NONLINEAR OPTICS

ization of the Π electrons is merely the result of the resonant tunne l ing of the part ic les th rough this f inite but per iod ic potent ia l w e l l , skewed in the presence of a static external (or internal) electric f ield. The parametrizat ion is straightforward: The resonance pulse of each harmonic oscillator is deter­m i n e d f r o m the e x p e r i m e n t a l b a n d g a p of the monomer and ld is fitted to that of the dimer. The length dependence of the optical properties agrees wel l wi th the experimental results 3 , 4 and wi th the ab initio calculations. 6

Moreover, our results indicate that care must be taken w i t h the we l l - known scal ing law γ ∞Eg

-6, a

precaution confirmed by recent experiments. 7 A s to the influence of conformation, we calculate the opti­cal bandgap as a funct ion of the d ihedra l angle θ between adjacent rings. A s θ increases, the overlap between the π orbitals of the two carbon atoms on either side of the bond w i l l decrease, resulting in an increased bond length and hence, an inhibi t ion of the tunnel ing mechanism. For the w e l l - k n o w n b ipheny l molecule, good agreement w i th experiments is obtained. 2

In conclusion, we presented a simple model where res­onant tunnel ing phenomena alone account for the length dependence of the optical properties of conjugated mole­cules. The very simple parametrization of the model makes it a powerful tool to estimate the opt imum length of newly designed conjugated molecules. It is clear that opt imized tunne l i ng resul ts i n a m a x i m i z e d non l i nea r response. Recent results of Marder et al.8 also indicate that the min i ­mization of the length difference between double and single bonds in the conjugated backbone results in an enhanced 7. Furthermore, experiments by Callender et al.9 show that the presence of side-groups on the conjugated backbone only enhances the optical nonlinearity by at most one order of magnitude. We therefore conclude that the search for organ­ic materials suitable for ultrafast photonic switching should focus more on the optimization of the macroscopic ordering of molecules and on the exploitat ion of new mechanisms such as resonant enhancement.

R E F E R E N C E S 1. Y. Verbandt el al., "Optical response of conjugated polymers," Phys. Rev.

B 48, 8651 (1993). 2. Y. Verbandt et al., "Optical response of conjugated systems: Length

Static second hyperpolarizability γ (10-47m5V-2) vs. number of repeat units N for thiophene (plain line) and phenyl oligomers (dotted line). The experimental values for the thiophene1 (triangles) and for the phenyl4

(diamonds) oligomers have been renormalized for dispersion by a simple two-level Drude model. Inset: Potential profile for the thiophene hep-tamer.

dependence and confirmation," Nonlinear Optics, in press. 3. H. Thienpont et al., "Saturation of the hyperpolarizability of oligothio-

phenes," Phys. Rev. Lett. 65, 2141 (1990). 4. M. Zhao et al., "Study of third-order microscopic optical nonlinearities in

sequentially built and systematically derivatized structures," J. Phys. Chem. 93, 7916 (1989).

5. Y. Verbandt et al. "Simple quasi-free electron model for the description of linear and nonlinear optical properties of conjugated oligomers," in Organic Thin Films for Photonic Applications Technical Digest 17, 129 (Opti­cal Society of America, Washington, D.C., 1993).

6. D. Beljonne et al., "Theoretical study of thiophene oligomers: Electronic excitations, relaxation energies, and nonlinear optical properties," J. Chem. Phys. 98, 8819 (1993).

7. C. Bubeck, "Third-order nonlinearities of dye molecules and conjugated polymers," Proc. of the 1994 IEEE Nonl inear Optics Conference (Waikoloa, Hawaii, July 25-29, 1994), paper MA7, 216.

8. S. R. Marder et al., "Relation between bond-length alternation and second electronic hyperpolarizability of conjugated organic molecules," Science 261, 186 (1993).

9. C. L. Callender et al., "Assessment of third-order optical nonlinearities in conjugated organic polymers," Opt. Eng. 32, 2246 (1993).

Beyond the Lorentz-Lorenz Formula

Bγ A.V. Ghiner and G.I. Surdutovich, Instituto de Fisica, Sao Carlos, Brazil

The general method of integral equations (MIE) is devel­oped for arbitrary nonlinear and anisotropic medium, tak­ing into account quadrupole and magnetic-dipole radiation. The main idea of this generalization stems from the suppo­sition that microscopic (local) fields E and H must satisfy

the macroscopic wave equation. The wel l -known relation­ships

between micro- and macrofields for isotropic electro-dipole media suggest that, in nonlinear and anisotropic media, the add i t iona l mechanisms of rad ia t ion , to the macroscopic wave equat ion must satisfy certain new quanti t ies con­structed f rom the local f ie ld, polar izat ion vector, quadru­pole, and magnetic-dipole volume densities and their gradi­ents. Because these new quantit ies must satisfy both the integral and wave equations, al l tensor factors of the quanti-

