Second-Order Nonlinear Optical Processes in Molecules and Solids

Chapter 2

Second-Order Nonlinear Optical Processes in Molecules and Solids

Downloaded by UNIV OF MISSOURI COLUMBIA on May 18, 2013 | http://pubs.acs.org Publication Date: March 11, 1991 | doi: 10.1021/bk-1991-0455.ch002

David J. Williams Corporate Research Laboratories, Eastman Kodak Company, Rochester, NY 14650-2110

In this paper, an overview of the origin of second-order nonlinear optical processes in molecular and thin film materials is presented. The tutorial begins with a discussion of the basic physical description of second-order nonlinear optical processes. Simple models are used to describe molecular responses and propagation characteristics of polarization and field components. A brief discussion of quantum mechanical approaches is followed by a discussion of the 2-level model and some structure property relationships are illustrated. The relationships between microscopic and macroscopic nonlinearities in crystals, polymers, and molecular assemblies are discussed. Finally, several of the more common experimental methods for determining nonlinear optical coefficients are reviewed. Interest in the field of nonlinear optics has grown tremendously in recent years. This is due, at least partially, to the technological potential of certain nonlinear optical effects for photonic based technologies. In addition, the responses generated through nonlinear optical interactions in molecules and materials are intimately related to molecular electronic structure as well as atomic and molecular arrangement in condensed states of matter. While much of the basic physics of nonlinear optics was developed in the 1960's and early 1970's, from both classical and quantum mechanical perspectives, progress in new materials designed to exhibit specific effects for various technologically important applications has been more recent and much remains to be done. While a variety of useful materials exist today for many of these applications, particularly inorganic crystals, many opportunities exist for new materials that can be fabricated and processed in thin film format for useful and potentially inexpensive devices. The potential for integration of devices on various substrates such as glass, Si, GaAs, etc., is a particularly attractive aspect of the organic polymeric approach. This tutorial deals with nonlinear optical effects associated with the first nonlinear term in expression for the polarization expansion described in the next section. The first nonlinear term is the origin of several interesting and important effects including second-harmonic generation, the linear electrooptic or Pockels effect,

0097-6156/91/0455-0031$06.00/0 © 1991 American Chemical Society In Materials for Nonlinear Optics; Marder, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

32

MATERIALS FOR NONLINEAR OPTICS: CHEMICAL PERSPECTIVES


various frequency mixing processes between coherent optical fields, and optical rectification where polarization of the medium at optical frequencies produces a D C response. Due to the scope of this chapter, the subject matter will be approached in a qualitative and conceptual manner, and the reader interested in learning the subject matter in greater depth will be referred to various sources of information from the literature on this subject (1). The tutorial begins with a description of the basic concepts of nonlinear optics and presents illustrations from simple models to account for the origin of the effects. The microscopic or molecular origin of these effects is then discussed in more detail. Following this, the relationship between molecular responses and the effects observed in bulk materials are presented and finally some of the experimental methods used to characterize these effects are described.

Concepts of Nonlinear Optics Lorenz Oscillator. Optical effects in matter result from the polarization of the electrons in a medium in response to the electromagnetic field associated with light propagating through the medium. A simple but illustrative model for these interactions is the Lorenz Oscillator described in a variety of texts (2). Here an electron is bonded to a nucleus by a spring with a natural frequency, co , and equilibrium displacement, r (Figure 1). The electric component of the optical field felt by the electron is represented as a sinusoidally varying field. Considering only the linear response of this object, or equivalently the harmonic oscillator approximation, the equation of motion can be written as 0

2

eE - mco r = m^- + 2T^ dt dt 0

(1)

2

The terms on the left are the electric force and the restoring force that is linear in the displacement. A damping term with a proportionally constant r is added to the right hand side to account for dissipation of energy during the polarization response. The solution to this equation leads to an expression for the displacement, r, as

2

m co -2irco-(o

2

Ee o

1C0t

+c.c.

