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Deciphering Pancharatnam’s discovery ofgeometric phase: retrospective
Luis Garza-Soto,1Nathan Hagen,1, *2

AND Dorilian Lopez-Mago1

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*nh@hagenlab.org
2

Received 12 January 2023; revised 17 March 2023; accepted 21 March 2023; posted 22 March 2023; published 18 April 2023

While Pancharatnam discovered the geometric phase in 1956, his work was not widely recognized until its
endorsement by Berry in 1987, after which it received wide appreciation. However, because Pancharatnam’s
paper is unusually difficult to follow, his work has often been misinterpreted as referring to an evolution ofstates of polarization, just as Berry’s work focused on a cycle of states, even though this consideration does notappear in Pancharatnam’s work. We walk the reader through Pancharatnam’s original derivation and show how
Pancharatnam’s approach connects to recent work in geometric phase. It is our hope to make this widely citedclassic paper more accessible and better understood. © 2023 Optica Publishing Grouphttps://doi.org/10.1364/JOSAA.485485

1. HISTORICAL INTRODUCTION
Sivaramakrishnan Pancharatnam was born in Calcutta in 1934
in a family of remarkable scientists. He joined his brother S.
Ramaseshan at Bangalore in the Raman Research Institute in1953 when he was 19 years old [1,2]. At that time, Ramaseshan’sresearch supervisor was C. V. Raman, his uncle, who received
the Nobel prize in physics in 1930. Under C. V. Raman’s
instruction, Pancharatnam studied the behavior of absorbing
biaxial crystals, as he points out in the opening of his nowfamous paper, “Generalized theory of interference and its
applications-Part I. Coherent pencils” [3]. This impressive
work was published in 1956 when Pancharatnam was 22. In it,
Pancharatnam proposes a definition for two beams of different
states of polarization to be in phase: “The phase advance of one
polarised beam over another (not necessarily in the same state ofpolarization) is the amount by which its phase must be retarded
relative to the second, in order that the intensity resulting fromtheir mutual interference may be maximum” [3]. Berry named
this Pancharatnam’s connection [4].
After providing expressions for the phase shift—what we now
refer to as the geometric phase—Pancharatnam goes on to relate
the expressions to the subtended solid angle on the Poincaré
sphere. In Sections 2 and 3 below, we retrace Pancharatnam’s
derivation using a more modern approach that should be easierfor modern readers to follow. Along the way, it becomes clearthat Pancharatnam nowhere considers the case that is widely
attributed to him, that the phase of polarized light changes aftera cyclic evolution of its polarization [4–11]. This misconceptionof his actual work appears to be a result of the difficulty one
encounters when reading Pancharatnam’s paper, which often
1084-7529/23/050925-07 Journal © 2023 Optica Publishing Groupfeels like it belongs in the 19th century, and also of confounding
Pancharatnam’s work with the closely related work of Berry.
Almost 30 years after Pancharatnam’s original work, Michael
Berry at the University of Bristol, unaware of Pancharatnam’s
work, discovered that an unexpected phase emerges after the
adiabatic evolution of a quantum state around a cycle in parameter space [12]. In 1983, before his initial paper was published,
Berry introduced the geometric phase to Barry Simon, who
immediately coined it as Berry’s phase [13]. By the end of 1986,
Ramaseshan and Nityananda revived Pancharatnam’s work andpresented it as an example of Berry’s phase [14]. Berry himself
received Nityananda’s manuscript and read it, but mentions
that it was not until he visited Bangalore in July 1987 that hecame to appreciate Pancharatnam’s work. One can surmise that
Nityananda’s interpretation of Pancharatnam’s phase was likely
greatly influenced by Berry’s personal approach to the geometricphase via cycles of states. This seems likely because Berry, on his
return from India, prepared a new manuscript that revealed theconnections between his and Pancharatnam’s work, in which
he explains Pancharatnam’s work in terms of cycles of states[15,16]. Subsequent researchers have almost invariably followed
this interpretation, with the result that Pancharatnam’s actual
achievement is obscured beneath the misconception.
As Berry himself pointed out, there have been several anticipations to geometric phase that arise before Pancharatnam’s
work [16]. Vinitskii et al., for example, mention the work of
Rytov (1938) and Vladimirskii (1941) as precursors [17]. Oriol
Arteaga has also pointed out that Fresnel and Arago in 1816
developed their “fifth” law of optical interference in such a waythat a geometric phase term (at the time not well understood)
had to be included in the interference equations [11]. However,926

