Download Light-matter Interaction

Light-matter Interaction
(Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 3)
When white light passes through a prism then under appropriate conditions a spectrum of colors is visible
in the transmitted light, as observed a long time ago by Newton. This phenomenon arises for two reasons:
one is that white light is comprised of a number of different frequencies, i.e. it is non-monochromatic, and
(ii) light-matter interaction leads to dispersion of the initial white light into its various components. Here we
will introduce basic concepts necessary to understand this and other phenomenon resulting from light-matter
interactions.
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Refractive index (n)
Light travelling in free space has a velocity c = 3 × 108 m/s which is obtained from the constants !o and µo
√
as c = 1/ !o µo . Matter consists of atoms which are in turn comprised of charged species like electron and
the positively charged ion cores. Therefore when an electromagnetic field is incident in matter significant
interactions can take place. One important consequence of this interaction is a change in the velocity of
√
light in a medium from its value in vacuum. This new velocity is represented as: v = 1/ !µ, where the
permittivity and permeability now differ from that of vacuum. To characterize this change, an important
dimensionless quantity known as the refractive index (n) is used and its magnitude is given by:
c
n= =
v
!
!µ
!o µo
(1)
Some typical values of n are given in table 1.
material
air
water
salt
glass (fused Silica)
n at λ = 5893 Å
1.0002 at oo C
1.333 at 20o C
1.5 at room temperature (R.T.)
1.458 at R.T.
Table 1: Refractive index of some typical material.
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Scattering
All problem in light-matter interaction can generally be understood in the context of scattering which is a
generic term describing the result of EM wave interaction with a charged particle. Consider the situation
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Light Matter Interactions
shown in Fig. 1 where an EM wave is incident on matter. Since we are primarily interested in the electromagnetic force, we can simplify the problem by using a simple harmonic form of the EM wave described as
E(t) = Eo cos(ωt)$i with the electric field oscillating in the x-direction. Let us picture such a wave incident
in the y-direction on matter comprising of charges bound by springs having spring constant K. This EM
field will interact with any charge q via a coulomb force given by Fe = qE which will in turn produce
an acceleration/deceleration of the particular charge. Consequently from basic electromagnetic theory of
radiation accelerating/decelerating charges will emit/absorb radiation and hence the incident light will be
scattered/absorbed. If the charge is bound to its medium by a spring-like connection then a restoring force
Fr = −Kx will act on the charge and hence the resulting time-dependent equation of motion will be:
Fe + Fr = m
d2 x
dt2
(2)
where m is the mass of the charge. Using the respective force terms we get:
qEo cosωt − Kx = m
d2 x
dt2
(3)
One possible simple solution to this equation is of type x = xo cosωt and when used in eq. 3 we get:
x(t) =
q/m
Eo cosωt
(ωo2 − ω 2 )
(4)
In the above solution of the displacement of the charge q the quantity ωo represents the
" natural frequency
of vibration of the charge determined by its mass m and spring constant K as ωo =
K
m,
which is what
would be expected for a simple harmonic oscillator. From eq. 4 we can come to some useful conclusions
about the interaction of light with matter:
1. The bound charge is forced to oscillate at the same frequency as the driving EM field
2. The amplitude (i.e. magnitude of x) and phase (sign of x) of oscillation depends on the natural and
applied frequencies:
(a) When ω < ωo : here the charge oscillates in-phase or in the same direction as E(t)
(b) When ω > ωo : here the charge moves out-of-phase or in the opposite direction as E(t)
(c) When ω = ωo : here the charge motion resonates with the E-field and corresponds to a large
amplitude resulting in efficient energy absorption.
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Dispersion
In a real solid which consists of rigidly bound positive (+ve) and negative (-ve) charges, the interaction
with light is better understood with a more elaborate description. Consider the interaction of EM waves
with an insulating (or dielectric) solid in which the charges are bound and cannot move (but can oscillate
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Light Matter Interactions
Figure 1: EM wave incident in a medium with bound charges represented as charges connected to the object
by springs. The wave is travelling in the y-direction with the linearly polarized electric field oscillating in
the x-direction.