(2)

0

This equation describes a sinusoidal response at frequency, co, to the electric field component at co. This is the basis for the linear optical response. To calculate the optical properties of the Lorenz oscillator the polarization of the medium is obtained as P =-Ner = x

( 1 )

L

E

(3)

where N is the density of polarizable units in the medium. The polarization is also often expressed in terms of a susceptibility x whose value can be readily obtained by comparison with equation 2. In the linear regime the relationship between the susceptibility and two other quantities of importance the refractive index, n, and dielectric constant, e, is given by (1)

2

e = n = l+47tx

(1)

(4)

and the optical properties of the Lorenz Oscillator as specified by the complex refractive index as

In Materials for Nonlinear Optics; Marder, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

2. WILLIAMS

2

n =l+ 2

4K

Ne 1 2

m J(D -2irco-co

2

0


33

Second-Order Nonlinear Optical Processes

' "

(5)

The real and imaginary parts of the refractive index are plotted schematically as a function of frequency in Figure 2. For the case where r= 0 there is no damping and therefore no absorption, n is real and corresponds to the refractive index of the medium. The situation where r is not equal to zero corresponds to optical absorption. This model reasonably describes the linear optical properties, in the absence of vibronic coupling, for typical organic molecules. Nonlinear optical effects can be introduced into this picture by postulating that the restoring force in equation 1 is no longer linear in the displacement and adding a term, say ar , to the left hand side of the equation, (3). The differential equation can no longer be solved in a simple way but, if the correction term is assumed to be small relative to the linear term, a straightforward solution follows leading to a modification of equation 3. 2

P=P +P L

(6)

N L

where PNL = -Ne(r (2co) + r (o)) (7) and r is the correction to the linear displacement r. The correction term oscillates at 2co rather than co and induces a D.C. offset to the displacement. The linear and nonlinear contributions to the polarization of a molecule of general structure is shown 2

2

2

schematically in Figure 3 (4). The oscillating polarization is clearly nonlinear relative to the driving field. This effect is due to the greater ease of displacement of electronic density in the direction from donor substituent, D, to acceptor, A , than vice-versa. The Fourier theorem states that a nonsinusoidal function can be expressed as a summation of sinusoidal responses at harmonics of the fundamental frequency with appropriate coefficients. A D C offset term is also associated with asymmetric functions. The polarization at the various frequencies, 2co for example, act as the sources of a new electromagnetic fields at the appropriate frequency which can lead to substantial conversion of energy at the fundamental frequency to the new frequency. The D.C. offset is the source of the optical rectification effect mentioned above. Since the coefficients of the subsequent orders in frequency tend to be a few orders of magnitude smaller than the previous one, the largest effects will be at the first harmonic (sometimes referred to as second-harmonic generation) and zero frequency. Constitutive Relations. A more general representation of the nonlinear polarization is that of a power series expansion in the electric field. For molecules this expansion is given by M-i =p = OfE + p:EE + y:EEE + ...

( 8 )

where p and E are the polarization and electric field vectors, respectively, the coefficients a, p, y are tensors, and uj is the induced dipole moment of the molecule. This expression is valid in the dipolar approximation where the wavelength of the optical field is large compared to the dimensions of the polarizable unit. The tensors a, p, and y relate the cartesian components of the electric field vectors to those of the


34


E=E coscut 0


Figure 1. Representation of the Lorenz oscillator.

Ren

Imn

Figure 2. Illustration of the optical properties of the harmonic oscillator.

L-

Fourier analysis of polarization response

\

/-

X-

P(2CJ) / \

P(0)

Figure 3. Plot of the polarization response to an incident electromagnetic field at frequency w and the Fourier components of that response. (Reprinted with permission from ref. 4. Copyright 1975 John Wiley and Sons.)


