Platform for Manipulating Polarization Modes Realized with Jones Vectors in MATHEMATICA

Journal of Astronomy and Space Sciences.
2015.
Jun,
32(2):
151-159

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Received : March 01, 2015
- Accepted : April 08, 2015
- Published : June 15, 2015

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The fundamental conception in physics of the propagation of the electromagnetic wave polarization in matter is newly understood as the cardinal keyword in free-space quantum communication technology and cosmology in astrophysics. Interactive visualization of the propagation mechanism of polarized electromagnetism in a medium with its helicity has accordingly received attention from scientists exploiting the protocol of quantum key distribution (QKD) to guarantee unconditional security in cryptography communication. We have provided a dynamic polarization platform for presenting the polarization modes of a transverse electromagnetic wave, converting the state of polarization through the arrangement of optical elements, using Jones vectors calculations in
Methematica
. The platform graphically simulates the mechanism of production and propagation of the polarized waves in a medium while satisfying Maxwell's equations.
and magnetic field vector
of the electromagnetic radfield vectoriation always oscillates parallel to a fixed direction in space. Light of such character is said to be linearly polarized, and maintains a constant direction of oscillation, and does vary spatially in a regular manner, producing either elliptically polarized or circularly polarized light (
Fowles 1975
;
Jackson 1975
;
Pedrotti & Pedrotti 1987
). Polarization characteristic have been used in radio transmission to reduce interference between channels, particularly at VHF frequencies and beyond (
Masayoshi 2007
;
Yao et al. 2007
). Free-space communication has forced the use of circular polarization, which has the fundamental advantage of precluding disturbance from the reflections of signals (
Leitch et al. 2002
;
Elser et al. 2009
). Free-space optical (FSO) communication utilizes a spatial diversity receiver to receive the binary signals, which are modulated by two circular polarizations. FSO communication employing a binary polarization shift keying coherent modulation scheme are utilized in atmospheric turbulence channel (
Tang et al. 2010
) and free-space quantum key distribution (QKD) by rotation-invariant twisted photons are used to guarantee unconditional security in cryptographic communication (
Vallone 2014
). On 17 March 2014, astronomers from the Harvard Smithsonian Center for Astrophysics announced their detection of signature patterns of polarized light in the Cosmic Microwave Background (CMB) (
Ade 2014
;
Calvin 2014
;
CfA 2014
). The team hunted for a special type of polarization called ‘B-mode’ which represents a twisting or ‘curl’ pattern in the polarized orientations of the ancient light. This is the strongest confirmation yet of the cosmic inflation theory (
Boyle 2006
). Gravitational waves squeeze space as they travel, and the squeezing produces a distinct pattern in the CMB. Gravitational waves have a “handedness” much like light wave and can have left- and right-handed polarization.
In the Standard Model, the weak bosons (W±, Z) mediate the weak interactions between different flavors (all quarks and leptons). Experimental results have shown that all produced and observed neutrinos have left-handed helicity, and all antineutrinos have right-handed helicity (
Aad et al. 2012
;
CfA 2014
). The helicity of the elementary particle could be a keyword to determine the Standard Model.
The scope of application of polarization has expanded explosively this decade with the development of communication technology. Accordingly, it is fundamentally important for application of a manipulation scheme to understand the physics of polarization’s conception and the process of producing polarization modes. Presently the definition of polarization has been modified, and its nomenclature upgraded, which can be confusing to students of physics and researchers. For instance, there are similar customary nomenclature for the right circularly polarization: right circularly (
Fowles 1975
;
Jackson 1975
), right-circularly (
Pedrotti & Pedrotti 1987
), right-hand circularly (
Reitz et al. 1993
), and right-handed (
Born & Wolf 1999
;
Georgi 1982
) circularly polarization. Right circularly polarization and right-handed polarization are different types of polarization, although with similar names. Recently, the visualization of polarization propagating in matter has drawn physicist's attention for potential applications in modern physics and information technology (
Tamm 1997
;
Mooleskamp & Stokes 2015
;
Yun & Choi 2013
).
We have provided a dynamic polarization modes platform for simulating polarization modes with Jones matrices calculations, corresponding to the physical arrangement of optical elements, in a
Mathematica
computing environment (
Mathematica 2015
).
and
of the electromagnetic waves interact with the electrons in a solid (
Pedrotti & Pedrotti 1987
;