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as is commonly the case in science, every discovery can be traced
to its anticipations, and we focus on Pancharatnam because hisachievement is widely recognized by the scientific community[18].
2. PANCHARATNAM’S STARTING POINT
At the beginning of his paper, after explaining briefly some ofthe properties of the Poincaré sphere and Stokes parameters,
Pancharatnam introduces the following theorem:
Theorem 1. When [an electromagnetic] vibration of intensity I in the state of polarisation C is decomposed into two
vibrations in the opposite states of polarisation A and A0 , theintensities of the A-component and the A0 -component are
I cos2 (AC /2) and I sin2 (AC /2), respectively.
Figure 1 shows an illustration of the geometry used in thetheorem. Here Pancharatnam makes use of the fact that theangle ∠A 0 C , between points A0 and C, subtended from thecenter of the Poincaré sphere, is complementary to angle ∠AC ,and so writes both components in terms of ∠AC alone. Since
we will be making use of these angles in many of the equations
below, we follow Pancharatnam and define the individual angles
a , b, and c as
c = ∠A B 
b = ∠AC .
(1)
a = ∠BC 
Note that these angles a , b, and c on the left-hand side of theequations have to be divided by two when used to describe thecorresponding angles of the electric fields in Cartesian space.
Since this theorem is the basis for much of Pancharatnam’s
subsequent results, we show how one can derive it usinga modern approach. Let us consider a monochromaticelectromagnetic wave propagating along the z axis, i.e., E =EC exp[i(kz − ωt)]. The electric field amplitude EC can be
written using elliptical polarization states {ê A , ê A0 } as a basis,∗
with the properties |ê A,A0 | = 1 and ê A · ê A0 = 0 (where ∗represents the complex conjugate). Therefore,
EC = E cos(α)ê A + E exp(iβ) sin(α)ê A0 ,(2)

Fig. 1. Polarization state C with respect to the two orthogonal states
A and A0 , represented on the Poincaré sphere. Point B represents an
SOP lying between A and A0 .

