The distributions of (or constraints for) amplitude and phase around Cpoints, including Lemon, MonStar and Star, are studied. A Cartesian coordinate system with origin at the Cpoint is established. Four curves, where the azimuthal angles of polarization ellipses are 0°, 45°, 90°, and 135° respectively, are used to determine the distributions. Discussions of these constraints illustrate why MonStar is rarer than Lemon or Star in experiments. The transformation relationships between these three polarization singularities (PSs) are also discussed. We construct suitable functions for amplitude and phase according to their constraints, and simulate several PSs of particular shapes. With the development of modulation techniques for amplitude and phase, it is clear that this work is helpful for generating arbitrarily shaped Cpoints in experiments.
I. INTRODUCTION
Since Nye and Berry first found dislocations in optical fields in 1974
[1]
, singular optics have been a subject of interest. The diverse singular patterns provide rich information about the fine structure of light. While phase singularities (wave dislocations, or optical vortices) are frequently encountered in the interference of scalar waves, they resolve into polarization singularities (PSs) when the vector nature of light is retained
[2

6]
. PSs include Cpoints (points where the light is circularly polarized) and Llines (lines where the light is linearly polarized)
[2]
. These PSs determine the distribution of polarization ellipses around them, such as how in nonparaxial fields the major axes (minor axes) of polarization ellipses surrounding Cpoints are shown to form Möbius strips
[3

5]
, while in paraxial fields polarization ellipses around Cpoints produce interesting structures named Lemon, MonStar, and Star
[2
,
6]
. Experiments have been successful in generating PSs
[7

13]
, but generation of PSs with a desired shape or structure, especially generation of a MonStar, is still a challenge. This is because the distributions of amplitude and phase around Cpoints are complex and have not been developed clearly.
In this paper, a Cartesian coordinate system with origin at the Cpoint is established. Then, four curves where the azimuthal angles of polarization ellipses are 0°, 45°, 90°, and 135° respectively are used to determine the distribution functions of amplitude and phase around Cpoints; these distributions include the Lemon, MonStar, and Star. Discussion of these distributions illustrates the difficulty of generating a MonStar in experiments. The probabilities of transformation of these three PSs are also discussed. According to the distributions of amplitude and phase, we give suitable functions and simulate several particularly shaped PSs. With the development of modulation techniques for amplitude and phase, it is clear that the work in this paper is helpful for generating arbitrarily shaped Cpoints in experiments.
II. THEORETICAL ANALYSIS
To analyze the distributions of (or constraints for) amplitude and phase around a Cpoint, we need to write the electromagnetic wave in term of the Jones matrix:
where
A_{x}
and
A_{y}
denote the amplitudes of
E_{x}
and
E_{y}
respectively, while
α_{x}
and
α_{y}
denote the corresponding phases. With appropriate calculation, the trajectory equation of the end of vector can be written as
where Δ=
α_{y}
−
α_{x}
is the phase difference between
E_{x}
and
E_{y}
. Eq. (2) denotes an internally tangent ellipse in a rectangle of dimensions 2
A_{x}
×2
A_{y}
, as shown in
Fig. 1
. The behavior of the ellipse depends on
A_{x}
,
A_{y}
, and Δ, so we analyze the phase difference Δ instead of the phases
α_{x}
and
α_{y}
. The azimuthal angle
θ
and the ellipticity
ε
are two important parameters of the polarization ellipse
[14]
:
Schematic illustration of the end of a vector. The ellipse is internally tangent to the rectangle.
where the azimuthal angle
θ
is defined as the angle between the major axis of the polarization ellipse and the +
x
axis
Combining Eqs. (2) and (3) and considering
Fig. 1
, we obtain the following relationship:
where ↔ is the symbol for “necessary and sufficient”. Eq. (5) is a very important conclusion to determine the distributions of the amplitude and phase difference, as follows.
