1. Nonlinear Optics Lab. Hanyang 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.

2. 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  with respect to the x-direction. Classical law of Malus : T=cos 2  When a single photon is incident on the polaroid, the incident photon is not split by the polarizer. It is “all or nothing.”

3. 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 cos 2  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.)

4. Nonlinear Optics Lab. Hanyang Univ. 9.3 Failure of Classical Wave Theory : Recoil and Coincidence 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 !

5. 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.

6. 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 !

7. Nonlinear Optics Lab. Hanyang Univ. 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.

8. 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)

9. Nonlinear Optics Lab. Hanyang Univ. 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. P n (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 where,: incident intensity : a constant depending on the detection Summary : Homework

10. Nonlinear Optics Lab. Hanyang Univ. (9.5.2) (9.5.3) or where, (9.5.14)

11. Nonlinear Optics Lab. Hanyang Univ. 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 : where, or (9.6.3) (9.6.4)

12. Nonlinear Optics Lab. Hanyang Univ. Properties of Poisson distribution

13. Nonlinear Optics Lab. Hanyang Univ.