where E is a real amplitude (thus, I = E 2 ), α controls the projection over the basis vectors, and β is the relative phase. With
the above equation, we can represent any state of polarization(SOP) by modifying α and β.
For the common definition of the Stokes parameters, wewould choose ê A = x̂ and ê A0 = ŷ , so that
S0 = |E x |2 + |E y |2 , 
S1 = |E x |2 − |E y |2 , (3)S2 = 2Re[E x E y∗ ], S = 2Im[E E ∗ ].
3xyHowever, this is only one specific choice. An equally valid choiceis the following general form of the Stokes parameters [19]:
S0 = |E A |2 + |E A0 |2 , 22S1 = |E A | − |E A0 | ,(4)∗
S2 = 2Re[E A E A0 ], ∗S3 = 2Im[E A E A0 ],where E j , with j ≡ A, A 0 , represents the components of EC ,i.e., E j = EC · ê j . Let us consider C = {S1 , S2 , S3 }, representing the SOP of EC , which lies on the surface of the Poincaré
sphere of radius S0 with main axes {S1 , S2 , S3 }. These definitions are illustrated in Fig. 1, where the Si are with respect to a
general elliptical basis and need not be with respect to the x -y
basis. By using Eq. (2), the Stokes parameters for EC are
S0 = E 2 = I ,S1 = I cos 2α,(5)S2 = I sin 2α cos β, S3 = −I sin 2α sin β.
Here, 2α and β are the polar and azimuthal angles, respectively,
in a spherical coordinate system, with S1 being the polar axis and{S2 , S3 } being the equatorial plane. Hence, if α = [0, π/2] andβ = [0, 2π ], we can cover the entire surface of the sphere.
From Eq. (5), we see that the normalized Stokes vector ofê A (shown in Fig. 1 as A) is equal to (1, 1, 0, 0). Therefore,
2α is the angle in Poincaré space that the SOP of EC makeswith the basis vector ê A (cf. Fig. 1). Moreover, if we define∠AC = 2α, we can use Eq. (2) to conclude that the intensity ofthe ê A -component is I cos2 (∠AC /2) and the intensity of theê A0 -component is I sin2 (∠AC /2), which is Pancharatnam’s
Theorem 1.
Pancharatnam next considers the problem of explaining
interference between two non-orthogonal states of polarization.
It was already well known that the interference of two beams of
the same SOP results in an intensity,p(6)
I (I1 , I2 , δ) = I1 + I2 + 2 I1 I2 cos(δ),
where I1 and I2 are the individual intensities, and δ is the phasedifference between the two beams. We can see that changing
I1 or I2 only modifies the modulation contrast (visibility),
but not the phase of the sum wave I [20]. In order to see what
happens when two beams of different SOPs are combined,
Pancharatnam considers the superposition of beams in states
A and B. He considers one of them, e.g., A, and its orthogonal
A0 as the polarization basis and finds the intensity of the sum byapplying the following steps:

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1. Adding the amplitude of the A-component of B to theamplitude of A, and squaring to give the intensity,
2. Adding the intensity of the A0 -component of B to the intensity obtained in step 1.
That is, the projection of B onto A gives the component of B that
directly interferes with A. The component of B orthogonal to A
only adds a bias to the intensity.
Considering Pancharatnam’s Theorem 1, we decompose
B into two beams of orthogonal polarizations, yielding thefollowing intensities:
I B,A = I B cos2 (c /2),

(7)

I B,A0 = I B sin2 (c /2),

(8)

where I B is the intensity of B, and c is the angle between thetwo states on the Poincaré sphere, and is, thus, twice the angle
between the states in Cartesian space. For step 1, we combine I A(the intensity of A) with I B,A using Eq. (6), giving ,I (I A , I B,A , δ) = I A + I B cos2 (c /2) + 2[I A I B cos2 (c /2)]1/2 cos(δ).

(9)
This expression represents the intensity that results from theaddition of the A-component of both beams.
Finally, with step 2, we add I B,A0 to obtain the following
intensity:
I = I A + I B cos2 (c /2) + I B sin2 (c /2)+ 2[I A I B cos2 (c /2)]1/2 cos(δ),(10)

horizontal, while that of the second beam is oriented at an angleof θ with respect to the horizontal. We also allow there to be a
propagation phase delay δ between the two beams. Therefore,
we have the two electric field vectors,
E A = |E A |x̂ ,
E B = |E B |e −iδ cos θ x̂ + sin θ ŷ .

(12)

The amplitude of their sum is
E A + E B = |E A | + |E B | cos θ e −iδ x̂ + |E B | sin θ e −iδ ŷ ,(13)so that intensity of the two beams’ interference is
I = (E A + E B ) · (E A + E B )∗which simplifies top
I = I A + I B + 2 I A I B cos(c /2) cos(δ).

Fig. 2. The “original wave” (solid curve) interferogram is obtainedin Eq. (11) when the two intensities I A and I B are equal and held fixed
as we vary the relative phase δ between them. The “attenuated wave”(dashed curve) occurs when we reduce the intensity of one beam andagain vary the phase between the two beams.