It is well known that there are three typical kinds of polarization singularities: Lemon, MonStar, and Star, shown in
Figs. 2(a)
,
(b)
, and
(c)
respectively
[2
,
6]
. In
Fig. 2
the gray ellipses surrounding Cpoint
present the polarization states of the fields. The gray solid curves are the envelopes of the azimuthal angles of polarization ellipses. Among these envelopes, there are some rays (the red rays in
Figs. 2(a)
~
(c)
) emanating from the Cpoints
. The azimuthal angles of polarization ellipses on these rays equal the angle between the ray and the +
x
axis. Obviously there is one ray emanating from the Lemon, while three emanate from the MonStar and Star
[2
,
15
,
16]
. To identify the type of PS, a very small circle
σ
with center at the Cpoint is drawn. Assuming that the azimuthal angle of a polarization ellipse at point
Q
on
σ
is
ϕ
, the value of
ϕ
varies as point
Q
moves along
σ
anticlockwise. Considering the total change in
ϕ
over
σ
, we have Δ
ϕ
=+
π
for Lemon or MonStar and Δ
ϕ
=−
π
for Star. Dividing Δ
ϕ
by 2
π
, we get the winding number
I
= +1/2 for Lemon or MonStar and
I
= −1/2 for Star. The function for the winding number is given as
[17]
. The winding number and number of rays jointly determine the type of the PS.
The gray ellipses present the polarization states of fields, and the gray curves are envelopes of the azimuthal angles of ellipses. The red curves are rays emanating from the Cpoints . A very small circle σ with center at the Cpoint is drawn. , , , and are places where the azimuthal angles of ellipses are 90°, 135°, 0°, and 45° respectively. (a)~(c) show respectively the patterns of Lemon, MonStar, and Star. (d) Lemon with only one ray emanating from the Cpoint. (e) MonStar with three rays emanating from the Cpoint. (f) Star with three rays emanating from the Cpoint.
Next a Cartesian coordinate system is established with origin
O
at the Cpoint
and 
y
axis coinciding with one of the rays, as shown in
Figs. 2(d)
~
(f)
. Thus the 
y
axis is where the azimuthal angles of polarization ellipses are 90°, and is marked as
. Then three other curves where the azimuthal angles are 0°, 45°, and 135° are drawn out and marked as
,
and
respectively, when they are rays. Because of the different distributions of polarization ellipses, rays
and
are to the right and left, respectively, of
AOC
in
Figs. 2(d)
and
(e)
, while they swap positions in
Fig. 2(f)
. We mark the other two rays in
Figs. 2(b)
and
(c)
as
and
, respectively, as shown in
Figs. 2(e)
and
(f)
. The field is divided into four regions, denoted by ∠
AOB
, ∠
BOD
, ∠
COD
, and ∠
AOD
. Establishing the Cartesian coordinate system and the four curves makes for concise analysis of the amplitude and phase of a PS, will be demonstrated as follows.
Now considering the amplitude distribution of the Lemon in
Fig. 2(d)
, we find the curve
BOD
to be a dividing curve, upon which azimuthal angles of ellipses change within the range of −
π
/4<
θ
<
π
/4, and under which the range is
π
/4<
θ
<3
π
/4, and at which the azimuthal angles are
π
/4 or 3
π
/4. According to Eq. (5), the amplitude distribution of the Lemon is given as:
Eq. (6) is the distribution function of amplitude around the Lemon. Now we analyze the distribution of phase around the Lemon. We assume that the field around the Cpoint is lefthandpolarized, namely 0<Δ<
π
. As shown in
Fig. 2(d)
, the azimuthal angle on ray
is
, while
on ray
. Substituting
and
for
θ
in Eq. (3), we obtain
Calculating the derivative of Eq. (3),
where
Taking a point that is infinitely close to and on the right of
, then the difference in
θ
is positive, namely d
θ
>0. Substituting
A_{x}
<
A_{y}
, d
A_{x}
> d
A_{y}
, and Δ=
π
/2 into Eqs. (7)~(9), we have dΔ>0. Using this analysis method, the distribution of the phase difference is given as
Eq. (10) means the curve
AOC
is also a dividing curve for the phase difference, while the curve
BOC
is the dividing curve for the amplitude. These two curves make the amplitude and phase around a Cpoint simple and clear, which is why we establish the Cartesian coordinate system and need the help of the four rays
,
,
, and
.