Fowles 1975
). In particular, although the electromagnetic waves are an harmonic plane wave, the fields may pull or push the electrons in the orbital of a solid, which is responsible for inducing dipole moment
and magnetization
im the solid. If we assume that the medium is not a magnetic material and
, the Helmholtz wave equation in a solid (
Fowles 1975
;
Jackson 1975
) is
where
. If we suppose the EM wave will be a form of solution such as
, we can rewrite Eq. (1) using a wave vector
It then becomes a vector equation for
, which indicates that the propagation process varies with the component of electric susceptibility
x
. Therefore, we will write the
solution of Eq. (2) in a solid as an inhomogeneous plane wave solution below (
Pedrotti & pedrotti 1987
;
Fowles 1975
;
Jackson 1975
)
Aa shown above,
field vector may be presented as the Jones vector (
Jones 1941
) with a complex vector amplitude {A, B ± iC} oscillating in an inhomogeneous plane. The wave vector
and form a real mutually orthogonal unit vectors
. Here
is a complex wave vector and N = n + iκ is the complex refraction index. For Eq. (3) to be a solution of Eq. (2) as a homogeneous plane harmonic wave, it should be,
. Then we get the relations:
, and
, which result in propagation speeds that are different along the direction in the medium. Therefore, there will be a cumulative phase difference Δ
ε
between the two components of the
field vector as they emerge in uniaxial crystals (Quartz, Calcite, etc.). After the wave has traveled a distance
d
, the phase difference is
between E
_{x}
wave and E
_{y}
wave when the radiation is propagating along the
direction. If the Δ
ε
=0 while the amplitude of
is real, the vector is responsible for the linearly polarized such as Jones vector {A, B}, otherwise the amplitude is the complex vector responsible for the elliptically polarized Jones vector as {A, B±C}. Specifically, if B=0 and A=C then the wave is a circularly polarized Jones vector such as {1, i}.
A quarter wave plate is a thin birefringent crystal the thickness of which has been adjusted to produce a ±
π
/4 phase difference between the ordinary and extraordinary rays at the operating wavelength. We desire a matrix that will transform the element E
_{0x}
e
^{iφx}
into E
_{0x}
e
^{i(εx+φx)}
and E
_{0y}
e
^{iφy}
into E
_{0y}
e
^{i(εy+φy)}
. The general form of a matrix representing a phase retarder will transform the elements by the matrix operation as follows
where ε
_{x}
and ε
_{y}
represent the advance in phase of E
_{x}
- and E
_{y}
- component of the incident light. As an example, consider a quarter-wave plate (QWP) which makes Δ
ε
=
π
/2. We may write the Jones matrix, M transforming the Jones vector which makes Δ
ε
=
π
/2 to produce right-handed circular polarizing light as
This is the case of fast axis vertical (FA vertical). Similarly, we can determine the corresponding Jones matrix for a half-wave plate (HWP) or eighth-wave plate (EWP) or arbitrary phase of retarded. Jones matrices derived for various wave plates are summarized in
Table 1
.
†Wave plate(V/H) stand for the fast transmission axis of vertical/horizontal.
We desire now to create a new simulation presenting a means of producing polarizing modes from a Jones calculation corresponding to the physical arrangement of optical elements in the
Mathematica
computing environment. In a plane wave, the electric field vector
always oscillates parallel to the fixed direction in space. Light of such character is said to be linearly polarized. If the linearly polarized light passes through a quarter-wave plate, elliptically polarized light emerges. The same can be said of the magnetic field vector
, which maintains an orientation perpendicular to the electric field vector such that the direction of
is everywhere the direction of wave propagation. Thus, the possibility of polarizing light is essentially due to its transverse character. Therefore, the
Mathematica
simulation should show the transverse character of the
with its vectorial behaviors dynamically satisfying Maxwell's wave equations. In addition to the transverse character of the polarizing wave, the helicity of a polarized wave is a critical factor in quantum cryptography communication technology or cosmology in moderen physics.
Mathematica
to implement interactive Jones matrix calculations and animations for the generation and propagation of the polarization modes in th solid state. First, we desire to confirm that the complex vector field
of Eq. (3) with Jones vectors in the matter satisfy the Maxwell's vector equations Eqs. (6) and (7) in both numeric and graphic simulations.
where,
,
, and
ard both complex vectors and
,
and
are real vectors with
where
is a magnetic intensity vector in the matter. We examine the process of generating polarzation modes with the normalized Jones vectors calculations in the
Mathematica
simulation as below. The
Mathematica
input code is shown below:
Mathematica
returns calculations;
The
Mathematica
calculations show the emerging polarization mode through the physical arrangement of optic elements, which promptly confirm those Jones vectors from the animating platform. For instance, in In[23]:= E0*LP∙QWV/.
