Download 7.8 Polarized light - one more excursion into optics 7.8.1 The

7.8 Polarized light - one more excursion into optics
7.8.1 The polarized plane wave
The electric field vector E(r, t) is observable in the real world, depending on space vector r and time t, and
thus we find it useful to write E as a real quantity
E(r, t) = i E 0 e 6 e i(kr−ωt) − e ∗ 6 e −i(kr−ωt) .
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Here E 0 is the field amplitude, e the unit polarisation vector (see below) and k the wave vector which is
related to wavelength λ and angular frequency ω by
k = 2π = ω
c .
λ
We note that in the literature, for convenience, often only one complex exponent is retained and we too
shall use this abbreviation whenever it appears appropriate. However, when describing optical transitions
both terms have their specific significance.
7.8.2 Unit polarization vector and linearly polarized light
Above the vector character of the field is cast completely into the unit polarisation vector e, which may be
complex. As a basis for the polarisation vector one may choose the unit vectors in the x, y and z
directions which we call e x , e y and e z , respectively, and which as column vectors are written as
1
ex =
0
0
, ey =
1
0
0
, ez =
0
0
1
with e x e y = e x e z = e z e y = 0.
Since light is transversely polarised it is usually convenient to choose a coordinate frame, say with its z
axis parallel to the direction of the light propagation, i.e. parallel to k. Any polarisation vector is then
described by a suitable combination of e z and e y .
To give some examples choose e = e z , which describes light linearly polarised in the z- direction and
insert in above equation.
sin(kr − ωt)
E(r, t) = −E 0
0
0
while linear polarisation in the z-direction e = e y gives
0
E(r, t) = −E 0
sin(kr − ωt)
0
7.8.3 Unit polarization vectors: circularly polarized light
An alternative and often more appropriate choice is the spherical basis (helicity basis):
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e +1
= −1 (e x + ie y ) = −1
2
2
1
i
0
0
e0 = ez =
0
1
1
e −1 =
1 (e x − ie y ) =
2
with e ∗q 6 e q v = δ qq v . The inverse relations are: e x =
−1
2
1
2
= −e ∗+1
−i
0
(e +1 − e −1 ) , e y =
i
2
(e +1 + e −1 ).
The unit polarisation vector e = e +1 describes circularly polarised σ + light as can be seen by inserting this
into E(r, t) = 2i E 0 e 6 e i(kr−ωt) − e ∗ 6 e −i(kr−ωt) :
E(r, t) =
i E
2 0
e 6 e i(kr−ωt) − e ∗ 6 e −i(kr−ωt)
E(r, t) =
−i E
0
2 2
1
1
6 e i(kr−ωt) −
i
0
i
∗
6 e −i(kr−ωt)
0
sin(kr − ωt)
E(r, t) =
1 E
0
2
cos(kr − ωt)
0
A vector illustration of this σ + light is shown in the figure where the E-vector at a fixed time t = 0 is
depicted along the x axis:
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As indicated, for a fixed point in space, E rotates clockwise around the light propagating vector, i. e. with
positive helicity, as the time increases.
Traditionally this light is called left-hand circularly polarised (LHC). Thus the terminology LHC refers to the
viewpoint of an observer looking into the light beam which propagates towards him; a position which is
highly dangerous when dealing with laser beams! Conversely, right hand circularly polarised light (RHC)
also called σ − light, has negative helicity, is described by e = e −1 and is explicitly written as:
sin(kr − ωt)
E(r, t) =
1 E
0
2
− cos(kr − ωt)
0
As is well-known, circularly polarised light can be composed by two sources of light linearly polarised
perpendicular to each other, with an appropriate phase difference.
7.8.4 Making circularly polarized light, λ/4 plate
It appears appropriate at this point to address some experimental aids for manipulating polarised light.
We first discuss the production of circularly polarised light from a linearly polarised laser beam.
The standard tool is a λ/4 plate. This is a plane parallel birefringent crystal plate (usually of quartz or
magnesium fluoride), i. e. , it has two different indices of refraction n f and n s > n f for light with its electric
vector coinciding with two orthogonal directions, the so-called fast (f) and slow (s) axes. The phase
velocity for light propagating perpendicular to these axes is v f = c/n f and v s = c/n s , respectively, with
v f > v s . Thus the wavelengths for the fast and slow directions within the crystal differ, λ f > λ s and the two
perpendicular components of the electric vector of a light beam propagating through the plate develop a
phase difference. Specifically, for a λ/4-plate the phase difference is 90 o and thus the thickness of the
plate § is given by
§ ns − nf = λ .
