3.7 Zero Goos–Hänchen Shift
79
determined to be d s = L cos θ i ≈ 0.2828 µm. The free-space wavelength λ 0 used
in this simulation is 1 µm, which reduced to λ 1 = λ 0 /1.5 inside medium 1. The
angle of incidence θ i = 45
o , the permittivity 2 = 0.001, and scattering boundary
condition have been used on all the four sides.
The leakage of power into the cladding results in a reduction of the intensity of
a reflected beam, which becomes a major drawback in systems involving multiple
reflections of light, where it propagates while losing some power (however small
it is) on every reflection. The use of zero-index metamaterials provides an efficient
solution for this problem.
3.8 Thresholdless Lasers
We know that in a crystal lattice, there are several energy levels available for atoms to
occupy. Atoms can absorb energy from outside and excite to higher energy levels or
release energy and de-excite to lower energy levels. According to Boltzmann’s law,
the number of atoms occupying higher energy levels is relatively lesser than those
occupying lower energy levels, at thermal equilibrium [170–172]. Figure 3.20 shows
two energy levels, E 1 and E 2 , having population densities N 1 and N 2 , respectively.
According to Boltzmann’s law, N 2 /N 1 = e
−(E 2 −E 1 )/K B T , hence N 2 < N 1 . Between
the two shown energy levels, there are three types of transitions possible—absorption,
spontaneous emission, and stimulated emission. Atoms of energy level E 1 can absorb
incoming radiation of frequency ω = (E 1 − E 2 )/ and excite to the E 2 . On the other
hand, an atom can de-excite from level 2 to level 1 by releasing energy E 1 − E 2 ,
or in other words, a photon of frequency ω = (E 1 − E 2 )/. While absorption can
only be of the stimulated type, emission can be either spontaneous or stimulated. The
spontaneous emission happens on its own, depends on the number of atoms in the
excited state only and does not require the presence of the optical field. Whereas the
stimulated emission requires radiation of a particular frequency to occur, and hence
depends on the relative power of various frequencies of the spectrum as well as on the
number of atoms in the excited state [11, 173]. In Optical Electronics (2011), Ghatak
and Thyagarajan have presented a detailed analysis of atomic transitions inside a laser
system, using Einstein’s A and B coefficients [11]. In a system illustrated in Fig. 3.20,
the number of stimulated transitions 1 → 2 is given by
12 = B 12 N 1 u(ω)
(3.18)
the number of stimulated transitions 2 → 1 is given by
21 = B 21 N 2 u(ω)
(3.19)
and the number of spontaneous transitions 2 → 1 is given by
79
determined to be d s = L cos θ i ≈ 0.2828 µm. The free-space wavelength λ 0 used
in this simulation is 1 µm, which reduced to λ 1 = λ 0 /1.5 inside medium 1. The
angle of incidence θ i = 45
o , the permittivity 2 = 0.001, and scattering boundary
condition have been used on all the four sides.
The leakage of power into the cladding results in a reduction of the intensity of
a reflected beam, which becomes a major drawback in systems involving multiple
reflections of light, where it propagates while losing some power (however small
it is) on every reflection. The use of zero-index metamaterials provides an efficient
solution for this problem.
3.8 Thresholdless Lasers
We know that in a crystal lattice, there are several energy levels available for atoms to
occupy. Atoms can absorb energy from outside and excite to higher energy levels or
release energy and de-excite to lower energy levels. According to Boltzmann’s law,
the number of atoms occupying higher energy levels is relatively lesser than those
occupying lower energy levels, at thermal equilibrium [170–172]. Figure 3.20 shows
two energy levels, E 1 and E 2 , having population densities N 1 and N 2 , respectively.
According to Boltzmann’s law, N 2 /N 1 = e
−(E 2 −E 1 )/K B T , hence N 2 < N 1 . Between
the two shown energy levels, there are three types of transitions possible—absorption,
spontaneous emission, and stimulated emission. Atoms of energy level E 1 can absorb
incoming radiation of frequency ω = (E 1 − E 2 )/ and excite to the E 2 . On the other
hand, an atom can de-excite from level 2 to level 1 by releasing energy E 1 − E 2 ,
or in other words, a photon of frequency ω = (E 1 − E 2 )/. While absorption can
only be of the stimulated type, emission can be either spontaneous or stimulated. The
spontaneous emission happens on its own, depends on the number of atoms in the
excited state only and does not require the presence of the optical field. Whereas the
stimulated emission requires radiation of a particular frequency to occur, and hence
depends on the relative power of various frequencies of the spectrum as well as on the
number of atoms in the excited state [11, 173]. In Optical Electronics (2011), Ghatak
and Thyagarajan have presented a detailed analysis of atomic transitions inside a laser
system, using Einstein’s A and B coefficients [11]. In a system illustrated in Fig. 3.20,
the number of stimulated transitions 1 → 2 is given by
12 = B 12 N 1 u(ω)
(3.18)
the number of stimulated transitions 2 → 1 is given by
21 = B 21 N 2 u(ω)
(3.19)
and the number of spontaneous transitions 2 → 1 is given by