You have learned in your electromagnetic course that an electromagnetic field is composed of an electric and a magnetic field, which obey Maxwell's equations. Such a field can exist even in the absence of charges and currents. And here I have written the corresponding Maxwell's equations without source terms. You may be surprised that a field can exist without any source. Of course, there needs to be a source. But when I speak of free radiation, I mean that the field has been launched by a source which may now be turned off. If the source is still emitting, I consider the field far enough from the source so I do not need to put the sources in the equations. You may think of a radio-wave emitted by a station at 100s of kilometers from the place where you listen to the radio. Or to the light emitted by the sun by a distant star, or even the lightbulb at the ceiling of the room where you are, if you consider the light falling on your desk. In all these cases, you are interested in radiation in a volume without sources, and you can use Maxwell's equations without sources as written here. To define the field in the volume of interest, you only need to know the field value at the boundary of that volume. The most elementary solution of the Maxwell's equation is called a mode. That is to say, a field that oscillates at a well defined frequency. Since it oscillates, it will be quantized as a harmonic oscillator. My goal in the next sequences is to show you how the canonical quantization procedure can be applied to that case. It will give us the opportunity to introduce important observables of quantum optics. There are many different kinds of modes, and here we will take the simplest possible mode a travelling plane wave which describes a freely propagating field. You certainly know, that the field written here is a solution of Maxwell's equations provided that some conditions are fulfilled. Note first that c.c., means complex conjugate. So the field E is real, as it should. Let us see the conditions for this field to obey Maxwell's equations. Firstly, the unique vector, epsilon_l, has to be perpendicular to the wave vector, k l to satisfy divergence E equals zero as shown on the figure. Epsilon L with an arrow defines a polarization of the field that is to say direction of the electric field. Usually I use a standard notation for vectors representing them with bold face fonts. Unfortunately I do not have a good bold face epsilon. This is why I use an arrow. Another condition for such a field to be solution of Maxwell's equations is that the complex number E_l in script font, characterizing the amplitude of the field, oscillates at an angular frequency, omega_l. Moreover, the angular frequency, omega_l, must be equal to the velocity of light, c, times the modulus of the wave vector k_l. The ensemble of the vector k_l that determines the direction of propagation and the frequency of the field with a unit vector epsilon_l perpendicular to k_l, characterizes a mode referred to as a polarized plane travelling monochromatic wave. There are other kinds of modes for instance standing waves as you will see in the homework of this lesson. In the general sense, a mode is an elementary oscillating solution of Maxwell's equation. To characterize the field it is not enough to know the mode. The characteristics of the mode k_l and epsilon_l only determines the structure of the field. You also want to know the complex amplitude E_l which characterizes the state of the field in the mode l. The complex number E_l is fully determined by two real variables for instance its real and imaginary parts or its modulus and phase. These variables are dynamic variables describing the state of the classical field in the mode l. To fully understand what I mean, let us come back for a while to the case of a pendulum. The mode is characterized by the oscillation frequency and the plane of oscillation. But to know the state of the pendulum, I also need to know the amplitude and the phase of the oscillation given by the modulus and the phase of the complex amplitude of the position. Or equivalently, I need to know the real and imaginary parts of this complex amplitude. They are the dynamic variables describing the state of the pendulum. Let us now return to our mode L of the electromagnetic field. Our goal is to identify two dynamic variables of the field, that are canonically conjugate of each other. To do that, we first need to know more details about how to find that form of the travelling wave, starting from Maxwell's equations. Let us start with that expression for the field, and use Maxwell's equation to obtain its property. We will take advantage of the simplicity of the formula describing the effect of the differential operator nabla on such a form. Remember that the vectorial differential operator nabla is a vector with components d over dx, d over dy, and d over dz. The divergence of E takes a simple algebraic form, i k_l dot complex vector E_l. Plus the complex conjugate expression. Epsilon_L times the complex amplitude E_l is a complex vector E_l. The first Maxwell equation then entails that epsilon_L is perpendicular to k_l. Similarly Rotational of E is i k_l cross complex vector E plus complex conjugate. Using the second and the third Maxwell equations, we can conclude that E, B, and k are three mutually orthogonal vectors forming a right-handed set. Finally, using the fourth Maxwell equation we find that B has a form with the same complex exponential as E. If you now take the time derivative of this expression of B and use the second Maxwell equation you obtain a second order differential equation for the complex amplitude E_l. This is equivalent to two first order equations. whose solutions are oscillatory, more precisely exponentials of plus or minus i omega_l t. One solution is associated to our wave traveling along k, while the other is traveling against k. We want one mode only. So we keep only one of the solutions. If we choose the one traveling along k, we must take the exponential of minus i omega_l t. And we find the expression of the field shown on the previous slide and written here again. We then deduce an explicit time-depending form for B showing not only that k, E, and B form a direct tri-orthogonal set, but also that B is in phase with E. More precisely, the complex amplitude B_l is equal to the complex amplitude E_l divided by c. Check that you can show that. To conclude, for a traveling electromagnetic wave propagating along k_l the dynamics is fully determined by d E_l over dt equal minus i omega_l E_l.
