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Figure 4.15: Polarization ellipse for a monochromatic plane wave.

the right thumb points in the direction of wave propagation then the fingers curl in the
direction of field rotation for increasing time. This is right-hand polarization (RHP). We
associate δ> 0 with left-hand polarization (LHP).
The polarization ellipse is contained within a rectangle of sides 2E x0 and 2E y0 , and
has its major axis rotated from the x-axis bythe tilt angle ψ, 0 ≤ ψ ≤ π. The ratio of
E y0 to E x0 determines an angle α, 0 ≤ α ≤ π/2:

E y0 /E x0 = tan α.

The shape of the ellipse is determined bythe three parameters E x0 , E y0 , and δ, while
the sense of polarization is described bythe sign of δ. These maynot, however, be
the most convenient parameters for describing the polarization of a wave. We can also
inscribe the ellipse within a box measuring 2a by 2b, where a and b are the lengths of
the semimajor and semiminor axes. Then b/a determines an angle χ, −π/4 ≤ χ ≤ π/4,
that is analogous to α:

±b/a = tan χ.
Here the algebraic sign of χ is used to indicate the sense of polarization: χ> 0 for LHP,
χ< 0 for RHP.
The quantities a, b,ψ can also be used to describe the polarization ellipse. When we
use the procedure outlined in Born and Wolf [19] to relate the quantities (a, b,ψ) to
(E x0 , E y0 ,δ), we find that

2
2
2
a + b = E 2 + E ,
x0 y0
2E x0 E y0
tan 2ψ = (tan 2α) cos δ = cos δ,
E 2 − E 2
x0 y0
2E x0 E y0
sin 2χ = (sin 2α) sin δ = 2 2 sin δ.
E x0 + E y0
Alternatively, we can describe the polarization ellipse by the angles ψ and χ and one of
the amplitudes E x0 or E y0 .

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