As it has already been mentioned silicon nanowires and silicon slab waveguides are characterized by their submicrometer cross-section and high-index contrast, which leads to strong optical confinement and waveguide dispersion. Therefore, great care needs to be taken during the design and fabrication of silicon waveguides. Moreover, the waveguide dispersion is derived from the propagation constant, β(ω), which is strongly dependent on the specific waveguide geometry. Note also that each coefficient of the Taylor series expansion of Eq. (2.23) has a different and a unique influence on the pulse evolution along the waveguide.
To be more specific, the first order parameter β1 determines the group velocity.
In particular,β1plays a crucial role on the interaction between optical pulses that co-
propagate in an optical medium. The significance of β1 is revealed by the analysis of
nonlinear phenomena, such as FWM or similariton collision, which will be described in the Chapters 3 and 4, respectively.
The most important dispersion coefficient isβ2, which represents frequency dis-
persion of the group velocity and is responsible for the variation of the pulse width. In order to quantify the effect of GVD, a dispersion parameter D is defined as follows:
D = dβ1 dλ =−
2πc
The coefficients β3 and β4 are known as third and fourth order dispersion coef-
ficients, respectively. Their effects are negligible for pulse widths of picoseconds and larger. Third order dispersion (TOD) coefficient is more important in the case of ultra short pulses whileβ4 plays a crucial role in many parametric processes in optics, such
as FWM.
The effects of GVD and TOD are presented in the subsections 2.5.1 and 2.5.2, respectively. However, we will not provide a comprehensive description of dispersive effects but only a brief qualitative presentation of the dispersive effects induced by GVD and TOD. More specifically, the fundamental phenomena induced by GVD and TOD properties are qualitatively the same for silica fibres, nanowires and photonic crystal slab waveguides. However, the extension to which the effects contribute to the pulse reshaping is significantly different in each of these cases. In particular, as the study of optical pulse propagation shifts from silica fibres to silicon nanowires and then to silicon photonic crystal slab waveguides the dispersive characteristic lengths are significantly reduced due to the increase of the optical pulse confinement. This means that all the linear dispersive effects are enhanced in the case of silicon nanowires and slab photonic crystal waveguides as compared to the case of silica fibres.
2.5.1
Group velocity dispersion effects
Group-velocity dispersion is the dependence of the group velocity of light propagating in a transparent medium on the optical frequency (or wavelength). Specifically, the phase of an optical pulse changes differently for each frequency component as the pulse propagates in the waveguide. Furthermore, there are cases where the phase also depends on whether the pulse experiences normal or anomalous dispersion. To be more specific, when a pulse is unchirped (C = 0), the phase is affected by the same amount for a pulse propagation in normal or anomalous dispersion regime. However, in the case of a chirped pulse (C̸= 0), the sign ofβ2has an important influence on the evolution of
the pulse shape. Many qualitative features arise from the dependence of GVD on the particular nature of dispersion, such as pulse compression or generation of parabolic pulses.
The effects of GVD are strongly dependent on the propagation distance of the pulse. A dispersion length, LD, is defined, in order to provide a length scale over which
dispersive effects become important as far as the pulse evolution is concerned:
LD= τ
2
|β2|. (2.35)
When the propagation distance is much shorter than the dispersion length (L≪
LD) GVD plays no significant role during pulse propagation. However, the opposite occurs when the length of the silicon nanowire is such that LD≪ L. In addition, ultra- short pulses in the femotsecond regime broaden much more than in picosecond regime because of a smaller dispersion length as Eq. (2.35) suggests. For instance, let us as- sume two pulses with the same value of β2=1.11 ps2 m−1 and different pulse widths
τ1=200 fs and τ2=2 ps have LD1=3.6 cm, and LD2=3.6 m, respectively. An interest- ing question is raised regarding to which extent GVD affects the pulse propagation. The answer depends on the profile of the propagating pulse. For an initially unchirped pulse profile, GVD broadens the pulse during propagation. Pulse broadening can be un- derstood better by taking into account that different frequency components of a pulse travel with different speeds through the silicon nanowire. For instance, red compo- nents (low frequency components) travel faster than blue components (high frequency
−6
−4
−2
0
2
4
6
−3
−2
−1
0
1
2
3
−8
−4
0
4
8
0
0.5
1
time [T
0]
time [T
0]
power [P
0]
frequency chirp
z=0
z=2L
Dz=4L
Dz=0
z=2L
Dz=4L
D(a)
(b)
Figure 2.2:a) Pulse intensity and b) chirp as functions of normalized time for a Gaussian pulse at z = 2LDand z = 4LDpropagating in the normal dispersion regime.