2. WILLIAMS

35


polarization vectors. The array of tensor elements is therefore intimately related to the electronic structure of the molecule. For the hypothetical molecule illustrated in Figure 3 it might be anticipated that PXXXE E » X

X

(9)

PxyyEyEy

so that (10)

E

Px ~ P x x x x E

x

since asymmetric charge displacement is the origin of the first nonlinear correction. Similarly p is expected to be zero since the molecule has a center of inversion symmetry along the y axis. An important point to note is that only molecules lacking a center of inversion can exhibit a nonzero value of p. The quantity y, however, is not related to the asymmetry of the polarization response but its departure from nonlinearity at large values of the displacement occurring in either direction. A l l molecules and atoms regardless of symmetry exhibit nonzero values of y tensor elements. A n expression similar to equation 8 can be written for the macroscopic polarization of a medium or ensemble of molecules as


y y y

P=x

U)

•E +

x

(2)

• EE

+

x

(3)

:EEE

(11)

For anisotropic media, the field and polarization terms must be treated as vectors and the coefficients as tensors for the reasons described previously. Even order terms such as x ^ are nonzero only in noncentrosymmetric media, whereas the x ^ and x ^ terms are nonzero in all media. The microscopic quantities a, p, y and related to x , (1)

v(2) v(3)

in a straightforward manner, as described in Section IV. Propagation of Light Through the Medium. The relationship between the phases of the electric fields and polarization responses as they propagate through a nonlinear medium determines the amplitude of the generated fields. The coupled amplitude equation (5) formalism is often used to illustrate the coupling of the various fields to each other which is required if they are to exchange energy with one another. While this treatment is beyond the scope of this chapter, a schematic illustration of phase dependent coupling is shown in Figure 4. The top trace is the fundamental k

field propagating in the z direction with wave vector ( o>=-^-) The harmonic polarization response at 2co propagates with wavevector 2co and its phase is inextricably linked to the fundamental field. This is sometimes referred to as the "bound wave." The electric field generated by P(2co) propagates with wavevector k2G>. Because of the dispersion in the refractive index as a function of frequency n2co * n ^ or equivalently k 2 c o * k©. As a result, the phase relationship between the "free wave" E(2co) and P(2co) varies along the z direction. The term "free wave" is used to signify that the electric field component at 2co propagates with its own phase velocity which is determined by the refractive index at 2co. The coupled amplitude equations predict the direction of power flow as a function of phase. As shown in the illustration, power flows into E(2co) in the first coherence length and back into E(co) during the second coherence length. In the example, the harmonic field amplitude is assumed to be a small fraction of the fundamental field so that the amplitude of the fundamental field does not vary significantly under these non-phase matched conditions. If phase matching can be achieved by some method, the field at E(2co) can



36


Figure 4. Illustration of the dependence of the amplitude of the harmonic field Ε(2ω) on its phase relationship with the polarization response Ρ(3ω).


2. WILLIAMS


37

become quite large and E(co) will be depleted. In a later section the circumstances under which the fundamental and harmonic fields propagate in phase will be illustrated.


Microscopic Description A number of approaches have been described in the literature for calculating the microscopic nonlinear response and are reviewed in detail elsewhere (1). In this section, these methods are briefly mentioned and a simplified version of time dependent perturbation theory is used to illustrate the intuitive aspects of the microscopic nonlinear polarization. The approaches to this problem follow along two general lines. In the first approach, one computes derivatives of the dipole moment with respect to the applied field and relates them to the terms in the polarization expansion of equation 8. Inspection of equation 8 suggests that the second derivative of the dipole moment with respect to the field gives p. The choice of the exact form of the Hamiltonian, which incorporates the optical field and the atomic basis set, determines the accuracy of this procedure. In one popular version of this approach, the finite field method, the time dependence of the Hamiltonian is ignored for purposes of simplification and the effects of dispersion on p, therefore, cannot be accounted for. A more widely used approach for organic molecules is based on second-order perturbation theory. Here the dipolar contribution to the field induced charge displacement is calculated by inclusion of the optical field as a perturbation to the Hamiltonian. Since the time dependence of the field is included here, dispersion effects can be accounted for. In this approach the effect of the external field is to mix excited state character into the ground state leading to charge displacement and polarization. The accuracy of this method depends on the parameterization of the Hamiltonian in the semi-empirical case, the extent to which contributions from various excited states are incorporated into the calculation, and the accuracy with which those excited states are described. This in turn depends on the nature of the basis set and the extent to which configuration interaction is employed. This method is generally referred to as the sum over states (SOS) method. In a simplified version of this approach a molecule is considered to have a ground and single-charge transfer excited state which dominate the hyperpolarizability