(11)

The above equation is Pancharatnam’s Eq. (1) in [3]. This
expression has the advantage that the factor containing thephase δ between the beams and cos(c /2)—the square root ofwhat Pancharatnam calls their “similarity factor”—are separated
as a product.
While δ is the phase delay between the A-component of bothbeams, Pancharatnam gives a couple of arguments to justify that
this phase is equal to the phase between A and B, supported by
Eq. (11). This equation shows that the intensity I of the beams’
superposition varies sinusoidally, with δ as the phase of I . This
phase is independent of what polarization states are involved
and is not even dependent on the relative similarity of the twostates. Rather, as the two states become more “dissimilar,” themodulation contrast (as illustrated in Fig. 2) decreases, but thephase remains unchanged. Second, if either of the two intensities I A or I B changes, this will modify I but does not change
the value of δ. We can see this in the interference of two beams,
regardless of their states of polarization: if we modify only theintensities of the input beams, we do not see a displacementof interference fringes. These results are what Pancharatnam
has in mind when he titles his paper the “generalized theory ofinterference.”
To compare with more modern approaches, we can analyze
the same case using the Jones calculus. In order to simplify themath, we will simply define the state of the first beam to be= |E A | + |E B | cos θ e +iδ+ |E B | sin θ e +iδ|E A | + |E B | cos θ e −iδ

|E B | sin θ e −iδ= |E A |2 + |E B |2 + 2|E A ||E B | cos θ cos δ,(14)which is exactly Pancharatnam’s expression [given as Eq. (11)above] since c = 2θ.
This can be generalized to an arbitrary pair of elliptical states,considering two waves with electric field amplitudes E A and E B ,and a phase difference δ. The intensity of their superposition is
I = (E A + e iδ E B ) · (E∗A + e −iδ E∗B )
= |E A |2 + |E B |2 + 2Re e −iδ (E A · E∗B ) .

(15)

The result of E A · E∗B can be inferred from the discussion about
Pancharatnam’s Theorem 1, which gives E A E B cos(c /2),
where E A , E B are the real amplitudes of the electric fields.
Therefore, we conclude that
I = |E A |2 + |E B |2 + 2E A E B cos(c /2) cos(δ)
p= I A + I B + 2 I A I B cos(c /2) cos(δ),
which agrees with Pancharatnam’s result.

(16)

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3. RELATIONSHIP BETWEEN INTENSITIES AND
LOCATIONS ON THE POINCARÉ SPHERE
Following his expression for the interference of nonorthogonal
beams, Pancharatnam requires only a few short steps (Section 4of his paper) to develop his famous solid angle formula for thegeometric phase. Here he considers the decomposition of abeam of SOP C into two nonorthogonal beams of polarization
states A and B. If Eq. (11) is a general equation for the interference of two beams with different SOPs, then one should be able
to rearrange the equation to solve for the phase delay δ between
them via the intensities of A and B (but which Pancharatnamlabels I1 and I2 ),I − (I A + I B ).
cos(δ) = √2 I A I B cos(c /2)(17)

We can interpret this result as saying that “whatever this phase δ
may be, it maintains a specific relationship between the intensities of the beam being decomposed (C) and the intensities ofthe beams that result from the decomposition (A and B).”
While he could have finished with this numerical formula forδ, he went a step further and realized that this equation expresses
a solid angle relationship between the SOPs of the three beams.
To explain this, he uses electric field vectors E A and E B as components of a sum vector EC = E A + E B . From vector analysis, it
is easy to see that the part of E B perpendicular to E A also has tobe equal to the part of EC perpendicular to E A . Writing this interms of the angles a , b, and c from Eq. (1), we havep√
I B sin(c /2) = I sin(b/2).
(18)
Taking the square of the components to find the intensities
results in the following proportions:
IA = I

sin2 (a /2)
sin2 (c /2)

and

IB = I

sin2 (b/2).
sin2 (c /2)(19)

