS
ON GEOMETRIC PHASE
To demonstrate how our wave-based approach to geometricphase can deal with optical elements, we use our wavefront

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Fig. 4. Diagram for visualizing the propagation of a polarized wave
through a linear retarder (retardance δ) with its fast axis oriented at 45◦ :(a) horizontally polarized (H) wave input and (b) a 22.5◦ linearly polarized wave input.

visualization technique (Fig. 3) and analyze the effect of a linear
retarder on the wavefront phase.
Figure 4(a) shows each step for calculating the phase fora horizontally polarized wave propagating through a linear
retarder whose fast axis is oriented at 45◦ .
Step 1 The input wave is horizontally polarized, so theV-wave component is zero.
Step 2 Transform the horizontally polarized wave into theeigenbasis of the retarder: the diagonal and antidiagonal (D, A)
linear polarization states. Since a projection from a linear
polarization state onto any orthogonal linear polarization basis
produces only zero phase shift, the D and A waves have to besymmetrically shifted with respect to the reference plane. While
it is also possible to use a −π phase shift for the A wave, this creates a distance of more than π between positive wave peaks. Ourconvention is to shift all wavefronts such that phase differences
between the positive wave peaks being added never exceed π .
Step 3 Apply the retardance to the two waves, by advancing the D wave and delaying the A wave symmetrically. (Thepropagation phase due to the retarder is incorporated into the
dynamic phase and, as we will see in Section 6, does not play arole in the final result.)
Step 4 Compose the D and A waves to determine the phase of
the resulting elliptical state. Since the peak of the resulting statecoincides with the location of the original reference plane, wefind γ = 0.
In this example, the D and A waves superposed after passing
through the retarder are of equal amplitude. As a result, thephase of their sum is located at the midpoint between the two
input peaks, and therefore, γ = 0. In fact, for a 45◦ orientation
linear retarder, this situation of zero geometric phase will occur
for any input polarization state that is located on the red circle
drawn in Fig. 5 [21].
To demonstrate the method by more conventional means,
we follow the same five steps above with numerical calculations.
Step 1 begins with a polarization state EH = (1 0)> . To apply
the wave plate retardance, we need to decompose this into the(D, A) basis. Doing so gives two waves, E1 and E2 , given by

Fig. 5. The great circle drawn in red on the Poincaré sphereindicates the states of polarization for which γ = 0 via Eq. (10)—i.e., when composing waves in the D-A basis. Dashed curves indicate
locations that are on the opposite face of the sphere. The axis labels(H, V , D, A, R, L) indicate the horizontal, vertical, diagonal,
antidiagonal, right-circular, and left-circular polarization states,
respectively.

1
E1 = √ E D ,21
E2 = − √ E A ,2(12)

where E D and E A are normalized Jones vectors for the D and A
polarization states. Once we apply the retardance (Step 3), these
two waves become
1E1 = √ e −iδ/2 E D ,21
E2 = − √ e iδ/2 E A .
2(13)

Here we have ignored the dynamic phase—an assumption thatis justified later in Eq. (25).
In Step 4, we compose the D and A waves into a single 2D
polarized wave using Eq. (10), with




1 2
−1 2
Ā 2 = √
+ √
=1
(14)
2
2
and

sin(δ/2) + sin(−δ/2)= 0,γ̄ = arctancos(δ/2) + cos(−δ/2)(15)

where the overbars indicate that the amplitude and phase referto the 2D wave rather than to either of its 1D component waves.
(Note that the equal amplitudes of D and A waves have allowed
the amplitudes to factor out from the expression.)
From Eq. (15), we see that the geometric phase γ̄ producedby the wave plate in this case is zero—a result that holds true
for any input polarization state that lies on the red circle drawn
in Fig. 5. For an input SOP that does not lie on the red circle,
the geometric phase will not be zero. This result agrees with theprevious findings of Ref. [21].
The step-by-step process outlined above produces the same
geometric phase as is obtained from the standard Jones vectors [16]. One drawback of the Jones vector approach is thatso far it has only been applied to calculate γ before and after
homogeneous optical elements, and not within them. In thisapproach, for the situation shown in Fig. 4(a), the Jones vectors

Research Article

Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America Aof the input and output waves are Ea = (1 0)> and√Eb = (1/ 2)(1 1)> ,(16)

respectively. With these two, the geometric phase obtained fromthe retarder is given as
γ = arg{Ea∀ Eb } = 0.

