102
4 Nonlinear Optics with Zero-Index Metamaterials
which encompasses the phase terms of the two field amplitudes as well as a normalized phase mismatch parameter
s = kl
(4.88)
Writing the coupled wave equation in terms of u 1 and u 2 , we get the following
equations:
du 1
dζ
= u 1 u 2 sinθ
(4.89)
du 2
dζ
= −u
2
1 sinθ
(4.90)
θ
dζ
= s +
cosθ
sinθ
d
dζ
(ln u
2
1 u 2 )
(4.91)
Solving the above equations for a perfectly, phase-matched (s = 0) situation yields
simple solutions as follows:
u 1 (ζ) = sechζ
(4.92)
and
u 2 (ζ) = tanhζ
(4.93)
Figure 4.12 graphically illustrates the growth of the second-harmonic field and the
reduction of the fundamental field with propagation distance. The development of
the second-harmonic occurs at the cost of the fundamental wave, i.e., power flows
from the fundamental wave to second-harmonic wave. For the perfect phase match,
the power transfer or, more accurately, the conversion efficiency is 100% as shown.
Better visualization of the growing u 2 and the diminishing u 1 has been presented
in Fig. 4.13, as an aid to the imagination. In general, efficiency of second-harmonic
generation at the end of the crystal of length L is given as
η =
u
2
2 (L)
u
2
1 (0)
(4.94)
To understand the field variation inside the crystal with imperfect phase matching,
Armstrong et al. provided an approximate formula according to which the secondharmonic amplitude for different values of mismatch is given by
p 2 =
4 p 1 (0)
s
sin
sζ
2
(4.95)