Figure: Illustrations of the mathematical description of electromagnetic radiation propagation

General Caption: Images illustrating the propagation of electromagnetic waves.

Further explication

1. Image 1 Caption: An image showing a beam of linearly polarized electromagnetic radiation (EMR) propagating to the right obliquely.

   The ray could be part of a continuum of rays making a beam of finite geometrical cross section such as a plane wave. See Image 2 below for an illustration of a plane wave.

   In fact, for definiteness, let's speak of the beam in the Image 1 as a plane wave.

2. The small Greek letter lambda λ is the common symbol for wavelength.

3. A wavelength is the length of a complete up-and-down spatial cycle of the wave motion.

4. The oscillations of the electric field (E-field) (shown in blue) and the magnetic field (B-field) (shown in red) have a directions in real space, but they extend in abstract electric field space and magnetic field space.

5. The lengths of the vectors are really the magnitudes or strengths of the electric field and magnetic field.

6. The electric field and magnetic field are perpendicular to each other and the propagation direction.

   The oscillations being perpendicular to the propagation direction makes EMR a transverse wave phenomenon like waves on a string.

7. In EMR, the electric fields and magnetic fields are coupled: i.e., are mutually interacting.

   In fact, a time-varying electric field creates a time-varying magnetic field and vice versa---they create each other to paraphrase Jack Nicholson (1937--).

   And they must do so for EMR to be self-propagating.

   

8. Image 2 Caption: A diagram of a square segment of linearly polarized electromagnetic radiation (EMR) plane wave.

9. Ideal plane waves, extend to infinity. Realistic ones have a finite extent with complicated edges in general. However, a small enough portion of any wavefront (e.g., of spherical waves as in Image 3 below) approximates a plane wave.

10. The directions of the electric field (E-field) (shown as red arrows) and the magnetic field (H-field) (shown as green arrows) are shown in in real space, but their extent is in, respectively, abstract electric field space and abstract magnetic field space.

    Note for magnetic fields either the symbol B or symbol H may be used. The two symbols do NOT mean quite the same thing, but our purposes they do.

11. In plane waves, the electric field and the magnetic field have a constant direction and magnitude throughout the any plane of the plane waves. These planes are perpendicular to the propagation direction given by black arrow S.

12. The electric field and the magnetic field are perpendicular to each other and are perpendicular to the propagation direction.

13. The plane waves form a monochromatic sine wave. The sine wave behavior (which represents extent in abstract electric field space and abstract magnetic field space) is shown by the red and green bands.

14. The sources of the EMR beams in Image 1 and 2 may be oscillating electric charges at the right, but this is NOT shown in the images.

15. Linear polarization:
    1. The beams in the Image 1 and Image 2 above are linearly polarized.

       This means that the electric field and magnetic field are confined to two perpendicular planes.

    2. Real finite beams are made up of many photons (the particles of EMR) with all orientations of the electric field and magnetic field perpendicular to the propagation direction being possible.

    3. The vectors of the photons do NOT cancel microscopically---or else there would be a lot less EMR than there is. This non-cancellation is consequence of quantum mechanics.

    4. Sources that radiate EMR from their own internal heat energy naturally radiate unpolarized light in which all orientations of the electric field and magnetic field perpendicular to the propagation direction occur with equal weighting.

    5. EMR can be linearly polarized through various processes like reflection and transmission through polarizing materials like those that are used in polarized sunglasses. A filter that linearly polarizes is called a polarizer.

    6. There is also circular polarization---but that is the story for another place.

    
16. Image 3 Caption: A illustration spherical wave of EMR propagating outward from a point source.

    Far from the point source, a small part of a wavefront will approximate a plane wave.

    The Poynting vector represents the energy flux of the EMR.