α
->1/4
π
produces Out[23]:= {I/2, I/2}, that is the right-handed circularly polarized light (RHCP) as shown in
Fig. 2
. From the arrangement of linearly polarizers in right angle no wave emerging shown as Out[29]:= {0, 0} while the Out[28]:={0.341506, 0.591506} which show the linear polarization (see
Figs. 3(g)
and
3(h)
). This enables us to switch on or off the polarization by the combination of arrangements of polarizers and wave plates.
Mathematica
(
Wolfram 2015a
). We can determine the helicity based on a calculation of the phase shift in the block respectively; phase shifts of the E
_{1}
=E0*Jv after passing through the polarizer and of E
_{2}
=Jm∙E
_{1}
after the wave plate respectively. For evaluation of the helicity of the polarized waves in progression, we have provided a
Mathematica
module, helicity [JvC_,JmC_]:=Module[{E1, E2, dp11, dp12, dp13, dp21, dp22, dp23}]. If you type in two numbers assigned to the polarizer JvC and wave plate JmC in the module, then helicity [JvC, JmC] will return helicity with a list of phase shifts. The outputs of the helicity [JvC, JmC] are shown as below:
The helicity[1, 2] returns two helicities with two lists of three phase differences in the two divisions for the optical device arrangement of linearly polarizer (JvC=1) and quarter-wave plate (JmC=2). The helicity[2, 7] returns the helicities of the optical arrangement of left-circularly polarizer (JvC=2) and eighth-wave plate (JmC=7). It needs to be noticed that the three phase differences are all 1.5708 (
π
/2) of the helicity[1, 2] which interprets the vertical component E
_{2z}
lead continuously the horizontal component E
_{2y}
with a constant phase shift (
π
/2) on the yz plane of
field vector propagating in x
_{1}
direction perpendicular to this plane with velocity
ω
/k. This results in
vector rotating in a counter colckwise direction
around the advancing x
_{1}
direction. That is, if we grasp our finger along the spin direction with right hand, then thumb directs the advancing direction of propagation wave, hence the wave is right-handed polarized wave regardless of viewer . This is very helpful in confirming the helicity of the polarized wave on the end block of the platform as shown in
Figs. 1
and
2
. We can correctly determine the helicity of the polarized wave in the E2 block by seeing the spin direction of the wave; either the spin direction is parallel
or anti parallel
to the advancing x
_{1}
direction (momentum direction). Helicity of E
_{2}
can changed simply by the choice of JmC optical element like as In[83] and In[84] codes. The half-wave plate (HWH) shifts phase -
π
in helicity[2, 5] while eighth-wave plate (EWH) shifts phase -
π
/4 in helicity[2, 7]. Furthermore we may easily calculate the helicities of the wave train on the helicity for any combination of optical elements in
Mathematica
, so that we can predict exactly helicity of the prod[JvC, JmC] moduleucing polarization modes. For more information for the helicity [JvC, JmC] module refer to citation (
Yun 2015
).
The starting platform of the dynamic polarization modes platform with Jones vectors (jdpmp). Unpolarized EM waves _{0} injected to the polarizer (LP) have not passed the polarizer yet. The simulation will run when you click the ▶ appearing while you spread the ⊕ of t1 panel of the platform.
Dynamic polarization modes platform (jdpmp). The picture shown is a snapshot of the right-handed circular polarized (RHCP) propagation EM wave train. Unpolarized EM waves _{0} injected into the linear polarizer (LP) with 45° to the horizontal transmission axis come from the vacuum and pass through the quarter-wave plate (QWPV), which result in the right-handed circular polarized EM wave train. This picture shows the E0*LP∙QWPV/.α ->1/4π process in the Jones vectors manipulation with a helicity of +1.
Graphics3D
in
Mathematica
, which simulates the polarizing modes dynamically while presenting the helicity of a running polarization mode. The platform manipulates three zones graphically using the Piecewise function of
Mathematica
depending on the polarizer and wave plate, with the Manipulate function of
Mathematica
. To present the transverse characteristic property satisfying Eq. (1) together with Eqs. (6) and (7), we used the Arrow function in
Mathematica
for drawing the vector array: Table[ Arrow [ { {x1, 0, 0}, {x1, Ey, Ez} } ] ] in the orthogonal {x1, Ey, Ez} coordinate system. In addition, the same was used for the magnetic field vector
calculated from the relation
of Eq. (6) also in solid.
Graphics3D
can draw the vector array of the
(blue) and
(red) fields of every point in the block as real polarization modes propagating in the advancing
direction, in our case, advancing in the x
_{1}
direction. This simulation scheme differs basically from that of an animation representing the envelope of the polarization propagation of
field only using the Animate function of graphic tools (
Harrison 2015
;
Mooleskamp & Stokes 2015
). Also, in this work, we simulated polarization process with vector fields satisfying wave equation and Maxwell's vector equations at every point during the process. Similar numerical method has been applied once to solve the Lagrange's equation in non-inertial frame (