This is referred to as the 2-level model (6) and was used in the past to understand trends in p with structural modifications of the molecule. With the computational methods and experimental accuracy available today this method is viewed as inadequate although it does serve to illustrate the essential features of molecular structure that control p. In the two level model, p (the component of the tensor along the charge transfer axis) is given by x

(O fb|i

2

3ft 2n

eg

K -co )(co 2

2

e g

2

(12) -4co ) 2

where co is the frequency of the molecular charge transfer transition, f is its oscillator strength, and Su. is the difference in dipole moment between the two states. Values of p calculated by the 2-level model and published experimental measurements of p are shown in Table I (5). The calculated number in eg


38


Table I. Measured and Calculated Values of ft Using Two-Level Model 30


30

p (10- esu) (2-level)

Px(10- ) (experimental)

19.6

16.2-345

10.9

10.2

x

Molecule

227 (69.1)

225

383

450

parenthesis was obtained by a SOS calculation and shows how the 2-level model over-estimates p in this case. In other cases under-estimation may occur. Nevertheless, the trends are clear and agree with the experimentally observed trends. Para-substitution provides for a strong resonance interaction and good charge separation. For ortho substitution, the resonance interaction is retained but the charge separation is clearly lower leading to the lower value of p. In meta substitution, charge transfer resonance is forbidden due to the symmetry relationship between the ground and excited state wave functions. A n increase in molecular length results in a substantial increase in p . In many cases it is relatively straightforward to make qualitative predictions, based on physical organic principles of the effect of a change in the nature of a substituent, or some other aspect of molecular structure, on p. x

Relationship of Macroscopic to Microscopic Nonlinearities Crystals. In the discussion of equation 11, it was pointed out that the macroscopic nonlinear coefficients could be related to the microscopic ones in a relatively straightforward manner. For the hypothetical crystal shown in Figure 5, the relationship is (8) X2

) K

=2/Vf

2 1

%»f^l

[f

ijk Vs=l

cose^cos0(fcos0^ Pijk

(13)

where V is the volume of the unit cell and N is the number of equivalent sites per unit cell (in this case 2), the cosGn etc. are direction cosines between the molecular and g


2. WILLIAMS

39

Second-Order Nonlinear Optical Processes 2

crystal frames of reference and fj ", ff, f£ are local field factors that alter the value of the external fields at the molecular site due to screening effect. Zyss and Oudar (8) have tabulated the specific form of the symmetry allowed X IJK tensor elements for all of the known polar space groups. With a knowledge of the molecular hyperpolarizability, p, which can be obtained from EFISH measurements as described in the next section, and a knowledge of the crystal structure it is possible to calculate the macroscopic coefficients using estimated local field factors (9).


(

Poled Polymers. Polymeric materials offer many potential advantages (as well as some disadvantages) relative to crystalline materials for second order nonlinear optical applications. The main problem with polymer films prepared by solvent casting or other film forming techniques is that they tend to be isotropic or at best have axial symmetry due to strain fields developed in the preparation procedure. Both of these are centrosymmetric arrangements and the films do not exhibit second-order nonlinear optical properties. A n approach to circumventing this problem discovered by Meredith et al. (10) is to apply an electric field to the polymer in a softened state and tends to align molecular dipoles associated with the chromophores in the direction of the field. The alignment forces are counteracted by thermal randomization associated with the kinetic energy of the system. A hypothetical situation is illustrated in Figure 6. When the field is applied and the orientation is quenched into the film by cooling or chemical crosslinking a polar axis is induced in the Z direction. The axis defines the symmetry of the medium as polar (uniaxial). Under these circumstances there are two unique directions; parallel and perpendicular to the polar axis. Using an oriented gas model with Boltzman thermal averages Meredith et al. (10) calculated the nonlinear coefficients applicable to poled polymers. These are (14)

and (15)

where N is the concentration of the chromophore. In these expressions, 6 is defined as the angle between the molecular dipole and the poling field direction. The brackets 3