According to the structures shown in
Fig. 2
, the ellipses surrounding Lemon and MonStar have the same tendency, apart from the number of rays (one for Lemon and three for MonStar). In other words, MonStar can be regarded as special Lemon. So the amplitude and phase of MonStar must be bound by Eqs. (6) and (10), respectively. On the other hand, there exist two other rays
and
in the field of the MonStar, as shown in
Fig. 2(e)
. Naming the azimuthal angles of ellipses on
and
as
and
respectively, and implementing the simple transform of Eq. (3), we have
This formula is a function of amplitude and phase simultaneously, which means the amplitude and phase on rays
and
should follow Eq. (11), while being restricted by Eqs. (6) and (10). It seems complex to construct amplitude and phase that satisfy Eqs. (6), (10), and (11) simultaneously. However, when the amplitude is set according to Eq. (6), the phase can be determined according to Eqs. (10) and (11). Similarly, a set phase can used to determine the amplitude according to Eqs. (6) and (11).
Compared to Lemon and MonStar, the Star has a different elliptic tendency. However, the analysis method used above is also suitable for acquiring the distributions of amplitude and phase around the Star. Compared to those for Lemon, the amplitude and phase of Star have the same expressions as Eqs. (6) and (10), respectively, but we emphasize that because of the opposite spatial positions of
and
in
Figs. 2(d)
and
(f)
, the distribution of phase for Star differs from that for Lemon. As for Monstar, there are also two other rays (
and
) in the field of Star, so the amplitude and phase of Star are likewise restricted by Eq. (11).
Now we see that the analyses of amplitude and phase for Lemon, MonStar, and Star become very concise. In addition, these distributions of the three PSs can be expressed by the same forms (Eqs. (6), (10) and (11)). These are the benefits of establishing the Cartesian coordinate system and the four rays
,
,
, and
.
Developing distribution functions of amplitudes and phases around Cpoints ultimately benefits the generation of PSs. According to the constraints, constructing amplitude and phase is a crucial step in generating the desired PSs. Constructing amplitude and phase functions following Eqs. (6) and (10) for Lemon is easy. Here we try to give expressions used to generate MonStar and Star when
and
coincide with −
y
and +
y
respectively. For convenience but without loss of generality, we assume that t he component
E_{x}
is a plane wave, i.e.
A_{x}
and
α_{x}
are constants. Assuming that we know the amplitude according to Eq. (6), denoted as
A_{y}
=
f
(
x
,
y
), the equation for
is
Then substituting
A_{y}
,
, and
into Eq. (11),
Eq. (12) is not the only expression that makes the amplitude and phase obey Eqs. (6), (10) and (11) simultaneously.
III. DISCUSSION
In section II we obtained distributions of amplitude and phase around the Cpoints Lemon, MonStar, and Star. We note that the work of
[7]
is a theoretical description of optical beams carrying isolated polarization singularity Cpoints. The PSs are formed by the superposition of a circularly polarized mode carrying an optical vortex and a fundamental Gaussian mode in the opposite state of polarization. By varying two parameters, Cpoints including asymmetric Lemon, MonStar, and Star can be generated. Compared to
[7]
, in this paper the electromagnetic waves are written in the more general terms of the Jones matrix, i.e. PSs are formed by the superposition of two linearly polarized modes that are orthogonal to each other. We analyze distributions of the amplitude and phase differences of the two polarized modes. According to the analyses, all three kinds of PSs can be generated (as shown in section IV). Moreover, the positions of the curves where the azimuthal angles of polarization ellipses are 0°, 45°, and 135° can be controlled.
For the Lemon, the amplitude must follow Eq. (6), while the phase must follow Eq. (10). These two formulas are independent constraints on the amplitude and phase, respectively. Thus there is a high probability to obtain amplitude and phase obeying Eqs. (6) and (10), which means generation of Lemon is relatively easy in experiment, such as the interference field of vector waves. As described above, MonStar is a special Lemon for which amplitude and phase are also restricted by Eq. (11), when following Eqs. (6) and (10) respectively. Eq. (11) is a function of both amplitude and phase, which means sophisticated control over phase (or amplitude) is needed to generate MonStar. However, in previous studies aiming to generate PSs
[8

13]
, there has been almost no purposeful control over amplitude and phase around Cpoints. More constraints on amplitude and phase also illustrate that there exists a smaller probability to generate MonStar than Lemon in experiments. As MonStar, the constraints for generating Star imply that it is also very rare. However, in many studies, fields with Stars are very common.
To explain this phenomenon, we consider a point
Q
moving on
σ
anticlockwise from the −
y
axis. Taking the Lemon in
Fig. 2(d)
for example, the angle Θ
_{Q1}
between
and the +
x
axis and the azimuthal angle Θ
_{Q2}
of the ellipse at
Q
are analyzed.
