Polarization Handedness
Convention

10 July 2017

Erik Gentzsch

Polarization Handedness Convention
When light is elliptically polarized, the electric field (E field) vector rotates with respect
to a Cartesian coordinate system as it propagates. The PAX1000 and PAX Polarimeter
System software plot the E field vector with respect to a right-handed coordinate
system with x, y, and z axes. The light wave propagates in the +z direction, and the(virtual) observer looks along the -z direction towards the light source. The handednessof the elliptically polarized light describes the direction of rotation of the E field vector
as seen by this observer. This convention is common in the technical literature.
Each state of polarization can be split into two linearly polarized orthogonalcomponents, in which one is oriented in the x direction and one in the y direction. If
both components have equal magnitudes and the phase shift of the y component
relative to the x component is +/2 or -/2, the light is circularly polarized. The sign ofthe phase difference determines the handedness of the rotation. A clockwise rotation
corresponds to a right-hand circular polarization state and a phase shift of -/2, while
a counterclockwise rotation refers to left-hand circular polarization state and a phaseshift of +/2.

Figure 1 Right-Circularly Polarized Light

Figure 2 Left-Circularly Polarized LightIn Figure 1 and Figure 2, the projection of the rotating E field vector on a virtual screen
generates a circle over time as the E field vector rotates clockwise (counterclockwise)
representing right (left) circular polarization.

Polarization Handedness
Convention

10 July 2017

Erik Gentzsch

Generation of Circularly Polarized LightA circular polarization state can be generated using linearly polarized light and aquarter waveplate. When the E field vector of the incident linearly polarized light isoriented at a 45° angle to the slow and fast axes of the quarter waveplate, the output
light is circularly polarized. It is convenient to mathematically describe thistransformation using matrix algebra, with Jones vectors representing the polarized lightand a Jones matrix representing the quarter waveplate.
The Jones matrix, 𝑀𝜆/4 , describing a quarter waveplate with its slow axis oriented alongthe x (horizontal) axis is :𝑀𝜆/4 = 𝑒 𝑖𝜋/4 [1 0]0 −𝑖The phase factor 𝑒 𝑖𝜋/4 can be omitted in almost all cases.
The Jones vector, 𝐽⃗+45 ,of a linear polarization state oriented at + 45° is written as:𝐽⃗+45 =

1 1[ ]√2 1

When this linearly polarized light travels through a quarter waveplate, the Jones vector,
𝐽⃗𝑅𝐻𝐶 , of the output light can be calculated as follows:
1 1
1 0 1 1𝐽⃗𝑅𝐻𝐶 = 𝑀𝜆/4 𝐽⃗+45 = [][ ]=[ ]
0 −𝑖 √2 1√2 −𝑖
The output polarization is right-hand circular.
This is illustrated by Figure 3, in which the slow and fast axes of the quarter waveplate
are aligned with the x- and y-axes, respectively, of the coordinate system. The purplevector at the origin indicates the orientation of the incident 45° linearly polarized light,
while the red and blue vectors represent this E field decomposed into its orthogonalcomponents. These two components are in phase. The x-axis component, which isrepresented by the blue vector, is aligned with the slow axis of the waveplate andtravels at a slower velocity through the waveplate than the y-axis component, which isaligned with the fast axis and represented by the red vector. Propagating through thewaveplate retards the phase of the x-axis component with respect to that of the y-axiscomponent. The amount of the retardation is determined by the thickness of the
waveplate, and quarter waveplates are designed to give a phase shift of -/2. This
retardation produces right-hand circularly polarized output light, whose E field vector
rotates clockwise as it propagates along the z-axis.

Polarization Handedness
Convention

10 July 2017

Erik Gentzsch

Figure 3

The input polarization is +45° linear and a right - hand circular polarization
state is generated. The slow axis of the quarter waveplate is oriented along
the x – axis.

When the incident beam has a -45° linear polarization, its Jones vector, 𝐽⃗−45, is:11𝐽⃗−45 = [ ]√2 −1

The output polarization state, 𝐽⃗𝐿𝐻𝐶 , is then expressed:
1 11 0 1 1
𝐽⃗𝐿𝐻𝐶 = 𝑀𝜆/4 𝐽⃗−45 = [][ ]=[ ]
0 −𝑖 √2 −1√2 𝑖
The output light is left-hand circularly polarized.

Polarization Handedness
Convention

10 July 2017

Erik Gentzsch

Figure 4 illustrates this case. The waveplate is oriented as in Figure 3, and the purple
vector at the origin again represents the orientation of the incident linearly polarized
light. As this is linearly polarized light, there is no phase difference between the x- andy-axis components of the E field. The -45° orientation of the polarization vector means
means it is azimuthally rotated. The red vector points in the +y direction and the blue
vector points in the –x direction. Propagating through the waveplate causes thecomponent aligned along the slow axis to add a phase shift of +/2 with respect to thecomponent aligned along the fast axis. The output light is left-hand circularly polarized
and has an E field vector that rotates counterclockwise as it propagates along the zaxis.

Figure 4

The input polarization is -45° linear and a left - hand circular polarization stateis generated. The slow axis of the quarter waveplate is oriented along the x –axis.

Polarization Handedness
Convention

10 July 2017

Erik Gentzsch

Jones Vectors and Matrices
The following tables contain the Jones vectors for selected polarizations of light andthe Jones matrices of standard optical elements, respectively.
State of Polarization
Horizontal Linear Polarization
Vertical Linear Polarization
Linear Polarization at +45°
Linear Polarization at -45°
Right Circular Polarization
Left Circular Polarization
Table 1

Jones Vector for Selected Polarization States
Linear optical Element
Horizontal Linear Polarizer
Vertical Linear Polarizer
Linear Polarizer at +45°
Linear Polarizer at -45°

Quarter Wave Plate, Slow Axis
Horizontal
Quarter Wave Plate, Slow Axis Vertical
Table 2Jones Vector
1[ ]
00[ ]
11 1[ ]
√2 1
1 1[ ]√2 −1
1 1[ ]√2 −𝑖
1 1[ ]√2 𝑖
Jones Matrix
1 0[]
0 00 0[]0 1
1 1 1[]
2 1 1
1 1 −1[]2 −1 11 0𝑒 𝑖𝜋/4 []0 −𝑖
1 0𝑒 𝑖𝜋/4 []0 𝑖

Jones Matrices for Standard Optical Elements

Further Reading
1. Serge Huard, Polarization of Light, Wiley, New York, 1997.
2. Eugene Hecht, Optics, Addison Wesley, New York, 2016.
3. Christian Brosseau, Fundamentals of Polarized Light: A Statistical OpticsApproach, Wiley, New York, 1998.