Visualization of electromagnetic-wave polarization evolution using the Poincaré sphere.

Abstract

For the first time to the best of our knowledge, we derive expressions for coordinates of the trajectory on the Poincaré sphere that represent polarization evolution in an arbitrary beam of completely polarized light. Our work substantially extends the mapping function of the Poincaré sphere, and opens up new possibilities for its use in optics. In particular, the obtained expressions allow one to visualize the results of the finite-difference time-domain modeling of light propagation through birefringent crystals, including simulations of polarization rotation experienced by ultrashort pulses in nonlinear media.

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Visualization of electromagnetic-wave polarization
evolution using the Poincaré sphere
Ivan D. Rukhlenko,* Chethiya Dissanayake, and Malin Premaratne
Advanced Computing and Simulation Laboratory, Monash University, Clayton, Victoria 3800, Australia
*Corresponding author: ivan.rukhlenko@monash.edu
Received April 13, 2010; revised May 21, 2010; accepted June 7, 2010;
posted June 11, 2010 (Doc. ID 126839); published June 24, 2010
For the first time to the best of our knowledge, we deriveexpressionsfor coordinatesof the trajectoryonthe Poincaré
sphere that represent polarization evolution in an arbitrary beam of completely polarized light. Our work substan-
tially extends the mapping function of the Poincaré sphere, and opens up new possibilities for its use in optics. In
particular, the obtained expressions allow one to visualize the results of the finite-difference time-
domain modeling of light propagation through birefringent crystals, including simulations of polarization rotation
experienced by ultrashort pulses in nonlinear media.© 2010 Optical Society of America
OCIS codes:
260.5430, 120.2130, 260.7120.
Since its development by Poincaré in the early 1890s [1],
the method of the Poincaré sphere has found remarkable
applications informulating, clarifying, and solving the nu-
merous problems in polarization optics [2]. Like other
graphical tools invented by Poincaré, this method allows
one to tackle physical problems without resorting to
complicated mathematics and provides a convenient
way to represent the findings. For example, the Poincaré
sphere substantially simplifies calculation of the polariza-
tion form of light passing through uniaxial or dichroic
crystals, polarizers, and phase plates; it helps to describe
the coherent and incoherent addition of light beams, the
interference phenomena in crystals, and the Faraday ef-
fect; it also enables one to graphically represent a par-
tially polarized light, as the interior of the sphere is
taken into consideration [3–6].
The study of polarization evolution traditionally in-
volves analyzing discrete changes of one elliptical polar-
ization form of a plane monochromatic wave by the
other, while eliminating all the transient polarization
states underlying the rapid evolution of the form itself
[7]. The results of this study are known to admit a simple
pictorial representation on the Poincaré sphere. With the
advent of lasers capable of generating femtosecond light
pulses [8], the problems of investigation and visualization
of ultrafast polarization evolution naturally arise. It is evi-
dent, however, that characterization of such evolution in
terms of the discrete elliptical polarizations is no longer
possible; the pulses are too short to even display a single
polarization form. Of course, one may still visualize the
variations of pulse polarization using its sectional pat-
tern. Unfortunately, in the majority of situations, this ap-
proach is inconvenient too, since it leads to obscure plots
and lacks the simple intuitive picture of the polarization
evolution of a propagating pulse. We suggest the solution
for the above problem by adapting the method of the
Poincaré sphere to the case of arbitrary polarization evo-
lution. With the relations derived in this Letter, the Poin-
caré sphere can be used to portray how the polarization
of extremely short optical pulse evolves.
Before derivation of the relations between the electric-
field components and the instantaneous coordinates on
the Poincaré sphere, we briefly discuss the definition of
the Poincaré sphere. One may graphically represent a
completely polarized light beam by one-to-one mapping
of its polarization form onto a point on the Poincaré unit
sphere. The mapping is illustrated in Fig. 1 and, for the
right-handed Cartesian coordinates, is expressed by the
relations
x ¼ cos2ψ cos2χ;
y ¼ sin2ψ cos2χ;
z ¼ sin2χ;
where the latitude 2ψ and the longitude 2χ of the point
P ≡ ðx;y;zÞ are proportional to the azimuth ψ ∈
ð−π=2;π=2? and the ellipticity χ ∈ ½−π=4;π=4? of the
beam’s polarization ellipse. The azimuth is the angle,
measured counterclockwise, between the x axis and
the major semiaxis. The tangent of the ellipticity is equal
in absolute value to the ratio of the length of the minor
semiaxis to that of the major semiaxis; the sign of the
ellipticity is assumed to be positive for right-handed po-
larization and negative for left-handed polarization (thus
tanχ ¼ b=a in Fig. 1, where it is assumed that the light is
coming in the þz direction toward the viewer). The prop-
erties of the Poincaré sphere defined as above, and the
significance of its various parts may be found in [5,9,10].
Consider a plane electromagnetic wave whose polari-
zation evolution in the xy plane is given in the parametric
form; the electric-field components ExðξÞ and EyðξÞ are
the functions of the parameter ξ, which may represent
time (ξ ¼ t) or propagation distance (ξ ¼ z), depending
on whether the time or spatial evolution is under inves-
tigation. Since the sectional pattern determined by the
Fig. 1.
sphere.
(Color online) Polarization ellipse and the Poincaré
July 1, 2010 / Vol. 35, No. 13 / OPTICS LETTERS2221
0146-9592/10/132221-03$15.00/0© 2010 Optical Society of America