(17)where the inverted A symbol represents a conjugate transpose(i.e., adjoint) operator.
From this we can see that all three methods—the graphicalapproach of Fig. 4, the mathematical approach of Eq. (10), andthe Jones-vector approach—agree on the result.
5. EFFECT OF POLARIZATION COMPONENTS
ON GEOMETRIC PHASE, II
Figure 4(b) shows a second example, this time of a 22.5◦ linear
polarization state propagating through the same linear retarderas in Fig. 4(a). This time, because the amplitude splitting is nolonger symmetric, we will have a geometric phase even when theelement is a quarter-wave plate.
Step 1 In the input wave, the H-wave component is smaller
than in Fig. 4(a), and the V-wave component is not zero.
Step 2 We project the 22.5◦ polarization state onto the(D, A) eigenbasis for the retarder.
Step 3 Apply the retardance: +δ/2 to the D wave, −δ/2 to
the A wave.
Step 4 Compose the D and A waves into a single ellipticallypolarized wave with peak location given by γ .
For the 22.5◦ input case, we see that the final wavefront
position shifts towards the component wave with the larger
amplitude, and this location is in general shifted by γ from theinitial reference plane. This is a “geometric” phase because it
does not depend on propagation distance z or time t, but rather
depends only on the relationships among the input SOP, theeigenbasis of the retarder, and the retardance.
As in Section 4, we can make the graphical approach above
quantitative as follows. When we decompose the initial 22.5◦
polarized wave into the (D, A) basis, we obtain
E1 = cos(π/8)E D ,E2 = − sin(π/8)E A .

(18)

Once we apply the retardance, these become
π 
π 
e −i(δ−π )/2 E D , E2 = sine −i(π −δ)/2 E A .
E1 = cos88(19)
The elliptical wave created by composing these two has
amplitude Ā and phase shift γ̄ given byĀ 2 = cos2 (π/8) + [− sin(π/8)]2 = 1

393

Therefore, for a quarter-wave plate retarder (δ = π/2), we findthat the geometric phase is γ̄ = −0.615 radians.
We also follow the Jones vector method, for which the input
and output waves Ea and Eb arecos(π/8)cos(π/8)e −i(δ−π)/2
Ea =, Eb =.
− sin(π/8)
sin(π/8)e +i(δ−π)/2(22)
For δ = π/2, these giveγ = arg{Ea∀ Eb } = −0.615 radians.

(23)

In this example, the input polarization state does not split into
equal amplitudes between the two eigenstates of the retarder. As
a result, we find that the geometric phase is no longer zero, butthat the peak of the elliptical wave emerging from the retarderis shifted by −0.615 radians with respect to the input reference
plane.
6. INTERFEROGRAM DETECTION OF
GEOMETRIC PHASE
Measurement of optical geometric phase requires an interferometer to detect the geometric phase as a shift with respect to areference phase. This involves combining the output from thesample arm of the interferometer—the one for which we are calculating the geometric phase—with a wave transmitted throughthe reference arm of the interferometer. Figure 6 shows the
optical layout for a Mach–Zender interferometer—the model
we use below for considering interferometric measurement. Ina typical setup, the polarization state of the reference arm of theinterferometer is designed to match the SOP of the light output
by the sample arm.
We write wave E1 as the electromagnetic wave from the sample arm, and E2 as the wave from the reference arm, using theJones vectors




−iφ2x
−iφ1x
iψ1 A 1x e
iψ2 A 2x e
, (24)
E1 = e, E2 = eA 2y e +iφ2y
A 1y e +iφ1ywhere ψ = kz − ωt is the propagation phase (also called the“dynamic phase”) obtained by the wave after passing throughthe interferometer, and φ encodes the phase delay between thex and y components of the wave [22]. At this point, it is likely
not at all clear that the sample arm encodes a geometric phase,
but Appendix C shows how the expression for E1 in Eq. (24) canbe expressed in a form to make the geometric phase e iγ explicit.
The phase difference 1 = ψ2 − ψ1 is the difference in optical
path length between the two arms of the interferometer—adegree of freedom that exists in the interferometer setup.