Kim & Yun 2014
).
The platform starts by clicking the button ▶ on pop up ⊕ of the
t1
panel on the platform in
Fig. 1
. We can choose an incoming angle to the polarizer and rotate the polarizer to compose an arrangement of devices for the polarization mode indent shown as
Fig. 2
. While the program is running, we will change the polarizer or wave plate, with the changed animation running continuously. Even if the animation is stopped, we can change the configuration of the platform and observe the polarizing mode of the changed state.
Fig. 3
shows snapshots of three kinds of polarization modes.
Figs. 3a
and
3b
shows a circularly polarization mode,
Figs. 3c
and
3d
linearly polarization,
Figs. 3e
and
3f
elliptically polarization according to the arrangement of optical devices. If we utilize linear polarizer JmLinear [
β
] instead of a wave plate, then either it passes a linearly polarized wave or block in the angle
β
. While making right angles between two polarizers no wave passes through the polarizer plate as shown in
Figs. 3g
and
3h
. Thus, controlling of phase of polarized light waves is available for the phase encoding in the quantum key distribution (QKD) of the unconditional security in quantum communication technology (
Rubenhok et al. 2013
).
Producing different polarized modes by the different physical arrangements of polarizer and wave plate: (a) RHCP of helicity +1, (b) LHCP of helicity −1, (c) LP, (d) LP, (e) RHEP of helicity +1, (f) LHEP of helicity −1, (g) LP, (h) No wave.
Fig. 4
presents snapshots of three kinds of polarization modes viewed in the ViewPoint (200, 0, 0) against the propagation direction for confirmation of the transverse behavior and progression of the Poynting vector. It shows the orthogonal transverse of the propagating process of the waves and confirms the flow of Poynting vector. This is the visualization of the graphic version of
satisfying Maxwell’s vector equations Eqs. (6) and (7). The traces of both
and
are dynamically shown in the polarization modes at the edge of the
jdpmp
platform.
Snapshots of the polarized modes at the ViewPoint (200, 0, 0); (a) LP of Fig. 3c , (b) No wave of Fig.3h , (c) RHCP of Fig. 3a , (d) RHEP of Fig. 3e . It presents fields by the blue arrows and fields by the red arrows.
The platform will run faster or slower and stop and restart again by clicking the pop-up menu of the tl panel for more analytical observations while the polarization mode is running. During the platform stop, you will change to another mode only if you click a panel, then the changed mode will be presented automatically. At that time you can save the presenting mode or print out the mode status. While the platform is running, you can confirm the helicity by observing the spin direction of the field vectors along the orange trace of the helical propagation of the
and
vector fields at the edge of the platform. The helicities on the
helicity
panel are correct in the case of E1=E0*LP(α) manipulation on the
jdpmp
platform. The helicities of other arrangements of optical devices are calculated promptly on the helicity (JvC, JmC) module in
Mathematica
.
As far as we know, other than
jdpmp
, there is no instructive platform that simulates a transverse EM wave satisfying Maxwell's equation in the vectorial version of polarization modes (
Harrison 2015
;
Mooleskamp & Stokes 2005
;
Tamm 1997
). The complete version of jdpmp.nb program including helicity [JvC, JmC] module is available from the site (
Yun 2015
). If your PC has not installed
Mathematica
, you can run the jdpmp.cdf instead on the Wolfram CDF Player (
Wolfram 2015
).
jdpmp
, for simulating and producing polarization modes with the Jones calculations, corresponding to the physical arrangement of optical elements, in
Mathematica
. The platform animates the polarization process of the
vector field together with
using the Arrow vectors, so that the transverse EM wave advance in the propagation direction, satisfying Maxwell's wave equation at evey point on the advancing axes of the platform. Consequently, the vectors
,
,
(in that order) form a right-hand orthogonal set.
jdpmp
can be accomplished graphically in
Graphics3D
of
Mathematica
based on the numerical vector calculations in
Mathematica
. The platform can be manipulated dynamically and interactively by advancing the polarized mode while clicking the panels of the polarizer and wave plate using the Manipulate function in
Mathematica
, so that the program will continuously simulate the changed mode. While the program is running the helicity of the polarized mode is displayed on the panel. The module helicity [JvC, JmC] will return the helicity of the process on
Notebook
of
Mathematica
when you're typing in a polarizer and a wave plate. The module is helpful for students or researchers to inspect the phase difference or helicity of a polarized mode for various kinds of physics arrangements. We expect the platform
jdpmp
will be a useful starting platform for physics students and researchers to explore polarizations in science.

