field at that order (i.e., a = dμ/dE). The higher-order terms become
important only if E is large or for specific, highly polarizable structures in
certain orientations. The constants b and g are known as the first and
second hyperpolarizability, respectively. These constants are essentially
zero for most systems when ordinary low-intensity light interacts with a
molecule. However, when light from an intense pulsed laser is used, these
nonlinear terms become significant. Equation 8.28 is the general equation
describing the effect of light on a molecule. It is important to realize that
all materials are nonlinear; it is a matter of the magnitude of the perturbing electric field that is needed to set in the anharmonicity and to
drive it to nonlinear behavior. Extending this treatment to a bulk material,
a similar equation is obtained, where the dipole (µ) is now replaced by the
average polarization (P) of the bulk material:
P = P 0 + c 1 E + c 2 E
2 + c 3 E
3 + …
(8.29)
The constants c 1 , c 2 , and c 3 are known as first-, second-, and third-order
susceptibilities. They are related to the sum components of the corresponding hyperpolarizabilities averaged over orientations of the molecules
in the bulk material. Like with the refractive index, each term in the
expansion has real and imaginary components, where the real components
represent refraction/scattering processes and the imaginary terms represent absorption/emission processes. The constant term P 0 is typically zero,
but a few materials such as lithium niobate (LiNbO 3 ) and barium titanate
(BaTiO 3 ) retain electric polarization at zero field, acting in an analogous
manner to permanent magnets. These are ferroelectric materials that are
crucial for many instruments that rely on nonlinear optical effects. The
complete mathematical descriptions of the various c terms are ignored
here, but it is should noted that at optical frequencies, c 1 is proportional to
the square of the refractive index. We can separate each polarization term
in Equation 8.29, such that P 1 = c 1 E represents the linear polarization,
P 2 = c 2 E
2 represents the second-order nonlinear polarization, and so on. In
fact, many of the nonlinear optical effects used to study nanomaterials are
based on the second-order nonlinear polarization. We focus on this term.
Let’s consider two incident intense laser sources with frequencies w 1 and
w 2 incident on a material. These two frequencies have two oscillatory
electric fields (with fields E 1 and E 2 ) simultaneously acting on the
material, given by the equation
E = E 1 cos w 1 t + E 2 cos w 2 t
(8.30)
CHAPTER 8: Surface Characterization and Imaging Methods
296
important only if E is large or for specific, highly polarizable structures in
certain orientations. The constants b and g are known as the first and
second hyperpolarizability, respectively. These constants are essentially
zero for most systems when ordinary low-intensity light interacts with a
molecule. However, when light from an intense pulsed laser is used, these
nonlinear terms become significant. Equation 8.28 is the general equation
describing the effect of light on a molecule. It is important to realize that
all materials are nonlinear; it is a matter of the magnitude of the perturbing electric field that is needed to set in the anharmonicity and to
drive it to nonlinear behavior. Extending this treatment to a bulk material,
a similar equation is obtained, where the dipole (µ) is now replaced by the
average polarization (P) of the bulk material:
P = P 0 + c 1 E + c 2 E
2 + c 3 E
3 + …
(8.29)
The constants c 1 , c 2 , and c 3 are known as first-, second-, and third-order
susceptibilities. They are related to the sum components of the corresponding hyperpolarizabilities averaged over orientations of the molecules
in the bulk material. Like with the refractive index, each term in the
expansion has real and imaginary components, where the real components
represent refraction/scattering processes and the imaginary terms represent absorption/emission processes. The constant term P 0 is typically zero,
but a few materials such as lithium niobate (LiNbO 3 ) and barium titanate
(BaTiO 3 ) retain electric polarization at zero field, acting in an analogous
manner to permanent magnets. These are ferroelectric materials that are
crucial for many instruments that rely on nonlinear optical effects. The
complete mathematical descriptions of the various c terms are ignored
here, but it is should noted that at optical frequencies, c 1 is proportional to
the square of the refractive index. We can separate each polarization term
in Equation 8.29, such that P 1 = c 1 E represents the linear polarization,
P 2 = c 2 E
2 represents the second-order nonlinear polarization, and so on. In
fact, many of the nonlinear optical effects used to study nanomaterials are
based on the second-order nonlinear polarization. We focus on this term.
Let’s consider two incident intense laser sources with frequencies w 1 and
w 2 incident on a material. These two frequencies have two oscillatory
electric fields (with fields E 1 and E 2 ) simultaneously acting on the
material, given by the equation
E = E 1 cos w 1 t + E 2 cos w 2 t
(8.30)
CHAPTER 8: Surface Characterization and Imaging Methods
296