Nonlinear Fiber Optics - 4 ed. Agrawal

12.5. Harmonic Generation 497
traps, or color centers. The main point is that such a redistribution of charges breaks the
inversion symmetry and is also periodic with the right periodicity required for phase
matching. In effect, the fiber has organized itself to produce second-harmonic light
such that the dipoles respond to the optical field with an effective value of χ (2) . In the
simplest case, χ (2) is proportional to P dc such that
χ (2) ≡ α SH P dc =(3α SH /4)ε 0 χ (3) |E p | 2 |E SH |cos(Δk p z + φ p ), (12.5.3)
where α SH is a constant whose magnitude depends on the microscopic process responsible
for χ (2) and φ p is a phase shift that depends on the initial phases of the pump
and the second-harmonic seed. Because of the periodic nature of χ (2) , the preparation
process is said to create a χ (2) grating.
This simple model in which a dc electric field E dc is generated through a χ (3)
process suffers from a major drawback [162]. Under typical experimental conditions,
Eq. (12.5.1) leads to E dc ∼ 1 V/cm for pump powers ∼1 kW and second-harmonic seed
powers ∼10 W, if we use χ (3) ≈ 10 −22 (m/V) 2 . This value is too small to orient defects
and generate a χ (2) grating.
Several alternative mechanisms have been proposed to solve this discrepancy. In
one model [166] a charge-delocalization process enhances χ (3) by several orders of
magnitudes, resulting in a corresponding enhancement in E dc . In another model [167],
free electrons are generated through photoionization of defects, and a strong electric
field (E dc ∼ 10 5 V/cm) is created through a coherent photovoltaic effect. In a
third model [168], ionization occurs through three multiphoton processes (involving
four pump photons, two second-harmonic photons, or two pump photons and one
second-harmonic photon). In this model, the χ (2) grating is created through quantuminterference
effects that cause the electron-injection process to depend on the relative
phase between the pump and second-harmonic fields. Such a charge-transport model
is in qualitative agreement with most of the observed features.
Conversion efficiencies realized in practice are limited to ∼1% [153]. The secondharmonic
power grows exponentially during initial stages of the preparation process
but then saturates. One possibility is that the light created through SHG interferes with
formation of the χ (2) grating because it is out of phase with the original grating [155].
If this is the case, it should be possible to erase the grating by sending just the second
harmonic through the fiber without the pump. Indeed, such an erasure has been observed
[159]. The erasing rate depends on the second-harmonic power launched into
the fiber. In one experiment [160], conversion efficiency decreased from its initial value
by a factor of 10 in about five minutes at an average power level of about 2 mW. The
decay was not exponential, but followed a time dependence of the form (1 +Ct) −1 ,
where C is a constant. Furthermore, erasure was reversible, that is, the fiber could be
reprepared to recover its original conversion efficiency. These observations are consistent
with the model in which the χ (2) grating is formed by ordering of charged entities
such as color centers, defects, or traps.
Thermal Poling and Quasi-Phase Matching
Because of a limited conversion efficiency associated with photosensitive fibers, the
technique of quasi-phase matching has been used to make fibers suitable for SHG.