in place). Such a material is shown in Fig. 2. The application of a harmonic wave like in the previous
section will result in an instantaneous displacement of the +ve charges in one direction and the -ve charges
in the opposite direction. This condition can be described by numerous dipoles having on average a dipole
moment p = qx where x describes the average separation between the +ve and -ve charges and is identical
to the quantity calculated in eq. 4. Consequently, inside the material a net electric field will be generated
due to these numerous dipoles and the material will have a net polarization P , which is the dipole moment
per unit volume, given by, P = qxN , where N is the number of dipoles/volume. Therefore the net internal
electric field Eint will now be expressed as:
Eint = Eapp + Epolarization
Using eq. 4, the polarization can be expressed as P =
theory, i.e. P = (! − !o )E we get:
! = !o +
q 2 N/m
E(t)
(ωo2 −ω 2 )
q 2 N/m
(ωo2 − ω 2 )
and using the results of Maxwell’s
(5)
Using this expression for the permittivity in the value of the refractive index expressed by eq. 1 we come
to the important conclusion that:
n2 = 1 +
N q2
m!o
#
1
2
ωo − ω 2
$
(6)
This important equation is known as the dispersion relation as it shows how light of a given frequency
ω behaves when it is incident in a material with N dipoles each having resonant frequency ωo . Some of the
consequences of dispersion are discussed below.
1. Wavelength-dependent velocity of light: Consider white light consisting of numerous frequencies,
all with ω < ωo , incident in a transparent medium. From eq. 6 we see that as the light frequency
increases towards ωo the value of n increases. Given the fact that n = c/v this implies that the
velocity decreases as the frequency increases. From knowledge of the light spectrum we know that
blue has higher frequency (or smaller wavelength) then red, and therefore blue light will travel slower
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Figure 2: Insulating material (a) before application of EM field showing that the charge-carrying constituents, like atoms, molecules, etc. are undisturbed or unpolarized. (b) On application of an EM field the
centers of +ve and -ve charges are displaced with respect to each other by average distance x resulting in a
net dipole moment.
then red. Also given the inverse relation between frequency and wavelength, dispersion relation can
be discussed similarly in terms of wavelength.
2. Wavelength-dependent bending of light (Refraction): When light is incident at an interface separating
two medium with different refractive indices, the light bends and this phenomenon is known as refraction. While the physical origin of refraction lies in the behavior of light velocity in different medium,
the explanation of bending can be made by evaluating the behavior of a wavefront as it passes across
an interface. Consider fig. 3 in which a ray of light, travelling in medium n1 and bounded by the
lines 1 and 2, is incident at the surface AB of a medium n2 . The wavefront is represented by ao and
is perpendicular to the ray. Part 2 of the ray reaches the surface at point o before part 1. Since the
velocity changes upon entering the medium n2 therefore, when part 1 reaches the surface at point
o" a distance ∆x1 away, part 2 would have traveled a different distance ∆x2 to point b in medium
n2 . Consequently, the ray bends upon entering the medium. This is refraction. We discuss this more
quantitatively in the notes on refraction.
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Optical path length
Because of the dependence of light velocity on the medium or refractive index n, it is useful to define a
quantity known as the optical path length given by:
OP =
%
ni Li
(7)
where ni and Li refer to the refractive index and length of each medium i in the path of travel. The
optical path length emphases the fact that although light might travel between similar end points, it might
take different times to reach. This can be understood with reference to Fig. 4 in which two rays originate
simultaneously from a monochromatic source S with the same phase θo and eventually arrive at point P .
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Light Matter Interactions
Figure 3: Bending of light ray or wavefront as it crosses an interface separating two medium with different
refractive index. The wavefronts are represented by ao in medium n1 and bo" in medium n2 .
Ray 1 travels through different material and/or optical components with various R.I. including n1 before
reaching point P . On the other hand ray 2 reaches P by travelling only through n1 . Since light travels at
different speeds through media with different n, therefore rays 1 and 2 will take different times to reach P .
This time difference is manifested as a phase difference between the two waves arriving at P and this phase
difference ∆θ can be readily evaluated in terms of the optical path length travelled by each beam as follows:
θ1 − θ2 = ∆θ =
%
2π
2π %
(OP 1 − OP 2 ) =
( ni L1i −
ni L2i )
λ
λ
(8)
Figure 4: Figure showing how the optical path length causes light to travel between identical points S and
P but along different paths resulting in a phase difference between the beams travelling along the various
paths.
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