74
3 Light–Matter Interactions for Photonic Applications
Fig. 3.6 Fundamental exciton–polariton model, freely sketched after [87]. Here, the harmonic
oscillator approximation of the macroscopic polarisation in a dielectric medium is the starting point
for the description of a coherent exciton’s dipole resonance with oscillator strength f according to the
material specific dielectric function This also yields the exciton–polariton dispersion branches.
a Imaginary part of in the vicinity of the transversal exciton resonance ω 0 = ω T indicating a peak
centered at ω 0 for different damping situations γ . For zero damping, a δ peak would be obtained,
for small damping, a Lorentzian line profile results which broadens with increased damping. ω 0
is found in the real part of in b as the frequency of the singularity
±
1 (ω → ω 0 ) → ±∞ for the
undamped oscillator (γ = 0). Note that negative 1 lead to the so-called Reststrahlenbande (i.e.
the lack of a propagating mode, cf. photonic stop band), which is smeared out with increasing
damping, and reduced with spatial dispersion (k-dependent LP adds a propagating mode with
high k in the range of ω T to ω L ). For small damping (i.e. smaller than the longitudinal–transverse
splitting LT ), the two opposing branches connect smoothly and ω 0 lies at the crossing of 1
with the background dielectric constant level b . The zero-crossing of 1 marks the longitudinal
frequency ω L . s = b + f /ω 0 is the static dielectric constant, an approximately constant value well
below ω 0 . In fact, the low and high frequency values b/s are linked to the two resonances by the
Lyddane–Sachs–Teller relation s // b = ω 2
L /ω 2
T > 1 for f > 0. For larger damping (γ → LT ),
the resonance smears out and no zero crossing of 1 occurs. This example of an isolated resonance
far away from other resonances of the material is well described by the classical Lorentz oscillator
model (for details see e.g. [87], Chaps. 4, 5, 13 are particularly inviting in this context). For vanishing
damping, f = (ω 2
L − ω 2
T )) b ∝ |M| 2 (the dipole transition matrix element squared). c and d show
the theoretical model for exciton–polaritons with spatial dispersion for two cases, without and
with small damping, respectively. Wide/short-dashed nearly-vertical lines sketch the light cone in
vacuum/medium (c 0/n k) and dotted curves the dispersion of the excitonic resonances ω L/T . Due to
the interaction of the light field with the polarisation in matter, the coupled system is represented by
the lower/upper polariton branches (LP/UP), with clear anticrossing of light and transverse exciton
mode for the undamped case. Note that the slope of the light cone changes from
√
s to
√ b
from LP to UP (i.e. below to above ω 0 ). (a–d) Horizontal dotted lines mark the levels of ω T , to which
the LP and UP (real and imaginary part, respectively, short-dashed curves) converge in c in the case
without spatial dispersion, and ω L , from where the UP begins in c at k = 0. For larger damping
(γ → LT , not shown here), the interesting and rather common situation leads to a dominant role of
the UP within the light cone for the matter resonance with real and imaginary parts of the dispersion
relation closely resembling the shape of b 1 and a 2 , respectively. This is, however, not surprising,
since Re/Im {k} = Re/Im { ˜
n} ωc −1 (i.e. ∝ refractive index n / extinction κ)
3 Light–Matter Interactions for Photonic Applications
Fig. 3.6 Fundamental exciton–polariton model, freely sketched after [87]. Here, the harmonic
oscillator approximation of the macroscopic polarisation in a dielectric medium is the starting point
for the description of a coherent exciton’s dipole resonance with oscillator strength f according to the
material specific dielectric function This also yields the exciton–polariton dispersion branches.
a Imaginary part of in the vicinity of the transversal exciton resonance ω 0 = ω T indicating a peak
centered at ω 0 for different damping situations γ . For zero damping, a δ peak would be obtained,
for small damping, a Lorentzian line profile results which broadens with increased damping. ω 0
is found in the real part of in b as the frequency of the singularity
±
1 (ω → ω 0 ) → ±∞ for the
undamped oscillator (γ = 0). Note that negative 1 lead to the so-called Reststrahlenbande (i.e.
the lack of a propagating mode, cf. photonic stop band), which is smeared out with increasing
damping, and reduced with spatial dispersion (k-dependent LP adds a propagating mode with
high k in the range of ω T to ω L ). For small damping (i.e. smaller than the longitudinal–transverse
splitting LT ), the two opposing branches connect smoothly and ω 0 lies at the crossing of 1
with the background dielectric constant level b . The zero-crossing of 1 marks the longitudinal
frequency ω L . s = b + f /ω 0 is the static dielectric constant, an approximately constant value well
below ω 0 . In fact, the low and high frequency values b/s are linked to the two resonances by the
Lyddane–Sachs–Teller relation s // b = ω 2
L /ω 2
T > 1 for f > 0. For larger damping (γ → LT ),
the resonance smears out and no zero crossing of 1 occurs. This example of an isolated resonance
far away from other resonances of the material is well described by the classical Lorentz oscillator
model (for details see e.g. [87], Chaps. 4, 5, 13 are particularly inviting in this context). For vanishing
damping, f = (ω 2
L − ω 2
T )) b ∝ |M| 2 (the dipole transition matrix element squared). c and d show
the theoretical model for exciton–polaritons with spatial dispersion for two cases, without and
with small damping, respectively. Wide/short-dashed nearly-vertical lines sketch the light cone in
vacuum/medium (c 0/n k) and dotted curves the dispersion of the excitonic resonances ω L/T . Due to
the interaction of the light field with the polarisation in matter, the coupled system is represented by
the lower/upper polariton branches (LP/UP), with clear anticrossing of light and transverse exciton
mode for the undamped case. Note that the slope of the light cone changes from
√
s to
√ b
from LP to UP (i.e. below to above ω 0 ). (a–d) Horizontal dotted lines mark the levels of ω T , to which
the LP and UP (real and imaginary part, respectively, short-dashed curves) converge in c in the case
without spatial dispersion, and ω L , from where the UP begins in c at k = 0. For larger damping
(γ → LT , not shown here), the interesting and rather common situation leads to a dominant role of
the UP within the light cone for the matter resonance with real and imaginary parts of the dispersion
relation closely resembling the shape of b 1 and a 2 , respectively. This is, however, not surprising,
since Re/Im {k} = Re/Im { ˜
n} ωc −1 (i.e. ∝ refractive index n / extinction κ)