208
7 Light in Biology and Medicine
vector and a selected direction in the same plane. Since the electric field must lie in
a plane perpendicular to the propagation of the wave, there are only two independent
directions of polarization of the wave (any polarization can be represented as a linear
combination of two vectors perpendicular to the propagation vector). The speed of
all electromagnetic waves in a vacuum is c = 2.99792458 × 10 8 m/s. Since the
distance between maxima along a given wave (i.e. its wavelength) is traversed by
the wave in its period T = 1/f , we have the universal relation relating frequency
and wavelength:
λ/T = f λ = c.
As Maxwell’s wave equations (7.1) and (7.2) are linear in E and B, a general
solution can be expressed as a Fourier series (or Fourier transform if there is no finite
period for the wave). A single term (of one frequency) will have the form of a ‘plane
wave’, i.e. the surfaces of constant phase for the wave are planes. Again because of
linearity, we can us the convenience of the complex form 5 for these waves, since
both the real and the imaginary parts will be solutions. The expressions
E = A ε exp (ik · r − iωt)
(7.8)
and
B = A k × ε exp (ik · r − iωt)
(7.9)
will be a plane wave solution of Maxwell’s equations (some distance away from the
source charges), provided k 2 = ω 2 /c 2 . Here, as usual, ω is angular frequency of the
wave, related to the ordinary frequency by f = ω/(2π). To get physical solutions,
one can take either the real or the imaginary part of the complex solution.
Besides the rather limited frequencies of light we humans detect with our
eyes, nature produces both higher and lower frequencies (longer and shorted
wavelengths) with biologic consequences. The common names for various regions
in this spectrum are (with increasing frequency): radio, microwave, infrared, visible,
ultraviolet, X-ray, and gamma ray. (See Fig. 7.2.) Approximate wavelength ranges,
as well as frequencies and photon energies, are given in Table 7.1. The ranges for
these names are rather arbitrary. The ranges for a given name overlap, with the
choice depending on the usage. Man-made radio waves were first detected by a spark
across a gap of a fraction of a millimeter in a circle of wire tuned to a corresponding
transmitter.
Humans initially produced and detected X-rays by scattering electrons off metal,
after the electrons were accelerated in a vacuum by an anode placed at thousands
of volts relative to a cathode. Gamma rays were first observed in the decay of
5 Here, ‘complex’ refers to the mathematical combination of a real number and an imaginary
number: a + ib, or in polar form,
√
a 2 + b 2 exp (i arctan (b/a)).
7 Light in Biology and Medicine
vector and a selected direction in the same plane. Since the electric field must lie in
a plane perpendicular to the propagation of the wave, there are only two independent
directions of polarization of the wave (any polarization can be represented as a linear
combination of two vectors perpendicular to the propagation vector). The speed of
all electromagnetic waves in a vacuum is c = 2.99792458 × 10 8 m/s. Since the
distance between maxima along a given wave (i.e. its wavelength) is traversed by
the wave in its period T = 1/f , we have the universal relation relating frequency
and wavelength:
λ/T = f λ = c.
As Maxwell’s wave equations (7.1) and (7.2) are linear in E and B, a general
solution can be expressed as a Fourier series (or Fourier transform if there is no finite
period for the wave). A single term (of one frequency) will have the form of a ‘plane
wave’, i.e. the surfaces of constant phase for the wave are planes. Again because of
linearity, we can us the convenience of the complex form 5 for these waves, since
both the real and the imaginary parts will be solutions. The expressions
E = A ε exp (ik · r − iωt)
(7.8)
and
B = A k × ε exp (ik · r − iωt)
(7.9)
will be a plane wave solution of Maxwell’s equations (some distance away from the
source charges), provided k 2 = ω 2 /c 2 . Here, as usual, ω is angular frequency of the
wave, related to the ordinary frequency by f = ω/(2π). To get physical solutions,
one can take either the real or the imaginary part of the complex solution.
Besides the rather limited frequencies of light we humans detect with our
eyes, nature produces both higher and lower frequencies (longer and shorted
wavelengths) with biologic consequences. The common names for various regions
in this spectrum are (with increasing frequency): radio, microwave, infrared, visible,
ultraviolet, X-ray, and gamma ray. (See Fig. 7.2.) Approximate wavelength ranges,
as well as frequencies and photon energies, are given in Table 7.1. The ranges for
these names are rather arbitrary. The ranges for a given name overlap, with the
choice depending on the usage. Man-made radio waves were first detected by a spark
across a gap of a fraction of a millimeter in a circle of wire tuned to a corresponding
transmitter.
Humans initially produced and detected X-rays by scattering electrons off metal,
after the electrons were accelerated in a vacuum by an anode placed at thousands
of volts relative to a cathode. Gamma rays were first observed in the decay of
5 Here, ‘complex’ refers to the mathematical combination of a real number and an imaginary
number: a + ib, or in polar form,
√
a 2 + b 2 exp (i arctan (b/a)).