By expressing the intensities I A and I B in terms of the total
intensity I , together with the angles between states on the
Poincaré sphere, we can rewrite Eq. (17) ascos(δ) =sin2 (c /2) − sin2 (a /2) − sin2 (b/2),
2 sin(a /2) sin(b/2) cos(c /2)(20)which is Pancharatnam’s Eq. (4). Now the equation is entirely interms of the angles between states, rather than their intensities,
and he recognizes that the form of this expression is almost thesame as that of a solid angle formula. In fact, if we re-express theabove relationships in terms of states A, B, and C0 rather than A,
B, and C, where C0 is the antipodal point to C on the Poincaré
sphere, we obtain [21]cos(δ) =1 − cos2 (c /2) − cos2 (a 0 /2) − cos2 (b 0 /2),
2 cos(a 0 /2) cos(b 0 /2) cos(c /2)(21)which does have the recognizable form of a solid angle formula. The primes indicate that the angles are to be taken withrespect to C0 rather than C. That is, b 0 = ∠AC 0 and a 0 = ∠BC 0 .
Making use of the solid angle formula, we obtain [22]
δ = π − 0 /2,(22)Retrospective

where 0 is the angle subtended by the spherical triangle
A, B, C0 from the center of the sphere. (Pancharatnam actually expresses this as δ = π − 12 E 0 , where E 0 is the sphericalexcess of the triangle and, therefore, also equal to the solid angle
0 = E 0 .) The sign of δ given here corresponds to describing
the sequence of states A → B → C0 in a clockwise sense. If thedirection of the sequence is reversed, then the sign flips.
These last statements, in which Pancharatnam works out thecorrect sign of the solid angle, is the only location in the paper
where he talks about a sequence of states. However, it is clear
from context that he is not referring to a cycle of polarization
states but rather to the phase relationships between the two
output states A and B, and the state C from which they were
decomposed.
4. EXTENDING THE REASONING TO THE
ORTHOGONAL CASE
Considering that Pancharatnam found the phase between
two beams using intensity of interference, his definition does
not apply for the case in which states A and B are orthogonal.
However, he provides a geometric argument to show that theformula can still be applied in the limit as the states approachorthogonality. Figure 3(a) shows an initial situation withstates of polarization A and B, and the state C obtained bytheir sum. This is, therefore, the inverse of the case treated in
Section 3 above, where we split a beam in two; this time weare adding A and B with a phase δ between them such thatEq. (11) in Section 2 holds. A point C0 lying on the geodesicarc AB describes a SOP that results from adding the beams ofpolarization A and B with no phase between them. Recallingthat the solid angle used by Pancharatnam is the one subtended
by triangle ABC0 rather than ABC, this situation gives a solid
angle of 0 = 2π . That is, when C0 lies on the smaller geodesic
between A and B, the lune enclosed between geodesic arcs AC0 B
and AC0 B spans a solid angle of half the sphere, while the lune
enclosed within ACB and AC0 B spans a solid angle of zero.
Next we modify the polarization state of B so that it moves
further away from A along the equator of the Poincaré sphere,
as shown in Fig. 3(b) and then Fig. 3(c). As B moves away fromA, the enclosed solid angle subtended by ABC increases. As B
approaches the point orthogonal to A (this point is labeled byPancharatnam as A0 ), the geodesic ACB becomes half a great
circle, as in Fig. 4. One might argue that in this case the enclosed
area becomes undefined since we can draw the geodesic connecting A and B in either a clockwise or an anticlockwise sense.
However, if we note that the geodesic from A to B has until thispoint always passed through the intermediate point C0 , thenit will for this limit case as well. The solid angle that relates thephase between the two beams is, therefore, the one enclosed
between the two geodesic arcs AC0 A0 and AC0 A0 (where A0 coincides with the point written as B in Fig. 4). This is a sphericallune (drawn in green in the figure) whose solid angle subtended
from the origin is exactly twice the value of the angle α formed
between states C0 , A, and C0 at the surface of the sphere.
Using Eq. (22) finally allows us to equate the phase difference
between beams A and B to the angle α: δ = α. That is, the angle
α (or half the solid angle ) is equal to the phase δ that one