2

indicate thermal averages. In an isotropic sample is 0 and % \ is likewise zero. If total alignment could be induced in the z direction, = >1 Z Z

3

2

and the maximum value of x \ would be determined by N and p . Under these z z

z

(2) 2

conditions, = 0 and the perpendicular component, XJL , would approach 0. The problem of calculating these coefficients is one of evaluating the thermal average 3

2)

}

. It should be apparent x[| and x± are not independent quantities and that a knowledge of one quantity implies a knowledge of the other. The Boltzman thermal average of cos 0 is given by 3

(16)

where f(0) is the orientational distribution function



40


Figure 6. Schematic representation of molecular orientation in a polymer film with respect to the poling field in the z direction.


2. WILLIAMS

41

Second-Order Nonlinear Optical Processes m = eWVkT

(17)

U(0) = U'(8) - u*E

(18)


and

U(0) is the orientation dependent local potential energy and U'(8) accounts for any anisotropy due to the local environment. For instance, i f the sample were liquid crystalline, U'(O) and U'fr) would represent the potential well associated with nematic or smectic director. In addition to local potentials it is clear that f(8) depends on the experimentally controllable quantities u, E , and T. The solution to equation 16 is the series expansion referred to as the third-order Langevin function L,3(p) (cos e) = (1+ 6 / p )Li(p) - 2p +... 2

3

(19)

and Li(p) = (cosG) = — X

'

^-s- + —^—r

3p 45p

3

945p

5

^ — ^ +... 7

9450p

(OQ) U

U

;

where p = uE/kT. A plot of L3(p) vs E is given in Figure 7. A linear relationship between x and L3(p) is predicted and has been verified experimentally (10,11) for a number of polymeric systems. (2)

Langmuir-Blodgett Films. In the Langmuir-Blodgett technique, monolayers of molecules containing a hydrophilic group at one end and a hydrophobic tail on the other are spread over the surface of water in an appropriate fixture. By controlling the surface pressure the molecules can be made to organize into highly oriented structures at the air-water interface. It is also possible to transfer the monolayers onto a substrate of appropriate polarity. Subsequent deposition cycles can be employed to fabricate multilayer films of macroscopic dimensions. With appropriate choices of head and tail units noncentrosymmetric films with polar cylindrical symmetry can be fabricated (Figure 8). Equations 14 and 15 apply to films fabricated by this method. Here however, the distribution of angles, 6, is expected to be much more tightly distributed around some central value, say 8 . This being the case, it is possible to determine the value of 9 experimentally from the polarization and angular dependence of the second harmonic intensity generated by these films. Since the films can be fabricated layer by layer the concentration N is given by 0

0

N = nN

s

(21)

where n is the number of layers. A linear dependence of yP-) versus n is predicted if the film deposition process proceeds as anticipated and this has been observed (12). Experimental Methods Electric Field Induced Second-Harmonic Generation. A n essential aspect of the development of materials for second-order nonlinear optics is the determination of the p tensor components. The technique that has been developed to accomplish this is called electric field induced second harmonic generation (EFISH) (13,14).


42



1.0

Electric field (MV/cm)

Figure 7. Plot of L (p) versus electric field for various values of the dipole moment. s

Figure 8. Schematic representation of a noncentrosymmetric Langmuir-Blodgett film. The nonlinear chromophore is incorporated into alternate layers, those represented by the squares, for example.