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parametric dependence EyðExÞ is an arbitrary, generally
open curve, the polarization form of the wave continu-
ously changes with the variation of ξ. The objective of
the following analysis is to find the trajectory on the Poin-
caré sphere that corresponds to this curve and repre-
sents the polarization evolution in a beam of arbitrary
polarized light.
One may establish a one-to-one correspondence be-
tween an arbitrary polarization pattern EyðExÞ and its
projection onto the Poincaré sphere by inscribing an el-
lipse into the pattern at each point ðEx;EyÞ. With this ob-
ject in view, we suppose that the functions ExðξÞ and
EyðξÞ satisfy the equation
FðEx;EyÞ ≡ðExcosφ þ EysinφÞ2
þðExsinφ − EycosφÞ2
a2
b2
− 1 ¼ 0;
ð1Þ
which, for the real a and b, describes an arbitrary polar-
ization ellipse passing through the point ðEx;EyÞ. Notice
that the angle φ ∈ ½−π=2;π=2? in Eq. (1) is generally not
the azimuth of the polarization ellipse, since it is always
measured from the semiaxis a, which may be less than b
for some ξ.
To find the parameters a, b, and φ, we require both the
slope and the curvature at the point of tangency to be
equal for the curve EyðExÞ and the ellipse in Eq. (1). This
implies that the first and second derivatives of Eywith
respect to Exare equal for the two curves. These deriva-
tives are easy to calculate, because the functions ExðξÞ
and EyðξÞ are known:
fðξÞ ≡dEy
dEx
¼E0y
E0x
;
ð2Þ
gðξÞ ≡d2Ey
dE2
x
¼E00
yE0x− E00
ðE0xÞ3
xE0y
;
ð3Þ
where the primes stand for the derivatives with respect
to ξ. For the curve in the implicit form of Eq. (1), the same
derivatives are given by [11]
dEy
dEx
¼ −∂F=∂Ex
∂F=∂Ey
;
ð4Þ
d2Ey
dE2
x
¼
?
−∂2F
∂E2
2
∂2F
∂Ex∂Ey
?∂F
∂F
∂Ex
?2??∂F
∂F
∂Ey
−∂2F
∂E2
?−3
x
?∂F
∂Ey
?2
y
∂Ex
∂Ey
:
ð5Þ
By solving Eqs. (2) and (4) together with Eq. (1), we
obtain
a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f cosφ − sinφ
ðfEx− EyÞðExcosφ þ EysinφÞ
s
;
ð6Þ
b ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f sinφ þ cosφ
ðfEx− EyÞðExsinφ − EycosφÞ
s
:
ð7Þ
By using this result in Eqs. (3) and (5), we find the
following solution for φ in the range −π=4 ≤ φ ≤ π=4:
?
Apparently, there is always another solution for φ in
the range −π=2 ≤ φ ≤ π=2, which leads to a different value
of latitude.According to the definition, the correct choice
of the latitude is unambiguously determined by the rela-
tive values of a and b: if a > b, the latitude must lie in the
range −π=2 < 2ψ < π=2; if b > a, the latitude must lie
either in the range −π < 2ψ < −π=2 or in the range π=2 <
2ψ ≤ π (see Fig. 1). It is easy to verify that these require-
ments are fulfilled if one takes
φ ¼1
2tan−1
2gExEyþ 2fðfEx− EyÞ
gðE2
x− E2
yÞ þ ð1 − f2ÞðfEx− EyÞ
?
:
ð8Þ
ψ ¼ φ − ðπ=4Þ½1 − signða − bÞ?signφ;
where sign x is the common signum function.
The longitude is determined not only by the relative
values of a and b, but also by the handedness, which de-
pends on the sign of the derivative f0¼ gE0x; the longitude
is negative for left-handed polarization (when f0> 0) and
positive otherwise. Therefore, we may write
ð9Þ
χ ¼ − sign f0tan−1ða=bÞsignðb−aÞ:
Equations (2), (3), and (6)–(10) solve the defined pro-
blem. They allow us to calculate the coordinates 2ψðξÞ
and 2χðξÞ of the trajectory on the Poincaré sphere, which
gives us the instantaneous polarization form for an arbi-
trary, parametrically defined sectional pattern.
It is easy to show that, in the case of elliptic polariza-
tion, when
ð10Þ
Ex¼ Axsinðβξ þ ϕxÞ;
where β, Aj, and ϕjðj ¼ x;yÞ do not depend on ξ, Eqs.
(6)–(8) lead to the well-known relations between the el-
lipticity and azimuth and the electric-vector-component
amplitudes and phase shift Δ ¼ ϕx− ϕy[6]:
tan2ψ ¼2AxAycosΔ
A2
y
Ey¼ Aysinðβξ þ ϕyÞ;
x− A2
;sin2χ ¼2AxAysinΔ
A2
xþ A2
y
:
It should be emphasized that these equations are not ap-
plicable even for the case of slowly varying Δ (such that
∂Δ=∂ξ ≪ β), which becomes obvious if Ax¼ Ay.
We first illustrate the derived relations and the capabil-
ities of the associated mapping using the Lissajous
patterns shown in Fig. 2. The projection of the left pattern
onto the Poincaré sphere clearly shows that a single by-
pass of the pattern results in two changes of the handed-
ness (two crossings of the equator), 12 oscillations of the
azimuth, and 12 oscillations of the ellipticity of the polar-
ization form. It also allows us to easily determine the
train of retarders that will cause polarization changes
presented by the whole pattern or any of its parts.
2222OPTICS LETTERS / Vol. 35, No. 13 / July 1, 2010

