86
4 Nonlinear Optics with Zero-Index Metamaterials
Fig. 4.1 Second-harmonic generation: a Schematic illustration b Energy band diagram. FF =
fundamental frequency, SH = second harmonic
2ω is the second-harmonic frequency. Figure 4.1 illustrates the process of secondharmonic generation by means of a block diagram and an energy band diagram.
The figure shows that when the fundamental frequency (ω) is fed as input into a
nonlinear medium of high second-order susceptibility, a fraction of it gets converted
into second harmonic (2ω) and the output is a mixture of two frequencies ω and
2ω. The band diagram shows the mechanism of the conversion process, in which it
can be seen that two photons of frequency ω are absorbed and an atom is excited
to energy level 2, and then the atom de-excites to the ground level, and one photon
of frequency 2ω is emitted. The percentage of the input power converted into the
second harmonic is called the conversion efficiency of the process. The mathematical
analysis of the second-harmonic generation process is given as follows.
Let the input electric field of the fundamental frequency ω be given as
˜
E(t) = Ee
−iωt
+ c.c.
(4.16)
where c.c. is the complex conjugate of the preceding quantity [11, 183]. The contribution of the second-order susceptibility into the polarization is
˜
P
(2)
(t) = 0 χ
(2)
[ ˜
E(t)]
2
(4.17)
= 0 χ
(2)
[Ee
−iωt
+ c.c.]
2
(4.18)
= 0 χ
(2)
[Ee
−iωt
+ E
∗ e
iωt
]
2
(4.19)
= 0 χ
(2)
[E
2 e
−2iωt
+ E
∗2 e
2iωt
+ 2E E
∗
]
(4.20)
= 0 χ
(2)
[2E E
∗
+ E
2 e
−2iωt
+ E
2∗ e
2iωt
]
(4.21)
= 0 χ
(2) 2E E
∗
+ [ 0 χ
(2) E
2 e
−2iωt
+ c.c.]
(4.22)
The first term on the right-hand side of Eq. 4.22 is the zero-frequency term and of not
much relevance here, while the second term represents the existence of the secondharmonic frequency. A common practical example of second-harmonic generation is
the production of 530 nm green laser from Nd:YAG [185–187] laser of infrared wavelength 1060 nm. The second-harmonic generation has been a very useful technique
to develop lasers of desired frequencies.
4 Nonlinear Optics with Zero-Index Metamaterials
Fig. 4.1 Second-harmonic generation: a Schematic illustration b Energy band diagram. FF =
fundamental frequency, SH = second harmonic
2ω is the second-harmonic frequency. Figure 4.1 illustrates the process of secondharmonic generation by means of a block diagram and an energy band diagram.
The figure shows that when the fundamental frequency (ω) is fed as input into a
nonlinear medium of high second-order susceptibility, a fraction of it gets converted
into second harmonic (2ω) and the output is a mixture of two frequencies ω and
2ω. The band diagram shows the mechanism of the conversion process, in which it
can be seen that two photons of frequency ω are absorbed and an atom is excited
to energy level 2, and then the atom de-excites to the ground level, and one photon
of frequency 2ω is emitted. The percentage of the input power converted into the
second harmonic is called the conversion efficiency of the process. The mathematical
analysis of the second-harmonic generation process is given as follows.
Let the input electric field of the fundamental frequency ω be given as
˜
E(t) = Ee
−iωt
+ c.c.
(4.16)
where c.c. is the complex conjugate of the preceding quantity [11, 183]. The contribution of the second-order susceptibility into the polarization is
˜
P
(2)
(t) = 0 χ
(2)
[ ˜
E(t)]
2
(4.17)
= 0 χ
(2)
[Ee
−iωt
+ c.c.]
2
(4.18)
= 0 χ
(2)
[Ee
−iωt
+ E
∗ e
iωt
]
2
(4.19)
= 0 χ
(2)
[E
2 e
−2iωt
+ E
∗2 e
2iωt
+ 2E E
∗
]
(4.20)
= 0 χ
(2)
[2E E
∗
+ E
2 e
−2iωt
+ E
2∗ e
2iωt
]
(4.21)
= 0 χ
(2) 2E E
∗
+ [ 0 χ
(2) E
2 e
−2iωt
+ c.c.]
(4.22)
The first term on the right-hand side of Eq. 4.22 is the zero-frequency term and of not
much relevance here, while the second term represents the existence of the secondharmonic frequency. A common practical example of second-harmonic generation is
the production of 530 nm green laser from Nd:YAG [185–187] laser of infrared wavelength 1060 nm. The second-harmonic generation has been a very useful technique
to develop lasers of desired frequencies.