2. WILLIAMS

43


In the EFISH method, the molecule of interest is dissolved in an appropriate solvent and put into a cell of the type shown in Figure 9. Electrodes above and below the cell provide the means for a D.C. electric field, which orients the solute (and solvent) molecules through its interaction with the molecular dipoles. Similar to the poled polymer approach, the average molecular orientation is increased along the field direction and an oriented gas model used to extract p. The EFISH experiment is formally a x process since it involves the (3)

interaction of 4 fields as indicated in argument XIJKL 0, co, co). The symbol D K L is a shorthand notation for this process. The polarization at 2co is given by


R

iCD T7*0 PI2C0 = fT*l>J K L"HEi0)"jCE E K

L

(

22)

An effective second-order nonlinear coefficient can be defined as D

IJK = IJKL L R

(23)

E

2co

so that the generation of Pj resembles a second-order under nonlinear official effect. For a pure liquid, a microscopic hyperpolarizability y can be defined by r r = N °f f^2w Y

(24)

0

where the f s are local field factors and I" — r^zzz — 3fzzyy

(25)

Designating the molecular axis parallel to the dipole moment as the z axis, -y° can be written as y°=y

+

^

(26) kT

The first term on the right, y, is the electronic contribution of y° to the polarization at 2co and the second term the contribution from p . Note that p cannot be determined from this experiment without a knowledge of the dipole moment. In compounds exhibiting significant charge transfer resonance |izP » y and the contribution of y is often ignored. The preceding discussion assumed a pure liquid was used for the measurement. Most molecules of interest, however, are not in the liquid state at room temperature. In this case it is common to dissolve the compound in an appropriate solvent and conduct the measurement. Contributions to the second harmonic signal are therefore obtained from both the solvent and solute. Since r and the local field factors that are related to e and n, (the dielectric constant and refractive index respectively) are concentration dependent, the determination of p for mixtures is not straightforward. Singer and Garito (15) have developed methods for obtaining r , e , and n , the values of the above quantities at infinite dilution, from which accurate values for p can be obtained in most cases. Referring to Figure 9 again, it should be noted that the path length varies across the cell in a manner determined by the angle a. As indicated earlier, second harmonic generation is a phase dependent process and dispersion in the refractive z

z

z

0

Q


0

44


index causes the phase velocities to be different at the fundamental and harmonic frequency. For a given pathlength, i, the phase difference, A(j>, is given by


Acj) = — A n c

(27)

where An is the difference in refractive index at the two frequencies. Typical coherence lengths for organic molecules are 10 to 20 um. For a path length in the cell of ~1 mm the bound and free waves, referred to in Figure 4, will change phase by 2n many times over the path length. As the nonlinear interaction is terminated at the glass-solution interface the phase relationship at that boundary will determine the amount of second harmonic intensity that is obtained. By translating the cell across the laser beam, the phase coherence dependence of the process can be mapped out. In the experiment, a sinusoidally varying signal is observed and is illustrated schematically in Figure 10. The intensity of the harmonic signal, l2©> is given as (14) I

=I^2sin ^H./ 2

2(0

(28)

By measuring i ^ relative to a reference HzP can be determined z

l i f t —

where Xrekrence *

1 v

st n e

(2)

l2m

,

O

O

A

2

effective value of x< ) for the reference. (2)

Characterization of Crystals. Two methods used to measure x in crystals are the Maker fringe technique and the wedge method (16). The methods are related to one another. In the Maker fringe method a crystal is rotated about an axis perpendicular to the laser beam. A n angular dependence of the phase mismatch occurs due to the differences in angles of refraction at the two wavelengths. As the crystal is rotated, fringes appear and x< tensor elements can be extracted if the signal can be compared with that from a crystal of known x . If the unknown crystal can be shaped into a wedge similar to the EFISH cell geometry, a similar sinusoidally varying signal is observed from which x ^ tensor elements can be extracted provided a reference sample of known properties is available. Quartz crystals have been characterized extensively (17) and can be obtained in the form of a wedge. They are often used as a reference material for both EFISH measurements and other crystals. Another technique for characterizing crystalline materials is the Kurtz powder technique (18). In this method, a sample is ground into a powder, spread into a thin layer, and irradiated with a laser beam. The intensity of the beam is compared with that of a known reference material, (quartz powder is often used) and conclusions are drawn. Before discussing the nature of those conclusions, the concept of birefringent phase matching is discussed. It was indicated numerous times that the SHG intensity is dependent on both the magnitude of x ^ tensor elements as well as the phase relationships between fundamental and harmonic fields in the crystal. Under certain circumstances, it is possible to achieve phase matched propagation of the fundamental and harmonic beams. Under these conditions, power is continually transferred from the fundamental to harmonic beam over a path length, which is only limited by the ability 2)

(2)

(2




2. WILLIAMS

Figure 10. Experimental EFISH trace for nitrobenzene.


