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Fundamentals of Polarization and Polarizability

Fundamentals of Polarization and Polarizability. Seth R. Marder Joseph W. Perry Department of Chemistry. Polarizability: A Microscopic View. F = qE (1). E 2. E 1. E = 0. Molecular polarization = µ =   (2).

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Fundamentals of Polarization and Polarizability

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  1. Fundamentals of Polarization and Polarizability Seth R. Marder Joseph W. Perry Department of Chemistry

  2. Polarizability: A Microscopic View F = qE (1) E2 E1 E = 0 Molecular polarization = µ =  (2)

  3. Effect of Application of an Oscillating Electric Field such as Light • Application of an oscillating electric field will induce an oscillating polarization in a material. • For linear polarizability, the polarization will have the same frequency as the applied electric field. • This induced polarization is a source of light will propagate through the material in the same direction as the light beam that created it. E or P 0

  4. Mechanisms of Polarization • The oscillating electric field of light affects all charges in the optical medium, not only the electrons. • Vibrational polarization and involves nuclear motion. • Dipolar molecules can rotate to create molecular polarization. • In ionic materials, the ionic motion can cause polarization.

  5. Dipoles in Electric Fields • For materials that contain electric dipoles, such as water molecules, the dipoles themselves reorient in the applied field.

  6. Anisotropic Nature of Polarizability • The polarization of a molecules need not be identical in all directions. dipole moment

  7. Each entry of the tensor is a component of the polarizability Tensorial Nature of Polarization Polarizability is a tensor quantity as shown below: mx Ex = my Ey mz Ez axx Ex + axyEy + axz Ez eg. mx =

  8. Polarizability: A Macroscopic View In bulk materials, the linear polarization is given by: Pi() = ij( Ej( (4) i,j where ij() is the linear susceptibility of an ensemble of molecules The total electric field (the "displaced" field, D) within the material becomes: D = E + 4P = (1 + 4E (5) Since P = E (Equation (4)), 4E is the internal electric field created by the induced displacement of charges (polarization).

  9. a i + a r + a v + a e a r + a v + a e a v + a e a e e VISIBLE IR RADIO MICROWAVE The Dielectric Constant The dielectric constant and the refractive index n(w) are two bulk parameters that characterize the susceptibility of a material. e(w) is defined as the ratio of the displaced internal field to the applied field (e = D/E) in that direction: ij() = 1 + 4ij() (6) The frequency dependence of the dielectric constant provides insight into the mechanisms of polarization. Frequency

  10. The Index of Refraction The ratio of the speed of light in a vacuum, c, to the speed of light in a material, v, is called the index of refraction (n): n = c/v. (7) The dielectric constant equals the square of the refractive index: e(w) = n2(w). (8) Consequently, we can relate the refractive index to the bulk linear (first-order) susceptibility: n2(w) = 1+ 4pc(w). (9) Index of refraction depends therefore on chemical structure.

  11. Harmonic and Anharmonic Potential Surfaces • Harmonic potentials gives rise to linear polarizability • Anharmonic potentials gives rise to nonlinear polarizability

  12. Asymmetric Polarization • With the addition of this anharmonicity the induced polarization depends on the direction of displacement. • For the covalent C=O bond in acetone, for example, one expects that the electron cloud would be more easily polarized towards the oxygen atom.

  13. Creation of an Asymmetric Polarization Wave • The application of an oscillating electric field to the electrons in an anharmonic potential leads to an asymmetric polarization response. • This polarization wave has diminished maxima in one direction and accentuated maxima in the opposite direction.

  14. Fourier Analysis of Asymmetric Polarization Wave • This asymmetric polarization can be Fourier decomposed into a DC polarization component and components at the fundamental and second harmonic frequencies. • Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

  15. Expression for Microscopic Nonlinear Polarizabilities • A common approximation is to expand the polarizability as a Taylor series: m = m0 + E (∂mi/¶Ej)Eo+ (1/2) E·E (¶2mi/¶EjEk)Eo + (1/6) E·E·E (¶3mi/¶EjEkEl)Eo + … (10) m = m0 + aijE + (ßijk/2) E·E + (gijkl /6) E·E·E + ... (11) • The terms beyond aijE are not linear in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects.

  16. Expression for Macroscopic Nonlinear Polarizabilities • The observed bulk polarization density is given by an expression analogous to (11): P = Po + c(1) ij ·E j + (c(2) ijk/2)·· E jE k + (c(3) ijkl/6)··· E jE kE l + ... (12) Where: • the c(i) susceptibility coefficients are tensors of order i+1 (e.g., c(2)ijk). • Po is the intrinsic static dipole moment density of the sample.

  17. Taylor Expansion for Bulk Polarization Or when all the fields are identical: P = Po + c(1)·E + (1/2)c(2)·· E2 + (1/6)c(3)···E3+ … (13) • Just as a molecule can only have a b if it is noncentrosymmetric, a material can only have a c(2) if the material is noncentrosymmetric. (i.e., a centrosymmetric arrangement of noncentrosymmetric molecules lead to zero c(2)).