(20)

and
tan γ̄ =cos2 ( π8 ) sin([π − δ]/2) − sin2 ( π8 ) sin([π − δ]/2)

cos2 ( π8 ) cos([π − δ]/2) + sin2 ( π8 ) cos([π − δ]/2)tan[(δ − π )/2]
=−
.
√
2
(21)

Fig. 6. Mach–Zender interferometer, showing the wave phases usedin Eqs. (24) and (25).

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Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America A

When the sample and reference waves combine to produce the interferogram, the resulting intensity distribution isexpressed by
I = (E1 + E2 )∀ (E1 + E2 )= A 21x + A 22x + 2A 1x A 2x cos[(ψ1 − ψ2 ) − (φ1x − φ2x )]+ A 21y + A 22y + 2A 1y A 2y cos[(ψ1 − ψ2 ) − (φ1y − φ2y )]= C + 2A 1x A 2x cos(1 − δx ) + 2A 1y A 2y cos(1 − δ y ),(25)
where C = A 21x + A 22x + A 21y + A 22y , δx = φ1x − φ2x , andδ y = φ1y − φ2y . As a result of the adjoint operation, theinterferogram I contains none of the time dependence or z
dependence that the optical waves themselves have. The partthat has disappeared from the expression is the mean dynamic
phase of the two arms.
It is a curious property of interferograms that, although they
operate with intensities rather than amplitudes, they give aphase result proportional to φ, the phase of the electric fields
themselves, and not to 2φ, the phase of the 2D oscillation wave.
The interferogram Eq. (25) consists of a constant C plus two
cosine waves that oscillate with respect to the variable 1. Thus,
as we tune the optical path difference between the arms of theinterferometer, the intensity modulates. Because we have a sumof two cosine waves, we can use the same approach that we didfor the 1D wave composition formulas to calculate the phase of
the interferogram peak with respect to 1. Taking the derivativeof I with respect to 1, setting the derivative to zero, and solving
for 1, we obtaintan 1 =A 1x A 2x sin δx − A 1y A 2y sin δ y.
A 1x A 2x cos δx + A 1y A 2y cos δ y(26)

Although the result is no longer a propagating wave but rather
a stationary interferogram, we find that the expression for theinterferogram phase 1 is nearly the same as that of the geometricphase γ of the 1D propagating wave, Eq. (3). Thus, when theA x /A y amplitude ratios of the two arms are the same, then theinterferogram phase 1 exactly matches the propagating wave’s
geometric phase: tan 1 = tan γ . Although it is not a requirement, this is easiest to see when the polarization states of the twoarms are also exactly matched (i.e., δx = δ y ). In this special case,the interferogram expression Eq. (25) simplifies to
I = C + C cos δ.

(27)

When the amplitudes in the two arms differ, and the polarization states of the two arms are not the same, then attenuating
one arm relative to the other will have an effect on the locationof the peak. In this case, the wave geometric phase and the interferogram phase will not be the same. This is a feature that is notwidely recognized in discussions about geometric phase, though
it is an element in Pancharatnam’s original paper [1].
The results presented in Eq. (26) allow us to describe what
happens in the case of “open loop” configurations of geometricphase measurements. Such open loop situations—where thepolarization state output by the interferometer’s sample arm
is not matched to that in the reference arm—are the subject ofResearch Article

