Download The Role of Electric Polarization in Nonlinear optics

The Role of Electric Polarization in Nonlinear optics
Sumith Doluweera
Department of Physics
University of Cincinnati
Cincinnati, Ohio 45221
Abstract
Nonlinear optics became a very active field of research in both experimental and
theoretical physics after the invention of LASER. Nonlinear nature is very important
for optical data processing applications and all physics of nonlinear optics can be extracted by studying the behavior of applied optical field-induced polarization of the
medium. In this article very basic properties of nonlinear electric susceptibilities are
discussed qualitatively.
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1
Introduction
In linear optics it is assumed that an optical disturbance propagating through an optical
medium can be described by a linear wave equation. As a consequence of that assumption
regardless of the intensity of light, the principle of superposition is valid in the regime of
linear optics. When the intensity of light becomes great enough, linear optics is no longer
enough to describe the situation observed experimentally [2]. It seems that light waves
interact with one another and optical medium. These nonlinear phenomena require an
extension of the linear theory, that allows for a nonlinear response of optical materials to
the electromagnetic radiation.
It was realized that the electric polarization induced in the medium is the key to study
physics of nonlinear optics and so in general the polarization of the medium should be
expressed as a power series of applied electric field E.
P = ²0 [χ(1) E + χ(2) EE + χ(3) EEE + ......].
(1)
Where P is the polarization,²0 is the free space permitivity and χ(i) is the i th order
susceptibility tensor(material coefficient) of a given medium. First approximation of this
equation gives linear optics and higher order terms contribute for nonlinear optics.
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Theoretical framework of Nonlinear optics
. Two major theoretical approaches can be employed in nonlinear optics as well as laser
optics. They are semi-classical theory and quantum electrodynamical theory. In the semiclassical theory the media composed of molecules and atoms are described by quantum
mechanics and light radiation is described by Maxwell’s theory.
The key issue of semi-classical theory is to give the expressions of macroscopic nonlinear
electric polarization for optical media. The density matrix method is used to derive the
expressions for various orders of susceptibilities and the expressions for various orders of
polarization components. Substituting the appropriate nonlinear polarization components
into the generalized wave equation, in principle ,it is possible to predict many possible
nonlinear optical responses of the medium for a given condition of the input intense optical
fields[1].
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Fundamental understanding of nonlinear polarization.
Under the interaction of an applied optical electric field, the atoms or molecules of the
medium may respond in two ways: (i) a real transition of a certain amount of atoms
or molecules from one quantum eigen state another, and (ii) the perturbation of normal
distribution or motion of internal electric charges within an atoms or molecules. The first
is predominant for resonant interactions and the second is predominant for non or near
resonant interactions. Optical field-induced electric dipole moment which, in turn, will
act as a new source to emit a secondary electromagnetic wave. This is the fundamental
process describing the optical field-induced dipole moment of a molecular system and the
re-emission of a secondary wave radiation. Assuming that the number of molecules in a
unit volume is N and induced dipole moment of i th molecule is pi then P is determined
by
P(t) =
N
X
i=1
2
pi (t)
(2)
. From Eq. (2) it can be seen that the electric polarization of a medium is determined
by two factors: one is the field induced dipole moment of each individual molecule of
the medium, and the other is the statistically averaged property of a great number
of molecules. That is the molecular dipole moment is determined by the microscopic
structure or their eigen functions; whereas the results of the summation depends on the
microscopic symmetry. Now taking the n th order Fourier component of polarization
P(n) (t)and the Fourier component of E(t) it can be shown that [1]
P(n) (ω) = ²0 χ(n) (ω1 , ω2 , ..., ωn )E(ω1 )E(ω2 )......E(ωn ).
(3)
.
The values and properties of χ(n) can be determined from the semi-classical theory
mentioned above and it is very complex and difficult. So By considering special features
of interaction between the incident light beam and the given medium and then simplifying
the above model it is possible to extract some useful conclusions.
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Various mechanisms causing Nonlinear polarization in a
medium.
Depending on the applied field frequency and the phase of the medium one or more of the
mechanisms discuss below become the major contributor to the nonlinear behavior of the
medium.
4.1
Distortion of electronic cloud
The change of the outer-shell electronic cloud of an atom, ions or molecule compare to
the undisturbed status without applying an external optical electric field. This mechanism has very fast response time and approximately less than (10−16 − 10−15 ) seconded:
most optical frequency mixing effects such as second harmonic and third harmonic generation, sum-frequency generation, optical parametric oscillation and four-photon parametric
interaction.
4.2
Intramolecular motion
It denotes the polarization contribution from an optical-field induced relative motion (vibration, rotation, etc .) between the nuclei(or ions) within a molecule. The response
time of this mechanism is around 10−12 second. Ex: some Raman resonance-enhanced
four-wave-mixing effects and Raman enhanced refractive index change.
4.3
Molecular reorientation
It denotes the additional electric polarization contribution from an optical field induced
reorientation of anisotropic molecules in a liquid. The response time of this process is
dependent on the rotational viscosity of molecules in the liquid and is approximately
(10−12 − 10−13 ) second. Ex: stimulated Kerr scattering and Kerr-effect related refractive
index change.
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4.4
Induced acoustic motion
It is the polarization contribution from an optical induced macroscopic acoustic motion
related to the so called electrostriction interaction. the response time of this mechanism
is (10−9 − 10−10 ) second depending on the phase state of the medium. Ex ; Brillouin
scattering, self focusing and optical breakdown damage.
4.5
Induced population change
The distribution of molecules in their different eigen states may be changed owing to
one-photon or two-photon absorption or Raman transition for resonant interaction. the
response time of this additional polarization of the medium is strongly depend on the
dynamic properties of molecular transition and relaxation of the medium. and in general
slower than that of first three mechanisms.