3 4 OPTICS & PHOTONICS N E W S / D E C E M B E R 1994

OPTICS in 1994

ties depending on the concrete structure of the med ium are set unambiguously. Our approach proves an extinction the­orem in a general and physical sense whi le remaining wi th­in the f ramework of the mo lecu la r opt ics. A s a resul t , instead of the Lorentz-Lorenz (LL) formula for the dielectric permitt ivity ε, we developed a tensor relationship that, in the simplest case of the electric -dipole medium, takes the form 1

where α is the polarizability of a separate radiator and tensor and is determined by the relative disposit ion of the radia­tors. For a random ( γ =O) distribution of the isotropic ( α becomes scalar) radiators, we come to the L L formula. So, Eq.(l) solves the long-standing problem of the L L formula denominator's divergency at

For the electric-dipole medium, the local f ield factor fp takes a form

where m takes on a value 3 or 2 for the 3-D or 2-D cases, cor­respondingly. For the random medium or cubic (quadratic) lattice tensor 7 becomes zero and then in the 3-D case fp

assumes the form

which is a tensor generalization of the known

factor, whereas in 2-D case it becomes the

factor, the two-d imensional analog of the L L factor. 2 Thus, two media wi th the same macroscopic tensor may have quite different relationships between macro- and microfields. The 2-D case implies an elementary radiator as the "elongated d ipo le " made of two charged threads. Simi lar situations arise in nonlinear media wi th long-oriented molecules (self-assembled f i lms), column-l ike mesostructure (porous s i l i ­con), or photonic crystals of two-d imens iona l arrays of dielectric rods or holes in the bulk material. 3

Over the past few years, an interest has emerged i n "discrete"media (cooled atom gas, photonic band gap, and quantum dots structures), in wh ich the distances b between the radiators (lattice's size) are not too small compared wi th the wavelength 2πk-1 The foregoing analysis ho lds true only in the l imit of a dense medium, when kb—>0. We avoid­ed the wel l -known l imitat ion on the Lorentz sphere's size by considering the variation of the polarization inside it.

The second principal problem of the M IE approach for k b ≠ 0 is the passage from summation to integration outside the sphere. Un l i ke the dense m e d i u m case, n o w summa doesn't coincide exactly w i th the integral inside the Lorentz sphere. To calculate their difference, we fol lowed a special procedure of " radiator sp l i t t ing. " The lattice was repro­duced w i t h twice lesser constant b was e laborated. B y means of the reiterative repetition of such a procedure, the

difference of the acting fields for the init ial lattice and simi­lar lat t ice w i t h an arb i t ra ry s m a l l b was ca lcu la ted . It a l lowed us to realized the passage to the integral equation.

The optically indistinguishable random and cubic lat­tice dense media showed v i v i d distinction w i th allowance for only just terms (kb)2. For kb ~1, it leads to change of the local f ield corrections' sign in comparison wi th the L L for­mula. 4 Even gas and l iqu id wi th the same discreet parame­ter kb have different optical properties due to the difference of their correlation functions. These considerations are suit­able for 2-D photonic crystals or quantum confined struc­tures as wel l . 5 Note that the quadratic lattice demonstrates an optical anisotropy w i th the birefringence angle θ deter­minable by the relation

where c is a constant of the order of unity, φ is the angle be tween wavevec to r and vector of the lat t ice, and the refractive index nO is determined by the two-dimensional analog of L L formula:

The effect is m a x i m a l at the angle Π/ 8. Photon ic gaps appear at larger values of the parameter kb .

Appl icat ion of this theory has shown excellent agree­ment w i th experimental measurements of the long-wave­length dielectric constants for two orthogonal polarizations in the 2-D photonic crystal made of dielectric rods. 2 , 3

R E F E R E N C E S 1. A . V . Gh iner and G.I. Surdutov ich, " M e t h o d of integral equations and an

extinction theorem in bu lk and surface phenomena in nonl inear opt ics," Phys. Rev. A 49,1313-1323 (1994).

2. A . V . Gh iner and G.I. Surdutov ich, " M e t h o d of integral equations and an ext inct ion theorem for two-d imensional problems i n nonl inear opt ics," Phys. Rev. A 50, 714-723 (1994).

3. W . M . Robertson et al., "Measurement of the photon dispersion relation in twodimensional ordered dielectric arrays," J. Opt . Soc. A m . B 10:2, 322-327 (1993).

4. A . V . G h i n e r and G. I .Surdu tov ich , "Suscep t ib i l i t y of coo led resonance gas," Laser Physics (in press).

5. G.I. Surdutov ich et al., " L o c a l f ield effects and optical properties of two­d imens iona l quantum-conf ined structures," XVI I Enconto Nac iona l de Fisica da Mater ia Condensada, Caxambu , 334 (1994).

OPTICS & PHOTONICS NEWS/DECEMBER 1994 35