Page 357 - Phase Space Optics Fundamentals and Applications

338 Chapter Eleven

framework for developing intuition about how such pulses propagate
and interact with matter and a set of rigorous calculation tools that
enable information to be extracted efficiently and accurately from ex-
perimental data. In this chapter, we develop the phase-space descrip-
tion of ultrafast processes and its application to the characterization of
ultrashort optical pulses. We make use of the strong analogy between
the propagation of pulses in time through dispersive optical elements
with the propagation of beams in space through paraxial optical sys-
tems, since this has played an important role in developing concepts
for measurement.

11.2 Phase-Space Representations
for Short Optical Pulses
The fundamental quantity describing an isolated, individual pulse of

light is its electric field vector ε(x, t). This is a function of time t and

space x or equivalently optical frequency and transverse wave vec-

tor k. In all but the most intense pulses, the magnetic field does not
affect the interaction of the pulse with matter and can be estimated di-
rectly from the electric field. Characterizing an optical pulse therefore
involves estimating the space-time dependence of the electric field.

Since the electric field ε(x, t) is the fundamental entity in Maxwell’s
theory, the ability to measure it precisely not only provides the ne
plus ultra of diagnostics, but also enables new experimental meth-
ods. When electromagnetic radiation interacts with matter, both its
amplitude and its phase can be altered. The changes induced by the
interaction can yield important information about the material dy-
namics: in fact, proper characterization of the temporal amplitude
and phase of the field can potentially lead to complete reconstruction
of the response function of the system. The spatiotemporal structure
of the input and output fields provides all the available information
from an optical experiment and therefore provides the data for the
most exacting tests of models of the process under consideration.
11.2.1 Representation of Pulsed Fields
Typically in ultrafast optics the problem is simplified by taking a scalar
approximation to the field vector. Simultaneous measurements of two
orthogonal polarizations can then be combined to give the full vector
field. Within this approximation, the real electric field, ε(t) underlying
∗
an optical pulse (suppressing, for brevity, the spatial dependence) is
twice the real part of its analytic signal E(t): ε(t) = 2 × Re[E(t)].
The appropriate dispersion relation for the medium or structure in which the
∗
pulse propagates provides a connection that reduces the number of variables to
three.