lengthy discussion in the literature, but we now have a straightforward means of quantifying what happens in this case, andhow this affects the geometric phase measurement. The closed
loop configuration, when attenuation in the sample arm is
negligible, will produce a phase value that agrees with geometricphase predictions, but if attenuation is significantly different in
the two arms, especially when the two polarization states differ,then the interferogram phase will diverge from γ .
The fact is that the interferogram is not a wave in the same
way that light is a wave. The interferogram is stationary, has different units, and is not a transverse wave. Keeping this in mind,
it is only natural to observe that they will not in general possess
the same phase, except under specific measurement conditions.
Finally, we can add that if the geometric phase γ is explicitly
represented in the wave expression, such as by rewriting Eq. (24)in the form
A 1x e −iφ1x
A 2x e −iφ2xi(ψ1 +γ )iψ2E1 = e, E2 = e,
A 1y e +iφ1y
A 2y e +iφ2y(28)then we can see that the result Eq. (26) merely has the geometricphase added to it,tan(1 − γ ) =A 1x A 2x sin δx − A 1y A 2y sin δ y,
A 1x A 2x cos δx + A 1y A 2y cos δ y(29)

so that the geometric phase γ from the sample arm produces one
shift, while the term on the right-hand side of the equation represents an additional phase shift produced by the interference of
two beams of differing polarization states.
Although Eq. (29) looks equivalent to Eq. (3), it has a subtle difference. Since the interferogram “wave” is stationary,
when we measure the wave, we can detect shifts of the patternwith respect to a fixed reference plane. This allows us to measure phases wrapped within the range of −π < γ ≤ +π. In
the case of summing waves as in Eq. (3), however, there is no
fixed reference plane available and so we can measure only thetwo wave peaks with respect to one another. This latter caseconstrains the geometric phase values to the smaller range of−π/2 < γ ≤ +π/2. This difference in geometric phase ranges
can cause confusion when comparing our analysis with theexisting literature, if an author does not make explicit whether
he/she is referring to the electric field wave phase or to theinterferogram phase.
The nonlinear behavior of the geometric phase has beensuggested for use in fast optical switching [23,24]. Although thishas been discussed using only 2D waves, the same behavior is already present in 1D, as is clearly demonstrated in
Visualization 1. For the sum of two waves of different amplitudes, the location of the sum peak not only shifts closer to thelocation of the higher-amplitude wave peak, but this shift “accelerates” as δ approaches ±π/2. At ±π/2, the value of γ jumps
by π as a result of the sudden change in reference location. Note,however, that this occurs at the point in which the two waves areclosest to their destructive interference point, where the output
intensity will be weak unless the two beams are substantially different in amplitude. For 2D waves, rather than being a locationof minimum brightness, this is instead a location at which thefringe visibility will be minimum [13,25]. Thus, there is a tradeoff between measurement precision (using the light intensity orResearch Article

Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America A

NP

fringe contrast) and switching speed, since a large difference inamplitude between the two beams blunts the sharpness of thetransition: with a larger difference in wave amplitudes, a largerchange in δ is needed to effect a phase change [26].
7. CONCLUSION
Whereas previous work in the literature has discussed geometric phase in the abstract formalism of matrix calculus or
spherical trigonometry, we have shown that the same results
can be derived entirely from considering the phases of waves
and analyzing how the wave peak position changes whenwaves are added together. This makes it possible to visualize
phase relationships from the waves themselves, with minimal
mathematical abstraction.
Our approach provides an argument (Section 3) for why
the unusual quantities of “instantaneous intensity” [27] (thesquare of the electric field, E 2 ) and “instantaneous Stokes vector” are used in the geometric phase literature. While previous
researchers have used these without justification, other than thatthey work, we see that these arise naturally from calculating theposition of the wave peak of a 2D polarized wave.
Whereas the existing literature calculates γ only before andafter a homogeneous optical element, it does not provide amodel for calculating γ continuously as a wave propagates
through a polarization element. The wave model that we present
here shows how this is to be constructed.
We also found that the likely reason that so much of the geometric phase literature assumes closed-loop cycles is that this is asituation for which the interferogram phase and the wave phase
are identical. If the reference arm polarization state and the sample arm state are different, then the interferogram will exhibit aphase shift in accordance with Eq. (26).
Finally, our results also provide a simple explanation for thedifferences between geometric phases of scalar waves and vector
waves [19]—differences entirely due to the disparity between
composing two parallel waves, Eq. (3), and composing two
orthogonal waves, Eq. (10).
APPENDIX A: SUMMING TOGETHER N WAVES
IN 1D
The harmonic addition theorem states that the sum of N waves
can itself be written as a wave [28]:Xψ=
A i cos(kz − ωt + φi ) = Ā cos(kz − ωt − φ̄),i(A1)with an amplitude and phase of
1/2

