−1 −0.5 0 0.5 1 1.5 2 0.7
0.75 0.8 0.85 0.9 0.95 1
Probe Delay ∆t [ psec ] Normalized Probe Transmittance T r
XX case XY case YX case YY case
Figure 7.7: Transmission coefficient Tr versus ∆t for different combinations of pump and probe polarizations; the first and second characters of each pair in the legend (e.g., “X”) specify the orientation of the polarization vector of the pump and probe pulses, respectively.
• The value ofTr(∆t) at both the dip and plateau increases with decreasing pump pulse energy.
• A smaller dip transmissionTrdipcomes about for parallel-polarized pulses (XX and YY) than for pulses polarized perpendicular to each other.
• The value ofTrat the plateau is determined by the probe polarization; extraordinarily polar- ized probe experiences a slightly larger plateau transmissionTrpl..
7.4.2 Modeling of collinear pump-and-probe experiment
We attribute the dip manifested by the probe transmissionTr(∆t) to the two-photon absorption process involving a pump and a probe photon; the sum of the photon energy in this case is∼4.8 eV, which is larger than the band gap of LiNbO3 (∼4 eV).
On the other hand, the plateau section in the dependenceTr(∆t) is attributed to the absorption of the probe pulse by charge carriers (electrons and/or holes) that are excited by the intense pump pulse; the concentration of the excited free carriers due to pump and probe photons is negligible in comparison. The polarization dependence of the plateau value is explained by the anisotropy inherent in the LiNbO3 crystal. We shall model the dip and the plateau sections ofTr(∆t) separately in the following.
7.4.2.1 Modeling the dip ofTr(∆t)
We can write down the following equation for the probe pulse intensity:
∂
∂z + 1 vr
∂
∂t
Ir=−βrIpIr. (7.4)
The similarity between Eq. (7.4) and Eq. (7.1) is obvious. The only difference is that the effective absorption coefficient in Eq. (7.4) is αr = βrIp, which is determined by a much more intense pump pulse, andβris the quadratic absorption coefficient characterizing the two-photon absorption participated by a pump and a probe photon. To solve this equation, it helps to make the change of
variables
ζ = z, τ = t−vzr, after which Eq. (7.4) becomes
∂
∂ζIr=−βrIpIr,
where
Ir(x, y, ζ = 0, τ) =Ir0exp
−(τ−∆t)2
t2p −x2+y2 D2p
,
and
Ip(x, y, ζ, τ) = Ip0
βpIp0ζ+ exph
(tτp−δ0dζ)2+x2D+y22 p
i.
according to Eq. (7.2). In these expressions, ∆t is the delay of the probe pulse, and δ0 = (np− nr)d/ctpis the parameter accounting for the difference of pump and probe velocities. We have also assumed that the pump and probe pulses have the same temporal duration and transverse spatial dimension.
The solution of the probe intensityIr(x, y, ζ=d, τ) is therefore Ir(x, y, ζ=d, τ)
= Ir(x, y, ζ = 0, τ) exph
−Rd
0 βrIpdζi
= Ir(x, y, ζ = 0, τ) exp
−ββrp
Rd 0
qpdu qpu+
»
(tpτ−δ0dζ)2+x2+y2
D2 p
–
, (7.5)
whereζ=udhas been used.
The transmitted probe energy can be calculated from Eq. (7.5), and the corresponding transmis- sion coefficient is
Tr= 1
√π Z ∞
−∞
exp
−(s−∆t/tp)2 Z 1
0
exp
−βr
βp
Z 1 0
qpfdu
qpf u+ exp[(s−δ0u)2]
dfds. (7.6)
If we neglect the velocity difference between pump and probe pulses,δ0= 0; the integrals involving variablesuandf can be carried out analytically, and we obtain from Eq. (7.6)
Tr= e−(∆t/tp)2
√πaqp
Z ∞
−∞
e2s∆t/tp[(1 +qpe−s2)a−1]ds, (7.7) where a = 1−βr/βp. In our experiment, the transmission at the dip can be approximated as Trdip= 1−Tr,min−1/2[1−Trpl.], which can then be used for the fitting ofβr.
7.4.2.2 Modeling the plateau ofTr(∆t)
Since the intense pump pulse excites carriers via direct two-photon transitions, it is able to induce an additional absorption for the probe. This pump-induced absorption is expected to remain until recombination of the photo-excited carriers occurs. To find out the concentration of excited carriers, we refer to Fig. 7.8. We define ρp(x, y, z, t) as the electromagnetic energy density carried by the
z z+'z N(z,t)
I(z,t) I(z+'z,t)
Figure 7.8: The pump pulse is passing through a short segment of nonlinear-absorptive medium.
pump pulse and
Ip(x, y, z, t) =vp×ρp(x, y, z, t).
Consider a short segment of length ∆z in the medium, as shown in Fig. 7.8. The rate of change of the pulse energy in the length ∆z is given by the energy flux into one end minus the energy flux out of the other end of the segment, plus the rate of pulse energy expended on the excitation of free carriers. LetN(x, y, z, t) be the free carrier concentration and neglect the free-carrier recombination;
we have
∂
∂t[ρp(z, t)∆z] =Ip(z, t)−Ip(z+ ∆z, t)−2~ωp
∂
∂tN(z, t)∆z.
After some straightforward manipulations, we end up with
∂
∂τN(x, y, ζ, τ) =− 1 2~ωp
∂
∂ζIp(x, y, ζ, τ). (7.8)
According to Eq. (7.8), the absorption coefficient experienced by the probe pulse due to the free carriers can be represented as
αfr(x, y, ζ, τ) =− σr
2~ωp
∂
∂ζ Z τ
−∞
Ip(x, y, ζ, τ0)dτ0, (7.9) whereσr is the effective absorption cross-section of the photo-excited carriers at the wavelengthλr. Combining the probe absorption due to the co-existence of the pump pulse and the excited carriers,
0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25
Pump Absorption Parameter q
p=βpI
p0d [1−T rdip ] or [1−T rplateau ]
Dip XX Dip YY Plateau XX Plateau YY
Figure 7.9: Dependence of the dip and plateau amplitudes on the productqp=βpIp0dfor sample 3 and two different pump-probe polarization states. The points are experimental data, and the solid lines are theoretical fits.
we have
Ir(x, y, ζ=d, τ) =Ir(x, y, ζ = 0, τ) exp
"
− Z d
0
(βrIp+αfr)dζ
#
. (7.10)
To computeTrpl., consider the case when the pump pulse has already passed through, i.e., the probe pulse does not overlap with the pump pulse. Mathematically, we setIp= 0 in Eq. (7.10) and τ=∞in Eq. (7.9) and the plateau value is
Trpl.= Z ∞
0
exp
−f−bqp
√ π 2 e−f−
Z ∞ 0
ds qp+ef+s2
df, (7.11)
whereb=σrtp/~ωpβpd.
Fig. 7.9 shows the experimental data for the dependence of the dip and plateau amplitudes on the pump absorption parameterqp, obtained for two different polarization cases, together with the theoretical fit. The fit parameters are the ratioβr/βp(for the dip amplitude) andb=σrtp/~ωpβpd (for the plateau amplitude).
The ratioβr/βp deduced from the fitting procedure is∼0.23 for the XX case and∼0.24 for the YY case. Thus, with an accuracy of (10∼15)%, the value ofβrcan be estimated as∼0.67 cm/GW.
The values ofbcan be estimated as∼0.21 and∼0.14 for the XX and YY cases, respectively. This gives us the values of the excitation cross-section: σr,xx∼1.60×10−17cm2andσr,yy∼1.08×10−17 cm2.