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We see that a λ/4-plate is much thicker than λ/4 (else it would be a very fragile object) since usually
n s − n f ¸ 1. The same effect is achieved by a plate of thickness 5§, 9§, etc. and one speaks of first and
higher order λ/4 plates. Normally, the equation holds strictly for only one specific wavelength,which
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somewhat limits the use of λ/4 plates. Here is an illustration:
linearly polarised light is incident perpendicular to the plate and the r vector is aligned at an angle of 45 o
between the fast and the slow axis. It may then be thought to consist of two components E ξ and E η
parallel to the two crystal axes. The figure shows how the optical path difference λ/4 on exit arises from
the different wavelengths for the two perpendicular components. From the outgoing components one may
reconstruct the total electric field which is seen to rotate in the sense indicated in tje figure. This
corresponds to our previous definition for σ + (LHC) light. Conversely, right circularly polarised σ − light is
obtained by aligning the E vector of the incident linearly polarised light at an angle of −45 o with respect to
the fast x-axis. Formally this is obvious by introducing a −π/2 phase shift into the E y component in the
defining equations with δ = ±45 o .
7.8.5 Fresnell rhomb
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Quarter wave plates have the disadvantage of being specifically cut for one particular wavelength.
Alternatively one may use the phase difference introduced by total reflection inside prisms. One particular
device is a so-called Fresnel rhomb, schematically illustrated in the figure, where two total reflections each
giving a phase difference of 45 o are used. For a refractive index n = 1. 5 the rhomb angle is 54 o 37 v . Also
shown in the figure is the sense of rotation of circularly polarised light for the two possible directions of the
incident E vector. One sees that with the above rule, the slow axis of this type of λ/4-plate is the vertical
direction of the rhomb. The Fresnel rhomb has the great advantage of being applicable in a broad
wavelength range. It has, however, the disadvantage of displacing the beam axis. Thus, it is most often
used in combination with a second Fresnel rhomb as a λ/2-plate, where the displacement is then
compensated.
7.8.6 Half wave plates
Half wave plates, made of two Fresnel rhombs or from a birefringent crystal with thickness §
§ ns − nf = λ
2
are also very useful objects and may be utilised for rotating the plane of polarisation of linearly polarised
light as illustrated in the figure. The fast axis of the λ/2-plate is assumed to be inclined by an angle α with
respect to the E vector of the incident light beam. As indicated in the left part of the figure one may again
consider E to be composed of two components along the fast and slow axes of the crystal. After passing
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through half of the plate, i. e. through the equivalent of a λ/4 plate, the beam will be elliptically polarised,
as indicated in the middle of the figure. The total phase shift after passage through the λ/2 plate is π, i. e.
the electric vector along the slow axis is reversed with respect to the fast component. Thus, as illustrated
on the right-hand side of the figure, the E vector is rotated through an angle δ = 2α. A full rotation of the
λ/2-plate through 2π rotates the electric vector through 4π, i. e. it will be parallel to its incident direction
four times. An alternative use for a λ/2-plate is to change the sense of orientation for circular polarised
light. The phase change of π clearly makes σ − light from σ + and vice versa.
7.8.7 Soleil-Babinet compensator
One may construct (and obtain commercially) plates which produce a variable phase shift by allowing two
wedges made of a double refracting crystal to slide on each other, thus varying the thickness of the whole
setup. Since such an arrangement has a considerable mechanical thickness one compensates the phase
difference between fast and slow components by a second plate of the same material in which the fast
and slow axes are interchanged. This so-called Soleil-Babinet compensator is depicted in the figure. Here
one obtains a total optical path difference
∆§ = n s − n f 6 (h I − h II )
where h I and h II are the thicknesses of the two plates.
It is obvious that ∆§ may be varied from positive to negative (typically from −λ/4 to 2λ) and is constant over
a large, useful area of the plate surface, as is necessary for manipulation of extended beams. Finally, we
mention that for many applications it is useful if the phase difference in such plates may be varied
electrically. The electro-optical double refraction effect in certain crystals like ADP allows us to achieve
this goal. Arrangements of this type are known as Pockels cells and are commercially available.
7.8.8 Beth’s experiment: Spin of the photon
N ph
I6A
I6A
Torque: T = ∆L
t =
t m j ¤ = 2 ¤ω m j ¤ = 2 ω m j for one transition of the photon beam (intensity I,
angular frequency ω) through area A. Torque is doubled by return beam. We can measure torque by
twisting the wire ϕ = D ϕ T and we find D ϕ from eigenfrequency of the disc: ν disc =
1
2π
Dϕ
MR 2 /2
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and m j =
ϕ ω
.
D ϕ I6A
7.8.9 The photon spin
Result of Beth’s experiment: m j = 1 for σ + and m j = −1 for σ − .
The photon has angular momentum quantum number j = 1 (Photonspin ¤) but only two orientiations are
possible:
LHC light (σ + ) spin orientation parallel to k (m j = 1) – positive helicity
RHC light (σ − ) spin orientation antiparallel to k (m j = −1) – negative helicity
The eigenstates are |e + ) and |e − ) (matrix/vector representation)
The component with m j = 0 is unphysical (photon propagates in k direction, rest mass is zero).
Linearly polarized light can be written as:
|e x ) =
−1
2
[|e +1 ) − |e −1 )] and |e y ) =
i
2
[|e +1 ) + |e −1 )].
The latter states are not eigenvectors of J z and < J z >= 0.
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