N X
N
X
Ā = A i A j cos(φi − φ j )i=1 j =1


=

N
X
i=1

A i2 + 2

N X
NX
i=1 j >i

1/2
A i A j cos(φi − φ j ), (A2)

i=1
tan φ̄ = N
P

395

A i sin φi.

(A3)

A i cos φii=1

APPENDIX B: COMPARING EQ. (13) TO AN
EXPRESSION IN THE EXISTING LITERATURE
Equation 7 in Ref. [16], and Eq. 1 in Ref. [21] give
8 = arg{µ p + µq + (µ p − µq )s p · sa }.

(B1)

From the definitions given in those papers, we can writeµ p = e −iδ , µq = e iδ . [Note that those references use thesymbol δ to represent what we define as 2δ in Eq. (5).] Also, we
can note that s · p = cos(2ψ), where ψ is an angle between thetwo points s p and sa on the Poincaré sphere. As a result, Eq. (33)
becomesγ = arg{e −iδ + e iδ + (e −iδ − e iδ ) cos(2ψ)}= arctan[tan(δ) cos(2ψ)],(B2)where we have also replaced 8 with our geometric phase variable
γ . To compare this expression with our own approach, we lookto Eq. (11). Although we use A 1 and A 2 rather than the angle ψ,
we can represent our amplitudes with A 1 = cos ψ, A 2 = sin ψ,
so that Eq. (11) becomestan γ = tan(δ)

cos2 ψ − sin2 ψ= tan(δ) cos(2ψ),
cos2 ψ + sin2 ψ(B3)which is in agreement with Eq. (B2).
APPENDIX C: HIDDEN ENCODING OF A GLOBALPHASE INSIDE LOCAL PHASESTo see how a global phase can be encoded inside the apparently
local phases defined in Eq. (24), we can consider the following
example. In our Mach–Zender interferometer, we place a single
retarder in the sample arm, with retardance δ. The light incident
on the sample arm is linearly polarized at 0◦ (i.e., an H wave),and the retarder is also oriented with azimuth 0◦ . Then the input
and output states in the sample arm will be
 
iψ1 1
E1 = e
,
(C1)
0


 
1 · e iδ/2iψ2i[ψ2 +(δ/2)] 1
E2 = e=e.
(C2)00 · e −iδ/2Thus, the polarization state is unchanged, but we have delayed
the H wave by half the retardance.
Next, we consider the case when the input polarization stateis oriented at 45◦ (i.e., a D wave), and the retarder fast axis isalso oriented with an azimuth angle of 45◦ . Thus, this setup iseffectively equivalent to the previous case, but with our reference
axis rotated. Still working in the x -y basis, the input and output
waves of this case will be
cos 45◦cos 45◦ e iδ/2iψ2
E 1 = e iψ1,E=e. (C3)
2
sin 45◦sin 45◦ e −iδ/2396Vol. 40, No. 2 / February 2023 / Journal of the Optical Society of America A

The retardance δ in the output wave no longer takes on theappearance of a global phase, but is instead encoded within thelocal phases of the x and y components.
Funding. Japan Society for the Promotion of Science (20K04516).
Disclosures. The authors declare no conflicts of interest.
Data availability. Data underlying the results presented in this paper are
not publicly available at this time but may be obtained from the authors upon
reasonable request.

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