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Fig. 3. Increasing area (drawn in orange) on the surface of the sphere, formed by the spherical triangle between points A, B, and the result of theiraddition, C. When beams in states of polarization corresponding to A and B are added in phase, the resulting polarization corresponds to C0 , lying in
the geodesic between them. When B approaches A0 , the antipode of A, then the orange area becomes that of Fig. 4.

Fig. 4. If we continue the evolution shown in Fig. 3 until state B
becomes orthogonal to A, then we form a spherical lune (drawn inorange) between geodesic arcs AC0 B and ACB. The complementary
spherical lune, drawn in green, between geodesic arcs AC0 B and AC0 B
corresponds to the solid angle  giving the geometric phase δ between
A and B needed to obtain C upon addition.

must retard state A0 from A in order for both to be correctly
decomposed from input state C.
5. INTERFERENCE OF THE COMPONENTS
TRANSMITTED BY AN ANALYZER
It was of practical importance for Pancharatnam to have an
expression for the phase of two arbitrarily polarized beams transmitted through an analyzer because this was the configuration
he was using to measure the pattern transmitted by biaxial crystals such as iolite. Finding such an expression is the purpose of
Section 8 of Pancharatnam’s paper.
Pancharatnam first defines D as the SOP transmitted by the
analyzer. Unfortunately, Pancharatnam once again reuses the
symbol C here. We write this as D to avoid confusion with thestates defined in the earlier sections. He also uses I1 and I2 inplace of I A and I B , but we have retained the latter for consistency. Pancharatnam also reuses the symbols a , b, and c for theangles between states, but since the definition of C has changed,

Fig. 5. Spherical surface elements used to calculate the intensity
transmitted by an arbitrary analyzer (state D) when two states A and B
are incident upon it.

we instead define θij to represent the angle between points iand j on the surface of the Poincaré sphere, as subtended fromthe center of the sphere. His goal is to relate the phase of theinterference transmitted by the analyzer to the phase difference
δ between the two input states (see Fig. 5).
When beams A and B are incident on the analyzer, the transmitted intensity I D will be the component of state A along direction D, plus the component of state B also along D, while incorporating an unknown phase difference δ 0 between A and B,
I D = I A cos2 (θAD /2) + I B cos2 (θBD /2)+p

I A I B cos(θAD /2) cos(θBD /2) cos(δ 0 ).

(23)

If we use the analyzer oriented at angle D0 orthogonal to D, thenwe would get a different intensity I D0 and an unknown phasedifference δ 00 ,I D0 = I A sin2 (θ A D0 /2) + I B sin2 (θ B D0 /2)+p

I A I B sin(θ A D0 /2) sin(θ B D0 /2) cos(δ 00 ).

(24)

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From Fig. 5, we can see that the area enclosed by DAD0 BD isa spherical lune. The phase δ 0 needed to generate state D fromadding A and B is given by δ 0 = π − 12 0 , where 0 is the solid
angle subtended by the spherical triangle ABD0 (drawn in green
in the figure). In a similar fashion, the phase δ 00 needed when
adding A and B to get D0 is given by δ 00 = π − 12 00 , where 00 is
the solid angle of ABD (drawn in orange).
Since both δ 0 and δ 00 are related to corresponding solid angles,
if we subtract the two, we obtain the solid angle subtended byquadrangular area F in the sphere [8,23],
F = ±(δ 00 − δ 0 ).