45

46


to maintain good overlap between the electric fields associated with the beams. Very efficient harmonic conversion can be obtained under these conditions. One method that will be discussed for doing this is birefringent phase matching. To understand this process refer to the refractive index ellipsoid, sometimes referred to as the optical indicatrix, in Figure 11. The intersections of the ellipse with the coordinates define the principal values of the refractive index tensor. For a positive uniaxial crystal the equation for the ellipse is 2


x + y

2

(30)

where rio and n are the ordinary and extraordinary values of the refractive index. The significance of the designations are as follows. For a light beam, S, propagating through the crystal at an arbitrary angle, 8, with respect to the optic axis only two propagating components are allowed. One is polarized orthogonal to S and the optic axis. The value of the refractive index governing its propagation is independent of 8 and is designated as n . The other allowed polarization component is orthogonal to S and n . Inspection of the diagram indicates that its value is angular dependent and it is designated as n (6). At some other wavelength, the shape of the ellipsoid will be different and a point might be found on the surface of the second ellipsoid where e

Q

0

e

2co

n (8) = n®. The angle 8 at which this occurs is the phase matching angle. Since the electric vectors of the fundamental and harmonic beams are orthogonal for the polarization directions, the primary contribution to the process will have to come from the off diagonal tensor elements x ^ and X %. As discussed earlier the magnitude of these coefficients depends in detail on the molecular packing in the unit cell. A number of other phase matching schemes are available but their discussion is outside the scope of this tutorial. Returning to the discussion of the powder measurements, a schematic illustration of the possible outcomes of this experiment is given in Figure 12. In the figure, the second harmonic intensity is plotted as a function of the average particle size and the degree coherence length . If the material is capable of birefringent phase matching the signal grows with particle size and eventually saturates. If it is not phase matchable the signal decays as the particle size increases substantially beyond the coherence length. In the phase matched case the behavior is easily rationalized. Those particles with the proper orientation relative to the beam for phase matched propagation will be the primary contributors to the signal as the particle size exceeds the coherence length. On the other hand, as the particles grow there will be less of them in the beam so that the increase in the intensity begins to saturate. In practice, the only meaningful information that this method can provide is whether the material is phase matchable or not, which can provide guidance for single crystal growth strategies. A single measurement on a sample of known or unknown particle size provides no such information and comparison with a reference is ambiguous. z

Electrooptic Measurements. The final characterization method to be discussed is the measurement of the electrooptic coefficient. The electrooptic effect is derived from the process indicated by XIJK(-W;O,CO) • Since this process is derived primarily from electronic polarization (as opposed to molecular reorientation) its value is expected to be very close to that for XIJK (-2co; co, to). This has in fact been observed by Morrell and Albrecht (19) for 2-methyl-4-nitroaniline crystals.



2. WILLIAMS


47

refractive index ellipse for cu

Figure 11. Schematic representation of the refractive index ellipsoid for a positive uniaxial material at frequency w. (Reprinted with permission from Williams, D. J. Angew. Chem. Int. Ed. Engl 1984, 23, 690. Copyright VCH Publishers.)

phase matchable

I(2cJ

/

Figure 12. Dependence of the second harmonic intensity on the ration of the average particle size to the average coherence length after Kurtz (16). (Reprinted with permission from Williams, D. J. Angew. Chem. Int. Ed. Engl 1984, 23, 690. Copyright VCH Publishers.) c