Page 3

It is important that not all parts of the curves drawn by
thetipoftheelectric-fieldvectorcanbegraphicallyrepre-
sentedwith thehelp ofthePoincarésphere.Forexample,
the two right panels in Fig. 2 show the situation where
changes in handedness are accompanied by discontinu-
ities of the trajectory on the sphere. The discontinuities
happenbecausetwosegmentsofthepolarizationpattern,
shown by the dashed curves, are approximated by hyper-
bolae (with b2< 0), rather than by ellipses.
Figure 2 also reveals a significant general feature of the
established mapping: the mirror symmetry of the sec-
tional pattern manifests itself in rotation and mirror sym-
metries of the trajectory on the Poincaré sphere. Indeed,
the Lissajous patterns on the left and right panels have,
respectively, two planes and one plane of symmetry. At
the same time, their projections onto the spheres has a
dyad X axis, and the trajectory on the left panel is sym-
metric with respect to reflections in the XY and XZ
planes.
The method of the Poincaré sphere can be used to
readily illustrate the polarization rotation for ultrashort
pulses comprising only a few oscillation cycles of the
electromagnetic field. In many instances, the trajectory
on the sphere demonstrates polarization changes much
clearer than the sectional pattern does. For example,
Fig. 3 shows several cycles of a pulse with fixed carrier
frequency and a slowly varying phase shift between the
polarization components. We can see that, even though
the sectional pattern has a sophisticated shape, its pro-
jection onto the Poincaré sphere is quite simple and
may be approximated by the arc of a circle. The arrange-
ment of the arc characterizes the trend of the polarization
evolution and reveals the change of the handedness.
As a concluding remark, we would like to note that the
converse problem of recovery field evolution for an arbi-
trary smooth curve on the Pioncaré sphere does not have
a unique solution, for identical changes in the polariza-
tion form may be caused by either the phase shift be-
tween perpendicular polarization components of fixed
amplitude, or by different attenuation of these compo-
nents with a fixed phase shift between them.
In this work, we extended the Poincaré-sphere method
for representation of polarized light to the case of beams
with ever-varying polarization form. We derived the ex-
pressions relating the electric-field components to the co-
ordinates of the instantaneous polarization type on the
Poincaré sphere. The extension of the method allows
one to get a concise and physically clear description
of any polarization evolution, which may prove useful
in modern ultrafast optics.
This work was funded by the Australian Research
Council through Discovery Grant DP0877232.
References
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3. B. A. Robson, The Theory of Polarization Phenomena
(Oxford U. Press, 1974).
4. G. N. Ramachandran and S. Ramaseshan, in Handbuch der
Physik (Springer Verlag, 1961).
5. W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).
6. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light
in Optics and Spectroscopy (Academic, 1990).
7. M. Born and E. Wolf, Principles of Optics (Cambridge U.
Press, 1999).
8. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
9. W. A. Shurcliff and S. S. Ballard, Polarized Light (Van
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Analysis (Pergamon, 1965), Vol. I.
Fig. 2.
presenting instantaneous polarization forms for two Lissajous
patterns: Ex¼ cos3ξ and Ey¼ sin4ξ on the left; Ex¼ cosξ
and Ey¼ sinð2ξ þ π=4Þ on the right; ξ ∈ ½0;2πÞ.
(Color online) Trajectories on the Poincaré sphere re-
Fig. 3.
and its representation on the Poincaré sphere.
(Color online) Polarization pattern of a few-cycle pulse
July 1, 2010 / Vol. 35, No. 13 / OPTICS LETTERS 2223