Page 375 - Phase Space Optics Fundamentals and Applications

356 Chapter Eleven

to the transfer matrix

1 + ,2 t ,1 + ,2 + ,1 ,2 t

T = (11.52)

t 1 + ,1 t
The particular case of = =−1/ yields the remarkable

,1 ,2 t

relations t out =− in / and out = t in between temporal and spec-
t
t
tral coordinates of a ray bundle, in a fashion similar to the Fourier
transform relation induced by a lens of focal length f between two
planes located a distance f apart from the lens. This is known as
the time-to-frequency converter because the temporal intensity of the
input pulse can be recovered from a simple measurement of the spec-
tral intensity of the output pulse. 39 Another interesting result is that
with 1/ + 1/ + = 0 (a condition referred to as temporal

,1 ,2 t
imaging), the upper right quadrant of Eq. (11.52) is zero, which leads
to t out = (1 + ,2 t
)t in . Such assembly therefore magnifies the ray
bundle in the time domain by the quantity 1 + . Temporal mag-

,2 t
nification following this formalism has been used to decrease the res-
olution required to measure an optical waveform. 36,40–43
A further useful analogy is that it is straightforward to propagate
Gaussian pulses using the temporal transfer matrices. The complex
pulse parameter
1
2
= − i (11.53)
2
where is the chirp parameter of the pulse at the reference plane and
is the corresponding pulse duration, is modified simply according
to the less well-known formula
2
A 0 /(c ) + B
0
= (11.54)
2
c 2 C 0 /(c ) + D
The elements of the temporal transfer matrix are then interpreted as
modifying the chirp and duration accordingly.
This formulation is useful in visualizing both the optical pulses
and the strategies that are used to measure them. They are in broad
agreement with formal definitions of phase-space distributions of the
pulsed fields, although they only agree in detail in cases, such as for in-
coherent ensembles, when all quantities are positive definite. Further,
it is useful in system design and analysis, because it provides a sim-
ple way to understand the first-order space-time couplings that occur
when geometrical dispersion (such as happens in a prism or grating)
is used to build a temporally dispersive delay line. In this case the
paraxial and temporal transfer matrices are combined into 4 × 4 ma-
trices that describe the coupling between the spectral and temporal