Fig. 3
is a graph of Θ
_{Q1}
and Θ
_{Q2}
with respect to the azimuth
θ_{Q}
of
Q
. The dashed red and solid blue lines are Θ
_{Q1}
(
θ_{Q}
) and Θ
_{Q2}
(
θ_{Q}
) respectively. The points
Q
making Θ
_{Q1}
(
θ_{Q}
) and Θ
_{Q2}
(
θ_{Q}
) intersect are places where the major axes of the ellipses point to
O
. Because the radius of
σ
is very small, the transmission line between
O
and
Q
is a ray emanating from the Cpoint. That is, the number of intersections of Θ
_{Q1}
(
θ_{Q}
) and Θ
_{Q2}
(
θ_{Q}
) determines the number of lines of the PS.
Fig. 3(a)
is the relationship for the Lemon, where Θ
_{Q1}
and Θ
_{Q2}
intersect at only one point,
θ_{Q}
=−
π
/2 (3
π
/2), which means there is only one line emanating from the Cpoint for the Lemon.
Fig. 3(b)
is the relationship for the MonStar. The two lines in
Fig. 3(b)
have the same tendency as in
Fig. 3(a)
, except that there are two other intersections, so there are three lines emanating from the Cpoints for the MonStar. The contribution of Eq. (11) is to modulate the slope of Θ
_{Q2}
(
θ_{Q}
) and make it intersect with Θ
_{Q1}
(
θ_{Q}
) at two additional points. This also illustrates the necessity of Eq. (11) for generating MonStar.
Fig. 3(c)
is the relationship for the Star. The solid blue line illustrates that Θ
_{Q2}
is a decreasing function, so Θ
_{Q2}
and Θ
_{Q1}
still intersect in three points, even if Eq. (11) is absent. Eq. (11) is not a necessary condition for generation of the Star; this is why the presence of Stars in experiments is very common. However, for specific Stars, such as when
θ
_{OL1}
and
θ
_{OL2}
need to be specific values, the condition of Eq. (11) is indispensable.
Graphs of Θ_{Q1} and Θ_{Q2} with respect to the azimuth θ_{Q} of Q. The dashed red curve is the angle Θ_{Q1} between and the +x axis, while the solid blue curve Θ_{Q2} is the azimuthal angle of the ellipse at Q. (a), (b), (c) show respectively the relationships for Lemon, MonStar, and Star.
Now we have seen that Eqs. (6) and (10) are necessary conditions for generating Lemon. Lemon can transform into MonStar while the amplitude and phase also follow Eq. (11). For generating Star, Eqs. (6) and (10) are also necessary conditions, while Eq. (11) is not. According to Eqs. (6) and (10) and
Figs. 2(d)
and
(f)
, the amplitudes of Lemon and Star have the same distribution, while the phase differences have opposite distribution trends. Thus Lemon (Star) can transform into Star (Lemon) if the phase difference of light is modulated by transmission media, such as anisotropic media or a spatial light modulator (SLM).
IV. SIMULATION
The purpose of this section is to simulate some specific PSs, such as
and
with given values. The construction of amplitudes and phases according to sections II and III is a crucial step in generating the desired PSs. We consider the polarization ellipses in a cross section of dimensions 2
b
×2
b
. For convenience but without loss of generality, we assume that the component
E_{x}
is a plane wave, i.e.
A_{x}
= 1 and
α_{x}
= 0; then Δ is equal to
α_{y}
. Following the distribution formulas for amplitude and phase, functions that are monotonic in the studied cross section can be very simple and useful.
For generating the Lemon, the simplest case is when
,
,
, and
coincide with −
y
, +
x
, +
y
, and −
x
, respectively. One set of expressions for this case is
A_{y}
= −
y
/
c
+1, Δ=
π
(
x
/
d
+1)/2 (
c
≥
b
,
d
≥
b
). Here, we present an asymmetric Lemon whose
has an angle of 60° with the +
x
axis, shown in
Fig. 4
. The background represents the light intensity, and the green ellipses show the distribution of polarization states.
is the only line of the Lemon. The azimuthal angles of ellipses on
is 0°. The rays
and
, where the azimuthal angles of ellipses are 135° and 45°, are also marked with dashed yellow lines. The amplitude used to generate
Fig. 4
is
A_{y}
= −
y
/
c
+1 (
c
≥
b
), while the phase difference is shown in
Fig. 5
. The phase difference is divided into two regions by a line
y
= tan(−15°)･
x
: the pink region, denoted as Δ
_{1}
, and the black region, denoted as Δ
_{2}
, expressed by
Simulation of Lemon. The background represents the light intensity. The green ellipses show the distribution of the polarization states. makes an angle of 60° with −x.