(25)

Confusingly, Pancharatnam yet again reuses the symbol C toindicate this quadrangular area. We will instead use F , as Fig. 5
does. The sign of F is determined by whether the sequenceof points ABD proceeds in a clockwise or anticlockwise fashion. Pancharatnam then takes both I D and I D0 and adds them
together to get the total intensity, equal to the intensity of C that
depends on δ but now expressed in terms of δ 0 and F ,p
I = I A + I B + 2 I A I B [cos(θAD /2) cos(θBD /2) cos(δ 0 )+ sin(θ A D0 /2) sin(θ B D0 /2) cos(δ 0 ± F )].
(26)
Now that we have the same angle δ 0 in both terms inside thesquare brackets, we can recognize that this is a standard expression for the spherical excess of a triangle, so that the equation
simplifies to [22]p1
I = I A + I B + 2 I A I B cos(c /2) cos(δ 0 + ).
2(27)This equation is now identical in form to Eq. (11), from which
we can then say that δ 0 = δ − 12 . Now we have a means ofcalculating δ 0 for a given pair of input states A and B, together
with the analyzer orientation giving D. Using this in Eq. (23),
we can now calculate the intensity transmitted by the analyzer
depending on the original phase difference δ between the two
input beams.

Retrospective

is often overlooked: the interferogram phase will not in general
be the same as the phase difference δ between the two beams.
If we want to measure the phase difference δ, we must pass the
addition of the beams through an analyzer to obtain the interferogram. We can use Eq. (27), however, to relate the measured
phase to δ.
The recently published “wave description of geometricphase” interprets the geometric phase as a shift in the wave peaklocation away from the midpoint between the peaks of the two
input waves [24]. One can see the close connection between the
wave description and Pancharatnam’s approach by considering
Eq. (11) [Pancharatnam’s Eq. (1)], which expresses the intensity
produced by adding two waves as being√a bias value (I1 + I2 )plus a single cosine term with amplitude I1 I2 cos(c /2). Thus,
Pancharatnam is also considering the case of two cosine waves
summing together into a single cosine output wave.
In Eq. (14), we were also able to show how Pancharatnam’s
results are derived using the familiar modern approach, notavailable to Pancharatnam, of the Jones calculus. The Jones
calculus explicitly forms a pair of 2D electric field vectors, andadding these two waves produces a cosine factor cos(δ) that
includes the same phase delay that Pancharatnam obtained,
but which does not seem to have been replicated until Michael
Berry’s work in 1987 [4].
Berry naturally interpreted Pancharatnam’s work throughthe lens of his own work, which approached the geometric phaseas a shift resulting from the evolution of a quantum state arounda cycle in parameter space. Upon reading Pancharatnam’s
approach, he saw that this could easily be a cycle of polarization
states generating this phase shift. Pancharatnam, however, wasfocused on interferometric measurement and not on modelingthe evolution of polarization states. As it turns out, the two areequivalent, and so the misunderstanding is not at all a serious
one. In the literature, many authors actually refer to this shift
due to polarization state evolution as a “Pancharatnam–Berryphase.” This seems the ideal choice since “Pancharatnam phase”
should perhaps be more narrowly defined as a shift resulting
from adding polarized waves.
Funding. Japan Society for the Promotion of Science (20K04516).

6. CONCLUSION

Disclosures. The authors declare no conflict of interest.

Through an impressive set of spherical trigonometry manipulations on the Poincaré sphere, Pancharatnam found thatpolarization states have specific phase relationships that are
generally not taken into account, unless one is performinginterferometric measurements. This was the case for him since
he was analyzing the light transmitted through dichroic biaxial
crystals. The correct analysis of these measurements requiredthat he incorporate this new phase, what we now refer to as thegeometric phase, into his equations.
Among his less-known results is Eq. (17), which shows that,
if one knows the intensities of two input beams as well as theintensity of their interference, one can infer the phase difference
δ between the two input beams from these simple intensity measurements alone. Also notable is Eq. (27), which demonstrates
that the phase transmitted by an analyzer is not in general equal
to the phase of a wave incident upon it. While it might seemthat Section 5 is not essential, it makes an important point that

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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