186
Chapter 3. Wave optics
tial wave front, with each point source radiating a spherical wave.
The point sources are assumed to be infinite in number, and infinitesimally separated along the initial wave front. They are also
assumed to radiate in phase, or coherently relative to one another.
At the time t + Δt the spherical waves have propagated to form
the envelope of the new wave front. This equivalent picture of
wave propagation is known as Huygens’ principle. It will prove
to be indispensible to formulating a mathematical description of
diffraction, which is derived in later sections.
Next, we consider each point source to form the initial point of
a wave function ψ(x a , t a ). From the preceding analysis, the final state wave function ψ(x b , t b ) is calculated from the solutions
(3.169, 3.186), depending on whether the electromagnetic potentials have explicit time dependence or not. The wave functions corresponding to the separate point sources add coherently to form
the composite wave front. The fact that the composite wave front
has a specific curvature says that the point sources have a corresponding relative position in space-time, as well as a specific
phase relationship to one another. This phase advances monotonically through space-time, as given by the action integral divided
by ¯
h. Each point source has an associated classical trajectory, as
depicted schemaically in Figure 3.4.
The mathematical description is exact in principle, and accounts
for all aberrations. In geometrical optics, the aberrations are manifest as a displacement of the classical particle trajectory from the
paraxial approximation. This displacement can be calculated in
principle to an arbitrary degree of accuracy. In wave optics, the
aberrations are manifest as displacements in the surfaces of constant phase.
The intensity is proportional to the probability density, which in
turn is related to the wave function as | ψ(x, t) |
2 . Obviously the
phase does not appear explicitly here. It is the relative phases of
neighboring trajectories that govern the shape of the wave fronts.
As a probability, the wave function for each point source must
Chapter 3. Wave optics
tial wave front, with each point source radiating a spherical wave.
The point sources are assumed to be infinite in number, and infinitesimally separated along the initial wave front. They are also
assumed to radiate in phase, or coherently relative to one another.
At the time t + Δt the spherical waves have propagated to form
the envelope of the new wave front. This equivalent picture of
wave propagation is known as Huygens’ principle. It will prove
to be indispensible to formulating a mathematical description of
diffraction, which is derived in later sections.
Next, we consider each point source to form the initial point of
a wave function ψ(x a , t a ). From the preceding analysis, the final state wave function ψ(x b , t b ) is calculated from the solutions
(3.169, 3.186), depending on whether the electromagnetic potentials have explicit time dependence or not. The wave functions corresponding to the separate point sources add coherently to form
the composite wave front. The fact that the composite wave front
has a specific curvature says that the point sources have a corresponding relative position in space-time, as well as a specific
phase relationship to one another. This phase advances monotonically through space-time, as given by the action integral divided
by ¯
h. Each point source has an associated classical trajectory, as
depicted schemaically in Figure 3.4.
The mathematical description is exact in principle, and accounts
for all aberrations. In geometrical optics, the aberrations are manifest as a displacement of the classical particle trajectory from the
paraxial approximation. This displacement can be calculated in
principle to an arbitrary degree of accuracy. In wave optics, the
aberrations are manifest as displacements in the surfaces of constant phase.
The intensity is proportional to the probability density, which in
turn is related to the wave function as | ψ(x, t) |
2 . Obviously the
phase does not appear explicitly here. It is the relative phases of
neighboring trajectories that govern the shape of the wave fronts.
As a probability, the wave function for each point source must