American Chemical Society Library I15S 16th St., N.w. Washington, DC. 20036


48


For the sake of illustration, the determination of the electrooptic coefficient for a uniaxial crystal is described below. Considering the nonlinear uniaxial medium of Figure 11, a D.C. electric field is applied in the z direction. The effect of the electric field is to modify the refractive index in the z direction by an amount proportional to the electric field, the modified ellipsoid is given as 2

x +y

2 r

zzz^z

o

nf

(31)


where A—o"- r n

z z z

E

(32)

z

and r is a component of the electrooptic tensor. It is related to corresponding components of the x tensor by z z z

(2)

8TT/

r

(2)

__^L

=

(33)

A way of measuring this tensor component is to split a coherent beam into two paths, directing one through the crystal and recombining them at a detector. The amplitude intensity pattern resulting from the interfering beams will vary according to their phase relationship, which in turn will depend on the applied field. Measuring the voltage required to shift from the minimum to maximum signal intensity is equivalent to measuring a n phase shift. Thus the coefficient r can be determined by z z z

3

u£n r y

n=

e

zzz

jL

( 3 4 )

where d is the sample thickness, V ^ is the voltage required to provide retardation of n, and t is the pathlength through the medium. This method is quite flexible and can be extended to other tensor components by the appropriate choice of field and polarization directions. For example, rotation of the polarization to the direction perpendicular to the z axis allows determination of r

zyy

r

~ zxx*

A number of variations of this basic measurement exist for materials in waveguide and fiber formats. A n understanding of the principles involved in the approach described above provides the basic framework for successfully understanding and utilizing these related approaches. Literature Cited 1. 2. 3. 4. 5.

Prasad, P. N . ; Williams, D. J., Introduction to Nonlinear Optics in Molecular and Polymeric Materials, John Wiley: New York, 1990. Hecht, E.; Zajac, A., Optics; Addison-Wesley, Reading, 1979; p 40. Zernike, F.; Midwinter, J. E., Applied Nonlinear Optics; John Wiley: New York, 1973; p 29. Yariv, A., Quantum Electronics; John Wiley: New York, 1975; p 419. Zernike, F.; Midwinter, J. E.; p 41.


2. WILLIAMS 6. 7.

8. 9.


10. 11. 12. 13. 14. 15. 16. 17. 18. 19.


49

Oudar, J. L., Chemla, D. S.; J. Chem. Phys. 1977, 66, 2664; Oudar, J. L.; Zyss, J., Phys. Rev. A. 1982, 26, 2016. For the sources of data in Table I see references sited in Williams, D. J.; Electronic and Photonic Applications of Polymers; ACS Advances in Chemistry Series No. 218, M. Bowden and S. R. Turner, Eds.; American Chemical Society: Washington, 1988; p 307. Zyss, J.; Oudar, J. L. Phys. Rev. A 1982, 26, 2025. For a discussion of local field factors see Prasad, P. N. and Williams, D. J.; Chapter 4. Meredith, G. R.; VanDusen, J. G.; Williams, D. J. Macromolecules 1982, 15, 1385. Singer, K. D.; Kuzyk, M. G.; Sohn, J. E. J. Opt. Soc. Am. B4, 968 (1987). Neal, D. B.; Petty, M . C.; Roberts, G. G.; Ahmad, M . M.; Feast, W. J.; Girling, I. R.; Cade, N. A.; Kolinsky, P. V., Peterson, I. R. Electron. Lett. 1986, 22, 460. Levine, B. F.; Bethea, C. G. J. Chem. Phys. 1975, 63, 2666. Oudar, J. L. J. Chem. Phys., 1977, 67, 466. Singer, K. D.; Garito, A. F., J. Chem. Phys., 1981, 75, 3572. Kurtz, S. K. Quantum Electronics; Editors, H. Robin and C. Tang, Academic Press, New York, 1975; Vol. 1, 209ff. Jerphagnon, J.; Kurtz, S. K. J. Appl. Phys. 1970, 41, 1667. Kurtz, S. K.; Perry, J. J. Appl. Phys. 1968, 39, 3798. Morrell, J. A.; Albrecht, A. C. Chem. Phys. Lett. 1979, 64, 46.

RECEIVED August 28, 1990


Second-Order Nonlinear Optical Processes in Molecules and Solids

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