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Manipulation of nonlinear susceptibilities
The main issue of nonlinear optics is to study the interaction between the laser field and
media. In many cases, the applied laser field is a coherent monochromatic electromagnetic
wave with narrow spectral line width and a small divergent angle. Since the laser field has
only one or more monochromatic Fourier components, the field-induced nonlinear polarization can also be recognized as a combination of limited number of Fourier components.
According to Eq.(3), the linear component is given by
P(1) (ω) = ²0 χ(1) (ω)E(ω)
(4)
Second-order nonlinear polarization is given by
P(2) (ω = ω1 + ω2 ) = ²0 χ(2) (ω1 , ω2 )E(ω1 )E(ω2 )
(5)
Physical meaning of this equation is that the radiation at new frequency ω = ω1 + ω2 can
be generated by two incident monochromatic waves with frequencies of ω 1 and ω2 .The χ(2)
is a third order tensor having 3 × 9 = 27 elements. The third order nonlinear polarization
is given by
P(3) (ω = ω1 + ω2 + ω3 ) = ²0 χ(2) (ω1 , ω2 , ω3 )E(ω1 )E(ω2 )E(ω3 )
(6)
.
This equation tells that in the third order approximation, the radiation at a new
frequency ω = ω1 + ω2 + ω3 can be generated by an intense incident field containing ω1 , ω2
and ω3 .The χ(3) is a fourth order tensor having 3×27 = 81 elements. The above equations
(4),(5) and(6) tell that various-order of susceptibilities are the key parameters to describe
the nonlinear coupling between different incident waves, as well as the generation of new
frequency radiation through the induced nonlinear electric polarization of the medium.
Based on the theoretical analyses [1] it is understood that various orders of susceptibilities of a medium possess the following basic properties, which help researchers
to considerably simplify theoretical work and have a better understanding the related
nonlinear optical processes.
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5.1
Relative magnitude of various orders of susceptibilities
If the major contribution for polarization is the distortion of electronic cloud of a molecular
system, the approximation
|χ(n) |/|χ(n−1) | ' 1/|E0 |
(7)
is valid and where |E0 | is the magnitude of average electric field strength inside an atom
( it is about 1011 V/m for a hydrogen atom.). according to Eq.(3) the ratio between
successive polarization can be roughly estimated by
|p(n) |/|p(n−1) | ' |χ(n−1) |/χ(n) | ' |E|/|E0 |.
(8)
Here |E| is the magnitude of an applied optical field. For ordinary light sources the ratio
of |E|/|E0 | is so small that all nonlinear terms can be neglected. If the applied incident
light is of high spectral intensity the ratio |E|/|E0 | is not small and second and third order
contributions may play vital roles.
5.2
Spatial symmetry restrictions on susceptibilities
The susceptibility tensors must remain unchanged upon the symmetric operation allowed
for the medium. This reduces the number of the independent and nonzero elements of the
susceptibility tensors. The most important conclusion from this property is that for all
centrosymmetric crystals and isotropic media (gases, liquids and amorphous solids), all
tensor elements of even-order susceptibilities (χ(2) , χ(4) ...) must be zero. Therefore no second order nonlinear effects can be observed for such media. But odd-odder susceptibility
tensors will not be zero and they provide nonlinear effects.
5.3
Resonance enhancement of susceptibility
When one or a combination of several frequency components of the applied electric field
approaches a resonant frequency of the medium , the magnitudes of the tensor elements
for appropriates orders of susceptibilities can be significantly increased.
5.4
Permutation symmetry of susceptibilities
The following permutation symmetry relations of tensor elements (in Cartesian coordinates) for second order and third order susceptibilities hold [1];
(2)
(2)
χijk (ω1 , ω2 ) = χijk (ω2 , ω1 )
and
(3)
(3)
(3)
χijkl (ω1 , ω2 , ω3 ) = χikjl (ω2 , ω1 , ω3 ) = χiljk (ω3 , ω1 , ω2 ) = ....
5.5
(9)
(10)
complex conjugation and time reversal symmetry
It can be proved that various orders of susceptibilities are complex quantities in general.
Since the polarization response must be a real quantity, the manipulation of complex
conjugation for χ(n) leads to
[χ(n) (ω1 , ω2 , .., ωn )]∗ = χ(n) (−ω1 , −ω2 , .., −ωn )
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equation Furthermore if we consider the non resonant case, χ(n) can be approximately
viewed as real quantities, and then we obtain [1]
[χ(n) (ω1 , ω2 , .., ωn )]∗ = χ(n) (ω1 , ω2 , .., ωn )
(12)
And finally considering both of above
χ(n) (ω1 , ω2 , .., ωn ) = χ(n) (−ω1 , −ω2 , .., −ωn )
(13)
This implies that susceptibility tensors are invariant for simultaneous change of the sign
for all frequency arguments. This is the time reversal symmetry and holds only for nonresonant interaction.
By using these symmetries the number of independent and nonzero elements of χ(n) can
be further reduced and greatly simplify the analyses.
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Conclusion
Nonlinear phenomena are ultimately due to inability of dipoles in the optical medium to
respond in a linear fashion to the alternating electric field associated with a light beam.
Understanding the various orders(mainly second and third) of susceptibility tensors and
the behavior of their elements of the optical medium subject to given condition is the
basic problem of nonlinear optics.
References
[1] Guang S. He and Song H. Liu, Physics of Nonlinear Optics.
[2] Frank L.Pedrotti, S.J.and Leno S. Pedrotti., Introduction to Optics.
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