Phase difference used to simulate the Lemon shown in Fig. 4. The phase difference is divided into two regions by line y = tan(−15°)･x: the pink region, denoted as Δ_{1}, and the black region, denoted as Δ_{2}. The phase difference is continuous.
where
. Although Δ
_{1}
and Δ
_{2}
have different expressions, we emphasize that they are equal to each other at the boundary
y
= tan(−15°)･
x
, which means the phase difference shown in
Fig. 5
is continuous. Eq. (13) is not the only formula that follows Eq. (10).
To generate MonStar, we can set the amplitude as
A_{y}
= −
y
/
c
+1 (
c
≥
b
) in the case of
and
. Then we simulate the PS shown in
Fig. 6
by substituting
and
into Eq. (12). In
Fig. 6
,
and
coincide with −
y
and +
y
, and
and
make angles with +
x
as expected. This certifies the correctness and usefulness of Eq. (12) for generating PSs with
and
of given values.
Simulation of MonStar. , , , and coincide with ^{–}y, +x, +y, and –x respectively. The rays and respectively make angles of –2π/3 and –π/3 with +x.
We know that the two conditions of Eqs. (6) and (10) determine the generation of a Star, while another formula, Eq. (11), is needed to generate a specific Star.
Figure 7
is a Star with
A_{y}
= −
y
/
c
+1 and Δ=
π
(−
x
/
c
+1)/2 (
c
≥
b
,
d
≥
b
). These distributions of amplitude and phase do not follow Eq. (11); however, we find that
Fig. 7
is still a Star. Here, we note that the curves
and
are no longer linear. This phenomenon raises a doubt: Must the curves emanating from the Cpoint be linear? We do not discuss this question here.
Simulation of Star with amplitude and phase following Eqs. (6) and (10) respectively. , , and coincide with ^{–}y, +x, +y, and –x respectively. The curves and are no longer linear.
There is a special case, namely that of
and
, in which Eq. (11) can be rewritten as
A_{x}
=
A_{y}
at
and
, which means
and
coincide with
and
respectively. Then the three constraints for generating a Star resolve into two constraints. In this case, Eq. (12) is no longer suitable. One expression for the amplitude and phase is
where
c
> 1 and
d
≥
b
.
Figure 8
is the simulation of a Star with
and
coinciding with
and
respectively. Actually, there is an analogous case in generating MonStar, in which
and
coinciding with
and
respectively
Simulation of Star with amplitude and phase following Eqs. (6), (10), and (11) simultaneously. and coincide with and .
V. CONCLUSION
By establishing a Cartesian coordinate system with origin at a Cpoint, and with the help of four curves where the azimuthal angles of polarization ellipses are 0°, 45°, 90°, and 135°, the distributions of amplitude and phase around Cpoints are analyzed and given. We also discuss the necessity of these constraints, which illustrates that there exists a smaller probability to generate MonStar than to generate Lemon or Star in experiments. The transformations of these three PSs are also discussed. Suitable modulation of phase difference can make Lemon (Star) transform into Star (Lemon). Compared to
[7]
, PSs with desired shapes can be generated using the constraints obtained in this paper. According to constraints on amplitude and phase, we construct suitable functions and simulate several particularly shaped PSs. These simulations certify the correctness of the analysis and discussions. With the development of modulation techniques for amplitude and phase, it is clear that the distributions of amplitude and phase around Cpoints are helpful to generate arbitrarily shaped Cpoints in experiments.
Color versions of one or more of the figures in this paper are available online.
Acknowledgements
This study was supported by a grant from National Science Foundation of China (61107011, 61077012, 11404086); The Ph.D. Programs Foundation of Ministry of Education of China (20123219110021); The Fundamental Research Funds for the Central Universities (30920130112005, J2014HGBZ0170, 2014HCGC0009).
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