1. INTRODUCTION

Polarization is a coherent characteristic of the electromagnetic (EM) wave in a medium. In a plane wave, both the electric field vector
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2.MATHEMATICASIMULATION FOR THE CONVERTING POLARIZATION MODES

- 2.1 Electromagnetic waves in solids

The propagating process of electromagnetic waves in solids is different from that in the vacuum, since
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Summary of Jones vectors in the most common and Jones matrices of the wave plates and phase retarders (Pedrotti & Perotti 1987).

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- 2.2 Polarization modesMathematicasimulation

We have used
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- 2.3 Helicity of the elliptically polarized wave

The handedness of an elementary particle depends on the correlation between its spin and its momentum (
Goldhaver et al. 1958
). If the spin and momentum are parallel, the particle can be said to be right-handed or have a helicity of 1. If they are anti parallel, the particle is left-handed or have a helicity of -1. We may also adopt this definition to modern optics, since the circularly polarized electromagnetic wave is just a helical motion with helicity. The helicity of the polarized electromagnetic wave is a critical factor in modern communication technology and photonics (
Aad et al. 2012
;
Goldhaver et al. 1958
;
Rubenhok et al. 2013
). However, determination of the helicity is a perplex issue because we need to aware of the vectorial behaviors accurately unless we may observe propagating polarized wave (
Goldhaver et al. 1958
). Therefore, it is very helpful to simulate advanced polarization modes in the medium, and further, it would be more helpful to evaluate the helicity operator in a physics state.
Here we simply estimate the helicity with a calculation of phase shift in the process in the
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3. DYNAMIC POLARIZATION PLATFORM WITH JONES VECTORS IN MATHEMATICA

We have provided a dynamic polarization platform with Jones vectors (jdpmp) in the
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4. SUMMARY

We have provided a dynamic polarization mode platform,
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Acknowledgements

This research was partially supported by the ReSEAT program funded by the Korean Ministry of Education, Science and Technology through, the National Research Foundation of Korea and the Korea Lottery Commission grants.

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Citing 'Platform for Manipulating Polarization Modes Realized with Jones Vectors in MATHEMATICA
'

@article{ OJOOBS_2015_v32n2_151}
,title={Platform for Manipulating Polarization Modes Realized with Jones Vectors in MATHEMATICA}
,volume={2}
, url={http://dx.doi.org/10.5140/JASS.2015.32.2.151}, DOI={10.5140/JASS.2015.32.2.151}
, number= {2}
, journal={Journal of Astronomy and Space Sciences}
, publisher={The Korean Space Science Society}
, author={Choi, Yong-Dae
and
Kim, Bogyeong
and
Yun, Hee-Joong}
, year={2015}
, month={Jun}