  18. Taylor Expansion with Oscillating Electric Fields-SHG Recalling that the electric field of a plane light wave can be expressed as: E = E0cos(wt), (14) equation (14) can be rewritten as: P = P0+ c(1)E0cos(wt) + c(2) E02cos2(wt) + c(3) E03cos3(wt) + ... (15) Since cos2(wt) equals 1/2 + (1/2) cos(2wt), the first three terms of equation (13) become: P = (P0+ (1/2) c (2) E02) + c (1)E0cos(wt) + (1/2) c(2)E02cos(2wt) + .. (16)

  19. Second Harmonic Generation (SHG) P = (P0+ 1/2c (2) E02) + c (1)E0cos(wt) + 1/2 c(2)E02cos(2wt) + … (16) Physically, equation (16) states that the polarization consists of a: • Second-order DC field contribution to the static polarization (first term), • Frequency component w corresponding to the light at the incident frequency (second term) and • A new frequency doubled component, 2w (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

  20. Sum and Difference Frequency Generation In the more general case, NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material. Consider two laser beams E1 and E2, the second-order term of equation (4) becomes: • (1/2) c(2)·E1cos(w1t)E2cos(w2t) (17) From trigonometry we know that equation (17) is equivalent to: (1/2)c(2)·E1E2cos [(w1+w2)t] + (1/2)c(2)·E1E2cos [(w1-w2)t] (18) thus when two light beams of frequencies w1and w2 interact in an NLO material, polarization(light) is created at sum (w1+w2) and difference(w1-w2) frequencies.

  21. Changing the Propagation Characteristics of Light: The Pockels Effect • It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. Consider the special case w2 = 0 [equation (17)] in which a DC electric field is applied to the material. The optical frequency polarization (Popt) arising from the second-order susceptibility is: (1/2) c(2)·E1E2(cos w1t) (19) where E2 is the magnitude of the electric field due to the voltage applied to the nonlinear material.

  22. Pockels Effect Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (15): c(1)·E1(cos w1t), (20) so the total optical frequency polarization is: Popt = c(1)·E1(cos w1t) + (1/2) c(2)·E1E2(cosw1t) (21) Popt = [c(1)+ (1/2) c(2)·E2] E1(cos w1t) (22)

  23. Changing the Propagation Characteristics of Light: The Pockels Effect (cont.) • The applied field in effect changes the linear susceptibility and thus the refractive index of the material. • This is, known as the linear electrooptic (LEO) orPockels effect, and is used to modulate light by changing the applied voltage. • At the microscopic level, the applied voltage anisotropically distorts the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

  24. Technological Applications Of The Pockels Effect • The Pockels effect has many important technological applications. • Light traveling through an electrooptic material can be phase of polarization modulated by refractive index changes induced by an applied electric field. • Devices exploiting this effect include optical switches, modulators, and wavelength filters.

  25. Design of Molecules for Nonlinear Optics A prototypical NLO chromophore was 4-(N,N-dimethylamino)-4'-nitrostilbene (DANS), which is shown below

  26. Role for Materials Chemists

  27. Bond Length Alternation

  28. Resonance Structures and BLA

  29. Electric Field Perturbation of Structure

  30. Linear Polarizability and BOA

  31. First Hyper-polarizability and BOA

  32. Factors Affecting Charge Separation

  33. Manipulation of BLA Through Topology

  34. Good, But Not Good Enough

  35. Second Hyper-polarizability and BOA

  36. Third-order Nonlinear Optical Properties of Polarized Polyenes

  37. Molecular Two-Photon Absorption--an Imaginary Side of Third Order Nonlinearity

  38. S n S 2 h  Electron Transfer A Energy Transfer S h  A 1 Photochemistry Two-photon Absorptivity Fluorescence  h  F  h  fl A S 0 Two-Photon Excited Processes

  39. -2 z I  2 TPA  I -4 TPA  z Two-Photon Processes Provide 3-D Resolution Excitation by two photons is confined to a volume very close to focus where intensity is highest, giving rise to pinpoint 3D resolution Excitation by one photon results in absorption along the entire path of the laser beam in the cuvette.

  40. TPA Provides Improved Penetration of Light Into Absorbing Materials Excitation by two photons of half the energy allows for penetration through the material, and then two photons can be absorbed by the sample Excitation by one photon results in absorption by surrounding medium before beam reaches sample

  41.  ≈ 10 x 10-50 cm4 s photon-1  ≈ 200 x 10-50 cm4 s photon-1 Effect of bis-Donor Substitution

  42. 0.1 0.05 0 -Charge Difference -0.05 -0.1  -0.15 N Phenyl Vinyl Phenyl N Proposed Model to Enhance TPA in Symmetrical Molecules • BDAS has large and symmetrical charge transfer from nitrogens to central vinyl group that is associated with large transition moment between S(1) and S(2). • These results suggest that a large change in quadrupole moment between S(0) and S(1) is leads to enhanced  Group

  43. Strategies for the Design of New Materials D--D Increase conjugation length Add electron acceptors to the backbone DD ADA Also:

  44. Chain-Length Dependence • Method: Two-photon induced fluorescence (TPF) • Pulse duration: ≈ 5 ns I II III • With increasing chain length: •  increases • (2)max red-shifts IV

  45. Design of TPA Chromophores p D-p-D D-A-D A-D-A 210 12 4700 995 1940 53 1250 2300 Albota et al., Science 1998 d in 10-50 cm4 s/photon

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