88
4 Nonlinear Optics with Zero-Index Metamaterials
Fig. 4.3 Difference-frequency generation: a Schematic illustration b Energy band diagram
output. How it happens can be understood with the help of the band diagram shown
in Fig. 4.3b. A photon of frequency ω 1 is absorbed and excites the atom to level
2. The excited atom in the energy level 2 is stimulated by the ω 2 field to decay to
ground level via emission of two photons. The de-excitation from level 2 to 1 gives
out frequency ω 2 , and from level 1 to level 0 is accompanied by the release of a
photon of frequency ω 3 = ω 1 − ω 2 , the difference frequency. It should be noted that
the emission of ω 3 from level 1 to 0 may be spontaneous at first, but once the photons
of ω 3 appear in the system, the transition 1 → 0 acquires stimulated nature [190,
191].
For the mathematical analysis of sum and difference generation, let us assume
two input waves of frequencies ω 1 and ω 2 , which add up to give the total optical field
inside the nonlinear medium as
˜
E(t) = E 1 e
−iω 1 t
+ E 2 e
−iω 2 t
+ c.c.
(4.23)
where c.c. is E
∗
1 e
iω 1 t
+ E
∗
2 e
iω 2 t . The second-order polarization due to the total electric
field ˜
E(t) is
˜
P
(2)
(t) = 0 χ
(2) ˜
E
2
(t)
(4.24)
Substituting Eq. 4.23 in Eq. 4.24, we get
˜
P
(2)
(t) = 0 χ
(2)
[E 1 e
−iω 1 t
+ E 2 e
−iω 2 t
+ E
∗
1 e
iω 1 t
+ E
∗
2 e
iω 2 t
]
2
(4.25)
= 0 χ
(2)
[(E 1 e
−iω 1 t
+ E
∗
1 e
iω 1 t
)
2
+ (E 2 e
−iω 2 t
+ E
∗
2 e
iω 2 t
)
2
+
(4.26)
2(E 1 e
−iω 1 t
+ E
∗
1 e
iω 1 t
)(E 2 e
−iω 2 t
+ E
∗
2 e
iω 2 t
)]
(4.27)
= 0 χ
(2)
[(E
2
1 e
−i2ω 1 t
+ E
∗2
1 e
i2ω 1 t
+ 2E 1 E
∗
1 ) + (E
2
2 e
−i2ω 2 t
+
(4.28)
E
∗2
2 e
i2ω 2 t
+ 2E 2 E
∗
2 ) + 2(E 1 E 2 e
−i(ω 1 +ω 2 )t
+ E 1 E
∗
2 e
−i(ω 1 −ω 2 )t
+
(4.29)
E
∗
1 E
∗
2 e
i(ω 1 +ω 2 )t
+ E
∗
1 E 2 e
i(ω 1 −ω 2 )t
)]
(4.30)
= 0 χ
(2)
[E
2
1 e
−i2ω 1 t
+ E
2
2 e
−i2ω 2 t
+ 2E 1 E 2 e
−i(ω 1 +ω 2 )t
+
(4.31)
2E 1 E
∗
2 e
−i(ω 1 −ω 2 )t
+ c.c.] + 2 0 χ
(2)
(E 1 E
∗
2 + E
∗
1 E 2 )
(4.32)
The amplitude of second-order polarization for various frequency terms in Eq. 4.32
can be individually written and interpreted as
4 Nonlinear Optics with Zero-Index Metamaterials
Fig. 4.3 Difference-frequency generation: a Schematic illustration b Energy band diagram
output. How it happens can be understood with the help of the band diagram shown
in Fig. 4.3b. A photon of frequency ω 1 is absorbed and excites the atom to level
2. The excited atom in the energy level 2 is stimulated by the ω 2 field to decay to
ground level via emission of two photons. The de-excitation from level 2 to 1 gives
out frequency ω 2 , and from level 1 to level 0 is accompanied by the release of a
photon of frequency ω 3 = ω 1 − ω 2 , the difference frequency. It should be noted that
the emission of ω 3 from level 1 to 0 may be spontaneous at first, but once the photons
of ω 3 appear in the system, the transition 1 → 0 acquires stimulated nature [190,
191].
For the mathematical analysis of sum and difference generation, let us assume
two input waves of frequencies ω 1 and ω 2 , which add up to give the total optical field
inside the nonlinear medium as
˜
E(t) = E 1 e
−iω 1 t
+ E 2 e
−iω 2 t
+ c.c.
(4.23)
where c.c. is E
∗
1 e
iω 1 t
+ E
∗
2 e
iω 2 t . The second-order polarization due to the total electric
field ˜
E(t) is
˜
P
(2)
(t) = 0 χ
(2) ˜
E
2
(t)
(4.24)
Substituting Eq. 4.23 in Eq. 4.24, we get
˜
P
(2)
(t) = 0 χ
(2)
[E 1 e
−iω 1 t
+ E 2 e
−iω 2 t
+ E
∗
1 e
iω 1 t
+ E
∗
2 e
iω 2 t
]
2
(4.25)
= 0 χ
(2)
[(E 1 e
−iω 1 t
+ E
∗
1 e
iω 1 t
)
2
+ (E 2 e
−iω 2 t
+ E
∗
2 e
iω 2 t
)
2
+
(4.26)
2(E 1 e
−iω 1 t
+ E
∗
1 e
iω 1 t
)(E 2 e
−iω 2 t
+ E
∗
2 e
iω 2 t
)]
(4.27)
= 0 χ
(2)
[(E
2
1 e
−i2ω 1 t
+ E
∗2
1 e
i2ω 1 t
+ 2E 1 E
∗
1 ) + (E
2
2 e
−i2ω 2 t
+
(4.28)
E
∗2
2 e
i2ω 2 t
+ 2E 2 E
∗
2 ) + 2(E 1 E 2 e
−i(ω 1 +ω 2 )t
+ E 1 E
∗
2 e
−i(ω 1 −ω 2 )t
+
(4.29)
E
∗
1 E
∗
2 e
i(ω 1 +ω 2 )t
+ E
∗
1 E 2 e
i(ω 1 −ω 2 )t
)]
(4.30)
= 0 χ
(2)
[E
2
1 e
−i2ω 1 t
+ E
2
2 e
−i2ω 2 t
+ 2E 1 E 2 e
−i(ω 1 +ω 2 )t
+
(4.31)
2E 1 E
∗
2 e
−i(ω 1 −ω 2 )t
+ c.c.] + 2 0 χ
(2)
(E 1 E
∗
2 + E
∗
1 E 2 )
(4.32)
The amplitude of second-order polarization for various frequency terms in Eq. 4.32
can be individually written and interpreted as