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At the end of this nice video, she says that electromagnetic wave is a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward.

I wonder where the photon is in this explanation. What is the relation between electromagnetic wave and photon?

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Please see my answer here. You can understand Willis Lamb's frustration and the waves and normal modes describe the electromagnetic field. Photons are then the changes of number state of each normal mode - they are like the discrete "communications" the whole EM field has with the other quantum fields of the World that make up "empty space". One can reinterpret this statement as Maxwell's equations being the propagation equation for a lone "photon", but only in terms of propagation equations for the mean of electric and magnetic field .... – WetSavannaAnimal aka Rod Vance Dec 19 '13 at 0:48
...observables when the EM field is in a superposition of $n=1$ Fock states (so it is "one photon propagating"). – WetSavannaAnimal aka Rod Vance Dec 19 '13 at 0:49
up vote 16 down vote accepted

Both the wave theory of light and the particle theory of light are approximations to a deeper theory called Quantum Electrodynamics (QED for short). Light is not a wave nor a particle but instead it is an excitation in a quantum field.

QED is a complicated theory, so while it is possible to do calculations directly in QED we often find it simpler to use an approximation. The wave theory of light is often a good approximation when we are looking at how light propagates, and the particle theory of light is often a good approximation when we are looking at how light interacts i.e. exchanges energy with something else.

So it isn't really possible to answer the question where the photon is in this explanation. In general if you're looking at a system, like the one in the video, where the wave theory is a good description of light you'll find the photon theory to be a poor description of light, and vice versa. The two ways of looking at light are complementary.

For example if you look at the experiment described in Anna's answer (which is one of the seminal experiments in understanding diffraction!) the wave theory gives us a good description of how the light travels through the Young's slits and creates the interference pattern, but it cannot describe how the light interacts with the photomultiplier used to record the image. By contrast the photon theory gives us a good explanation of how the light interacts with the photomultiplier but cannot describe how it travelled through the slits and formed the diffraction pattern.

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This is news because all QM teachers told me that photons abstractions, proposed by QED, which is more exact than wave discription. However, this should not stop us from figuring out how two are related. Actually quanta = particles. – Val Dec 20 '13 at 18:09
@Val The way we actually calculate things in QED is with a perturbative expansion that involves photons. The underlying exact theory is one of several completely quantum fields. – Kevin Driscoll Dec 20 '13 at 19:26

In this link there exists a mathematical explanation of how an ensemble of photons of frequency $\nu$ and energy $E=h\nu$ end up building coherently the classical electromagnetic wave of frequency $\nu$.

It is not simple to follow if one does not have the mathematical background. Conceptually watching the build up of interference fringes from single photons in a two slit experiment might give you an intuition of how even though light is composed of individual elementary particles, photons, the classical wave pattern emerges when the ensemble becomes large.

single photon

Figure 1. Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

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In 1995 Willis Lamb published a provocative article with the title "Anti-photon", Appl. Phys. B 60, 77-84 (1995). As Lamb was one of the great pioneers of 20th century physics it is not easy to dismiss him as an old crank.

He writes in the introductory paragraph:

The photon concepts as used by a high percentage of the laser community have no scientific justification. It is now about thirty-five years after the making of the first laser. The sooner an appropriate reformulation of our educational processes can be made, the better.

He finishes with these comments:

There is a lot to talk about the wave-particle duality in discussion of quantum mechanics. This may be necessary for those who are unwilling or unable to acquire an understanding of the theory. However, this concept is even more pointlessly introduced in discussions of problems in the quantum theory or radiation. Here the normal mode waves of a purely classical electrodynamics appear, and for each normal mode there is an equivalent pseudosimple harmonic-oscillator particle which may then have a wave function whose argument is the corresponding normal-mode amplitude. Note that the particle is not a photon. One might rather think of a multiplicity of two distinct wave concepts and a particle concept for each normal mode of the radiation field. However, such concepts are really not useful or appropriate. The "Complementarity Principle" and the notion of wave-particle duality were introduced by N. Bohr in 1927. They reflect the fact that he mostly dealt with theoretical and philosophical concepts, and left the detailed work to postdoctoral assistants. It is very likely that Bohr never, by himself, made a significant quantum-mechanical calculation after the formulation of quantum mechanics in 1925-1926. It is high time to give up the use of the word "photon", and of a bad concept which will shortly be a century old. Radiation does not consist of particles, and the classical, i.e., non-quantum, limit of QTR is described by Maxwell's equations for the electromagnetic fields, which do not involve particles. Talking about radiation in terms of particles is like using such ubiquitous phrases as "You know" or "I mean" which are very much to be heard in some cultures. For a friend of Charlie Brown, it might serve as a kind of security blanket.

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