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Tutorial
Light Polarization
In classical physics, light of a single color is described by an electromagnetic
field in which electric and magnetic fields oscillate at a frequency,
that is related to the wavelength, by the relation c = , where
c is the velocity of light. For example, visible light has wavelengths
in the range from 400-750 nm, while longer wavelength radiation (invisible
to the eye) is known as infra- red.
Polarized Optical Waves
An important property of optical
waves is their polarization: we will define a vertically polarized
("V") wave as one for which the electric field is restricted
to lie along the z-axis for a wave propagating along the x-axis, and
similarly a horizontally polarized ("H") wave will be defined
as one in which the electric field lies along the y-axis. It is a remarkable
property of light that any other polarization state of light propagating
along the x-axis can be resolved into a linear superposition of vertically
polarized and horizontally polarized waves with a particular relative
phase. In the case of linearly polarized light the amplitude of the
two components is determined by the projections of the polarization
direction along the V or H polarization axes. For instance, light linearly
polarized along the +45† direction in the y-z plane is an equal
amplitude, in-phase superposition of V and H, while light polarized
along the -45° direction in the y-z plane is an equal amplitude,
opposite-phase superposition. For obvious reasons V and H polarization
states (or +45° and -45° polarizations) will be referred to
as orthogonal polarizations, while two polarizations (such as V and
+45°) that have a non-zero projection will be called non-orthogonal
.
Crossed Polarizers
Light of a particular linear polarization can be produced by sending unpolarized
light through a polarizing medium (Polaroid) whose polarizing axis is oriented
along the direction of the desired linear polarization. When this light is
passed through a second polarizer, only the component polarized parallel to
the polarizing axis emerges, while the orthogonal component is absorbed. For
example if V light impinges on a polarizer oriented at +45° the emerging
light is reduced in amplitude by a factor of 2^-1/2 (one over the square root
of two), has the +45° polarization and an intensity (proportional to the
square of the amplitude) which is 50% of the incident intensity. Likewise,
if V light impinges on an H polarizer, no light emerges, and we refer to this
configuration as having "crossed polarizers."
Foundations of Information Science
These are the essential features of classical polarized light, but in quantum
cryptography we deal with very low intensity light where quantum mechanics
must be used. Specifically, during propagation such light has wave-like properties,
but on detection exhibits particle-like behavior: the optical energy is quantized
into indivisible units of size h, called photons (the elementary particles
of electromagnetic radiation) where h is a fundamental constant of Nature
known as Planck's constant. The indivisibility of photons raises the interesting
question of how a photon of polarized light behaves when it encounters a
polarizer. Clearly, there is no difficulty of interpretation if we are only
concerned with orthogonal polarizations: a V-photon would pass a V-polarizer
with certainty, but be absorbed by an H-polarizer with certainty, for instance.
However, once non-orthogonal polarizations are introduced the peculiarities
of quantum mechanics become evident.
Consider a +45°-polarized photon
impinging on a V-polarizer: there is no such object as a "half photon" by
analogy with the classical 50% transmission intensity.
Collapsing the Photon's Wave Function Instead, quantum mechanics predicts that there is a 50% probability that the photon will be absorbed,
and a 50% probability that it will be transmitted, with V-polarization, in
any given trial (experiment). Of particular relevance for quantum cryptography
is that beyond these probabilities we cannot predict how a particular +45°-photon
will behave at a V-polarizer. (Of course the result of a given experiment
is either a definite absorption or transmission, but with many repetitions
of the experiment we would build up a set of results reflecting the 50-50
absorption-transmission probabilities.) A further relevant quantum peculiarity
is that if a +45°-photon passes the V- polarizer it loses all of its "+45°-ness." Specifically,
if the emerging photon is made to impinge on a second polarizer oriented
at +45° it will be absorbed or transmitted with 50% probability in each
case, even though it was originally a +45°-photon. This randomization
of properties by non- orthogonal measurements is a crucial element in the
detectability of eavesdropping in quantum cryptography. In the terminology
of quantum mechanics one says that the V-polarizer has "collapsed the
photon's wavefunction."
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