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Chapter 9. Wave-Particle Duality of Light
9.1 What is a Photon ?
Whether light consists of particles or waves ?
- ~1700, Newton : Particles !
- 1700~1890, T. Young, J.C. Maxwell, H. Hertz, Lorentz : Waves !
- 1900~1913, Planck, Einstein, Bohr : Light quantum (photon)
=> Quantum mechanics : Nature is fundamentally statistical rather than deterministic
According to quantum mechanics the wave and particle views of light are both
oversimplifications. Radiation and matter have both wave and particle attributes or a
“wave-particle duality.”
What is a photon ?
By considering a few “thought experiments” we will try to explain the essence of the
answer given by quantum mechanics.
Nonlinear Optics Lab.
Hanyang Univ.
9.2 Photon Polarization: All or Nothing
A plane EM wave is incident upon a polaroid sheet oriented so that it passes radiation of
polarization along the x-direction. The incident light is assumed to be linearly polarized
at an angle q with respect to the x-direction.
Classical law of Malus : T=cos2q
When a single photon is incident on
the polaroid, the incident photon is
not split by the polarizer.
It is “all or nothing.”
Nonlinear Optics Lab.
Hanyang Univ.
If we repeat one-photon experiment (by reducing the intensity of incident light)
many times, always with the same source and arrangement,
we find that sometimes the photon passes through the sheet and sometimes it does not.
=> Repetition of the experiment many times reveals that cos2q is the “probability”
that a photon polarized at an angle q to the polaroid axis will pass through. The
situation here is akin to coin flipping.
Quantum mechanics asserts that the statistical aspect of our one-photon experiment is
a fundamental characteristics of Nature.
And, we approach the classical law of Malus when the number of incident photons is
large. (“All or nothing nature of photon polarization may be ignored.)
Nonlinear Optics Lab.
Hanyang Univ.
9.3 Failure of Classical Wave Theory : Recoil and Coincidence
<Spontaneous emission>
2) Quantum theory
1) Classical theory
- Photon energy : hn
- Photon momentum : hn/c
Classical theory treats spontaneous emission
as a smooth process, with radiation being
continuously emitted more or less in all
directions.
=> Photon recoil is not accounted.
Consevation of linear momentum demands
that an atom that undergoes spontaneous
emission must recoil with a linear momentum
of hn/c !
Nonlinear Optics Lab.
Hanyang Univ.
Experimental results
1) Spread out of an atomic beam
In 1933, O.R. Frisch inferred the recoil of spontaneous emitting atom, and it has been
been confirmed with greater accuracy.
A well-collimated atomic beam of excited atoms will spread laterally
because of the recoil associated with spontaneous emission.
Nonlinear Optics Lab.
Hanyang Univ.
2) Coincidence measurement
When A undergoes spontaneous emission, is there any probabilty that a coincidence occurs,
i.e., that the emitted radiation from A can be detected at both B and C ?
: Yes in classical theory, but No by the quantum theory.
In quantum theory, a photon is an indivisible unit of energy that can trigger the emission of
only one photoelectron. Therefore, the radiation from A can register a count at either B or C,
or neither, but never at both B and C ! <= Experiment supports this fact !
Nonlinear Optics Lab.
Hanyang Univ.
<Stimulated emission / absorption>
The direction of atomic recoil follows exactly from the direction of propagation of the incident
radiation. => Doppler effect in the emission or absorption of radiation by a moving atom.
Nonlinear Optics Lab.
Hanyang Univ.
9.4 Wave Interference and Photons
How quantum mechanics reconciles the wave and particle aspects of light ?
Example) Young’s double slit experiment
A single photon by itself does not produce an
interference pattern. Instead, there is a relatively
high probability of detecting the photon at points
satisfying the constructive interference condition,
and the photon will never be found at points
satisfying the destructive interference condition.
The wave and particle aspects of two-slit
interference are reconciled by associating a
patricle(photon) probability distribution function
with the classical (wave) intensity pattern.
The entire interference pattern must be “known”
to each single photon. => A photon interferes
only with itself. (P.A.M. Dirac)
Nonlinear Optics Lab.
Hanyang Univ.
9.5 Photon Counting
Summary : Homework
Effect number (not an exact number) of photons can be counted by counting the number of
photoelectrons. Let us suppose nevertheless that we can count photons anyway.
We open the shutter at some time t and close it at a time t+T, and record the number of
photons that were counted while the shutter was open.
Pn(T) : The probability of counting n photons in
a given time interval T ?
The probability of ejecting one electron in time
interval Dt, is given by
p (t )Dt  I (t )Dt
where, I (t ) : incident intensity
 : a constant depending on the detection
Nonlinear Optics Lab.
Hanyang Univ.
Pn (t Dt )
Pn 1 (t ) p(t )Dt  (probabilit y of n  1 photons in time t )
 (probabilit y of 1 photon in Dt )
(9.5.2)
or
Pn (t )[1  p(t )Dt ]  (probabilit y of n photons in time t )
 (probabilit y of no photon in Dt )
(9.5.3)
 Pn (t Dt )Pn1 (t ) p(t )Dt  Pn (t )[1 p(t )Dt ]
1
 Pn (t ,T ) [ X (t ,T )]n exp[  X (t ,T )]
n!
(9.5.14)
where, X (t , T )  T I (t , T )
t T
1
I (t ,T ) I (t ')dt '
T t
Nonlinear Optics Lab.
Hanyang Univ.
9.6 The Poisson Distribution
Summary : Homework
Suppose the intensity I(t) is constant, then
I (t )  I  constant
t T
t T
1
1
I (t ,T ) I (t ')dt ' I dt ' I
T t
T t
The probability of counting n photons in a time interval T :
(IT ) n IT
Pn (T ) 
e
n!
(9.6.3)
or
(n ) n  n
Pn (T ) 
e
n!
(9.6.4)
where, n  IT
Nonlinear Optics Lab.
Hanyang Univ.
Properties of Poisson distribution



(n ) n  n
(n ) n
n
Pn 
e e 
1

n 0
n 0 n!
n 0 n!


(n ) n  n
n(T )   nPn (T )  n
e
n!
n 1
n 1
(n ) n 1
 ne 
n 1 ( n  1)!
n


(n ) n
 ne 
 n e n e n
n  0 ( n)!
 n  IT
n
Nonlinear Optics Lab.
Hanyang Univ.

n(T )
2
  n 2 Pn (T )  n 2  n
n 0
Dn(T ) 2  n(T )  n(T )

2
 n(T ) 2 2n(T ) n(T )  n(T )
 n(T ) 2 2 n(T )  n(T )
2
 n(T ) 2  n(T )
2
2
2
 Dn(T ) 2 n
Dn(T )rms  n
Nonlinear Optics Lab.
Hanyang Univ.