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 distance [cm] broadening fac tor c<0 c>0 c=0
Figure 2.3:Broadening factor as a function of distance propagating at anomalous dispersion regime.
components) for β2> 0 while the opposite happens forβ2< 0. Therefore, the pulse
will not maintain its shape when all the spectral components do not arrive at the same time. In any other case, an unchirped pulse will broaden. One should note that GVD produces a linear chirp across the pulse even though the initial pulse was unchirped. The dispersion induced pulse broadening for an unchirped Gaussian pulse as well its dispersion induced linear chirp are shown in Fig. 2.2.
Figure 2.2 shows that the dispersion-induced linear chirp switches from positive to negative values. This feature plays a crucial role on pulse evolution in case of initially chirped pulses. More specifically, chirped pulses may broaden or compress depending on whether β2 or C have the same or opposite signs. A chirped pulse broadens at a
faster rate than that of an unchirped pulse whenβ2C > 0 . On the other hand, a chirped
pulse can be compressed for a certain propagation distance ifβ2C < 0. The evolution
of the broadening factor, final pulse width divided by the input pulse width, determined for Gaussian pulse is presented in Fig. 2.3.
where we provide the analytical expressions that describe the evolution of different pulse profiles in silicon nanowires.
2.5.2
Third order dispersion effects
As it has been mentioned in the previous sections, TOD can be neglected in many cases of practical interest. However, there are certain cases where it is necessary to take into account the TOD effects. For example, if the pulse wavelength corresponds to zero group velocity dispersion (ZGVD,β2= 0) wavelength, the TOD becomes the dominant
dispersive effect influencing the pulse propagation. In addition, when the pulse width (τ) is in the femtosecond range, it is necessary to incorporateβ3into the mathematical
model because ∆ω ≪ω0 is no longer valid so that the truncation of Eq. (2.23) is no
longer justified.
Similar to the case of the dispersion length LD, it is useful to introduce a dispersion length associated with TOD as
L′D= τ
3
|β3|. (2.36)
According to this equation, TOD plays an important role when L′D≈ LD. Thus TOD contributes significantly to pulse reshaping, making the pulse profile to be assymetric. In such cases, TOD is the main source of pulse distortion. Specifically, whenβ3> 0 an
oscillatory structure appears at the trailing edge of the pulse whereas in the case when
β3< 0 it appears at the leading edge of the pulse. The pulse assymetry is elliminated
when LD= L
′
D. The physical effects of TOD on the pulse shape are depicted in Fig. 2.4.
−80 −6 −4 −2 0 2 4 6 8 0.5 1 power [P 0 ] time [T0]
output pulse profile input pulse profile
(β2=0)
Note, that for the case of extremely short pulse widths of a few femtosecondsβ4
and higher order terms in Eq. (2.23) should be taken into consideration. Regarding some of the problems studied in this thesis, higher-order terms up to the fourth-order term (β4) have been included in the rigorous theoretical model of Eq. (2.22) even when
hundreds of femtoseconds or a few picoseconds pulse widths have been used. In addi- tion,β4plays a significant role in the nonlinear process of FWM. This is due to the fact
that β4 is one of the terms that defines the phase-matching condition for an efficient
FWM. A detailed study of FWM in silicon nanowires and silicon photonic crystal slab waveguides is provided in Chapters 6 and 7, respectively.
It is important to mention, that not only the linear effects determine the evolution of an optical pulse but also the nonlinear properties of the silicon nanowire. Thus, a description of nonlinear effects in the silicon waveguides is presented in the following section 5.2.