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The initial phase of an alternating field has played a minor role in the force exerted by the field over free or bound charges. The main reason being that in most cases the transient behaviour has a slowly varying envelope.

However, in recent times either with passive or active schemes, very short optical pulses have been achieved. These pulses may contain only a few oscillations and the slowly varying envelope condition is no longer fulfilled.

The Schrodinger equation has been intensively studied in recent years without invoking the slowly varying envelope approximation (SVEA).[1]

We wish to reconsider the effect of a harmonic field firstly on free charges and secondly on bound charges.

Most work has been performed on bound charges using a nonlinear Shrödinger equation.

Free charges

adiabatic approximation

Let us consider a slowly varying envelope harmonic field incident on a free charge.

The usual approximation is to consider the small velocities so that the magnetic contribution of the Lorentz force may be neglected.

This is reminiscent of the Lorentz - Drude model in the limit where the binding force is zero thus leading to free charges.

Evaluate the relative phase between the force, the velocity and the position.

consider three cases:

a) harmonic field with one minus exponential growth \(1-\exp (-gt)\)

b) harmonic field with one minus exponential growth and eventually, at time \(t_d\), exponential decay \(\exp [-g(t-t_d)]\)

c) Gaussian temporal envelope \(\exp(\frac {(t-t_c)^2}{\gamma ^2})\)

abrupt approximation

On the other hand, we shall consider the opposite limit where the external field is abruptly turned on.

What we really want is to establish the initial conditions that should be imposed on a constant field.

Bound charges

solutions to the differential equation with and without damping

Titles for the effect we are describing

transient charge spreading

This title would describe the collective phenomenon from shot to shot displacement or different displacements due to different positions within the beam.

charge scattering due to transient harmonic fields

charge motion under critical initial conditions

charge net displacement due to transient harmonic fields


  1. Nakamura S., Sakei T., Koyamada Y., Observation of slowly varying envelope approximation breakdown ..., Japanese Journal of applied Physics, 43, 10, 2004, 7015-7025