Читать книгу Stigmatic Optics - Rafael G González-Acuña - Страница 23

We start the one-dimensional case, we assume that there is electric filed in the z direction.

E⃗(r⃗,t)=Ez⃗(z)e−ikct.(1.46)

Replacing equation (1.46) in the Helmholtz equation,

∂2Ez⃗∂z2+k2Ez⃗=0.(1.47)

The solution of equation is given by

Ez⃗=Aeikz+Be−ikz.(1.48)

Therefore, E⃗(r⃗,t) is given by

E⃗(r⃗,t)=Aeikz+Be−ikze−ikvt=Aei(kz−kvt)+Bei(kz+kvt)(1.49)

simplifying,

E⃗(r⃗,t)=Acos(kz−kvt)+Bcos(kz+kvt).(1.50)

The minus sign of the first cosine means that the wave is travelling to the right of positive z. The plus sign of the second cosine implies that the wave is moving to the right of negative z.

Also, notice that the time is being multiplied by the angular frequency,

w=kv.(1.51)

Notice that is the speed inside a medium and c is the speed in vacuum. Therefore, if we pick the wave that is traveling to positive z, E⃗(r⃗,t) is given by

E⃗(r⃗,t)=E0cos(kz−wt)(1.52)

where we set A→E0. The last expression is the equation of the plane wave.