Nonlinear Optics Lab. Hanyang Univ. Chapter 9. Wave-Particle Duality of Light 9.1 What is a Photon ?...

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.

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.


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 with respect to the x-direction.

Classical law of Malus : T=cos2

When a single photon is incident on the polaroid, the incident photon is not split by the polarizer. It is “all or nothing.”


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 cos2 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.)


9.3 Failure of Classical Wave Theory : Recoil and Coincidence

<Spontaneous emission>

1) Classical theory

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.

2) Quantum theory

- Photon energy : h- Photon momentum : h/c

Consevation of linear momentum demands that an atom that undergoes spontaneous emission must recoil with a linear momentum of h/c !


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.


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 !


<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.


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 aninterference 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)


9.5 Photon Counting

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 t, is given by

ttIttp )()( where, )(tI : incident intensity

: a constant depending on the detection

Summary : Homework


)in photon 1 ofy probabilit(

) in time photons 1 ofy probabilit()()(1

t

tnttptPn

(9.5.2)

)in photon no ofy probabilit(

) in time photons ofy probabilit(])(1)[(

t

tnttptPn

(9.5.3)

or

)( ttPn

])(1)[()()()( 1 ttptPttptPttP nnn

)],(exp[)],([!

1),( TtXTtXn

TtP nn

where, ),(),( TtITTtX

Tt

t

dttIT

TtI ')'(1

),(

(9.5.14)


9.6 The Poisson Distribution

Suppose the intensity I(t) is constant, then

Summary : Homework

The probability of counting n photons in a time interval T :

constant)( ItI

Tt

t

Tt

t

IdtIT

dttIT

TtI '1

')'(1

),(

ITn

n en

ITTP

!

)()(

nn

n en

nTP

!

)()(

ITn where,

or

(9.6.3)

(9.6.4)


Properties of Poisson distribution

1!

)(

!

)(

000

n

n

n

nnn

nn n

nee

n

nP

ITn

eenn

nen

n

nen

en

nnTnPTn

n

nnn

n

n

nn

n

nn

nn

)!(

)(

)!1(

)(

!

)()()(

0

1

1

11


nnTPnTn n

n

2

0

22 )()(

22 )()()( TnTnTn

22

222

22

)()(

)()(2)(

)()()(2)(

TnTn

TnTnTn

TnTnTnTn

)( 2 nTn

nTn rms)(

Nonlinear Optics Lab. Hanyang Univ. Chapter 9. Wave-Particle Duality of Light 9.1 What is a Photon ?...

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