This comes from Maxwell's equations.  You can substitute a plane wave solution into the divergence equations and find that k (wave vector) and D are perpendicular as well as k and B.  Then there are the curl equations.  One curl equation states that if the electric field tends to rotate, it will induce a magnetic field at the center of rotation and in a direction perpendicular to the rotation.  The other curl equations states the opposite, that an electric field will be induced at the center of rotation of a magnetic field.  The curl equations put electric and magnetic fields perpendicular to each other.  Now you have electric and magnetic fields perpendicular to k and perpendicular to each other.  Thus a traditional TEM wave.

I recently wrote a book chapter where in part of that paper I go into this in detail.  There are cases where things are a little more strange, like inside anisotropic materials.  Here it is...

Raymond C. Rumpf "Engineering the Dispersion and Anisotropy of Periodic Electromagnetic Structures," Solid State Physics, Vol. 66, pp. 213-300, 2015.

Hope this helps!

This comes from Maxwell's equations.  You can substitute a plane wave solution into the divergence equations and find that k (wave vector) and D are perpendicular as well as k and B.  Then there are the curl equations.  One curl equation states that if the electric field tends to rotate, it will induce a magnetic field at the center of rotation and in a direction perpendicular to the rotation.  The other curl equations states the opposite, that an electric field will be induced at the center of rotation of a magnetic field.  The curl equations put electric and magnetic fields perpendicular to each other.  Now you have electric and magnetic fields perpendicular to k and perpendicular to each other.  Thus a traditional TEM wave.

I recently wrote a book chapter where in part of that paper I go into this in detail.  There are cases where things are a little more strange, like inside anisotropic materials.  Here it is...

Raymond C. Rumpf "Engineering the Dispersion and Anisotropy of Periodic Electromagnetic Structures," Solid State Physics, Vol. 66, pp. 213-300, 2015.

Hope this helps!

That the electric and magnetic fields are perpendicular is only true in empty space and for the case that the emitting source is at rest relative to the observer (i.e the measruing device).  These fields, that are convenient quantities but cannot be measured directly, are solutions to the Maxwell equations and these funamental law of physics require the fields to be perpendicular to each other.

      Notice, also one important point in Physics, that if they were no orthogonal, then there would be frames that could distinguish their state of motion respect to the light or find their relative velocity, in contradiction with the principles of relativity. The invariant E.B is zero respect to ever system of coordinates or observer placed in vacuum for the light.

One more doubt! According to Faraday's/Ampere's Law, time rate of change of Magnetic/Electric Flux produces spatially varying and at the same time circulating Electric/Magnetic fields. So, does time rate of change of any one of these fields (E/H) produce the other (H/E) changing w.r.t. space in all the co-ordinates (x, y, z) or only in a single coordinate?

Before Maxwell's equations, the researchers absorbed during their experimentation that, for an TEM wave the electric and magnetic fields are perpendicular to each other. Later on the same thing was explained by MAXWELL in terms of mathematical formulas.

Dear Suhas,

The orthogonality of the electric and magnetic fields is independent of the system of coordinates, thus it must be in any system of coordinates. It is important to understand that if they were not orthogonal, then there could be one observer to see only electricity or magnetism, which would be catastrophic for one electromagnetic wave (i.e. which follows Maxwell equations).

By the way, it seems that this was the first difficulty that Einstein's found if one observer could be on the electromagnetic wave system.

The mathematical analysis in the link attached proves how E,H and k (propagation constant) are all perpendicular to each other and how EM radiation propagates at the speed of light.

http://farside.ph.utexas.edu/teaching/em/lectures/node48.html

By definition. The electric field - is that part of the electromagnetic field, which acts on the fixed charge; correspondingly, the magnetic field - that is the part of the EM field that acts on a moving charge, that is  current. From experiments we know that the forces generated by these parts of one (EM) field perpendicular. .............

Re-read my answer and did not understand what is written. I must confess.

In the plane wave E and H are always perpendicular to each other. Easily follows from Maxwell's equations.

However, if the electromagnetic field is a sum of two plane waves (E1, H1) and (E2, H2), it is obvious that not necessarily E1 perpendicular to  H2.

  Arbitrary EM field can always be represented by the sum (integral) of plane waves.

           Consequently:

  In an arbitrary EM field E and H are not perpendicular.

Moreover, you can find that E and H fields in a rectangular waveguide (with conductor walls) are also orthogonal to each other, both in a TE mode or in a TM mode. However, of course, you can imagine a situation in which you have excited simultaneously a TE mode and a TM one. In this case, only accidentally the resulting E and H fields will remain orthogonal.