Determination of the eiectrooptic coefficient of a poled polymer film Jay S. Schildkraut

Eastman Kodak Corporate Research Laboratories, Roch­ester, New York 14650. Received 6 June 1989. 0003-6935/90/192839-03$02.00/0. © 1990 Optical Society of America.

The use of the rate of change of the relative phase shift of p- and s-polarized light with voltage to determine the eiec­trooptic coefficient of a poled polymer film is discussed.

Polymer films which contain chromophores with a high second-order molecular hyperpolarizability are of increasing interest because of their application in guided-wave nonlin­ear optics.1 Films are normally formed by spin coating a polymer solution onto a substrate. An electric field must be applied to the film heated to near its glass transition tem­perature to align the chromophores and obtain a film with a large eiectrooptic coefficient r. This process is referred to as poling.

The eiectrooptic effect can be observed by passing through the film an oblique light beam that is linearly polarized at an angle of 45° with respect to the plane of incidence. A voltage applied across the film changes the phase of p- and s-polar-ized light by a different amount. This Letter is concerned with the use of the rate of change of the relative phase shift of p- and s-polarized light with voltage, δΔ/δ V, to determine r of a poled polymer film. This method is advantageous to wave­guide methods of determining r in that buffer layers and waveguide couplers such as a prism or grating are not re­quired.

Khanarian et al.2 state a relationship between δΔ/δVand r which neglects the birefringence of the polymer film caused by the poling process and the effect of reflection of the light beam at interfaces. We present a derivation of a more gener­al relationship between δΔ/δV and r which includes the effect of poling birefringence but not of reflections. This relation­ship reduces to that of Khanarian et al. in the limit of zero poíing birefringence. Then, to determine the importance of reflections and associated interference effects, we compare δΔ/δV obtained from this equation with δΔ/δV determined numerically using anisotropic Fresnel equations.

First, we consider two idealized situations involving only a poled polymer on a substrate with no other layers present: a reflection experiment in which no reflection occurs at the air-film interface and light is completely reflected at the film-substrate interface and a transmission experiment in which no reflection occurs at the air-film interface and light is completely transmitted at the film-substrate interface. In these conditions the change in phase of a light beam of polarization v on propagation through a film of thickness d is

where kvz is the component of the wave vector of the light

normal to the plane of the film and d is the thickness of the film; m is equal to 1 (2) in a transmission (reflection) experi­ment.

Mosteller and Wooten3 have derived equations for the case of an uniaxial film with its optical axis normal to the plane of the film which relate kv

2 to nt and nn, the in-film-plane and normal-to-film-plane index of refraction, respectively. For s-polarized light

and for p-polarized light

where 0 is the angle of incidence and k = 2π/λ. Equations (2) and (3) also define an effective propagation angle of s- and p-polarized light in the film φs and φp, respectively.4

Using the definition of Δ as the difference in the phase shift of p- and s-polarized light the voltage-induced phase shift δΔ is defined as

where Ep(s) denotes the complex amplitude of the optical electric field of p(s) polarization, and a zero subscript de­notes zero applied voltage.

From Eqs. (l)-(4) we obtain

With the use of Eqs. (2) and (3) the differential on the right-hand side of Eq. (5) may be written solely in terms of δnt and δnn. The first term in the square bracket on the right side of the equation is

The second term is

Inserting Eqs. (6) and (7) into Eq. (5) we obtain

Equation (8) is a general relationship between δΔ, δnt and δnn, which applies to strongly poled films for which poling birefringence cannot be neglected. δnt and δnn are related to the elements of the eiectrooptic tensor r13 and r33 by

where δ V is the change in voltage across the film. We now specialize to the case of weak poling and make the

approximations nn ≈ nt and r33 ≈ 3r13· With these approxi­mations Eq. (8) becomes

Equation (10) is equivalent to Eq. (4) in Ref. 2. The affect of poling birefringence on the value of r33 calcu­

lated from measured δΔ/δV values can be determined by comparing the results of Eqs. (8) and (10). In this calcula­tion we assume that the affect of poling is to change the index ellipsoid5 from spherical to an ellipsoid of rotation about the major axis (positive uniaxial) in such a way that its volume remains unchanged. This implies that after poling

1 July 1990 / Vol. 29, No. 19 / APPLIED OPTICS 2839

Fig. 1. Percent deviation of r33 calculated using Eq. (10), which assumes an isotropic film from r33 calculated using Eq. (8), which includes the affect of poling birefringence as a function of δ (see

text).

Fig. 2. Poled polymer film on ITO covered glass with a top gold electrode. The electrooptic coefficient of the polymer film is deter­mined from the voltage-induced relative phase shift of p- and s-

where n is the refractive index before poling and δ is in general an unknown function of the poling field. This as­sumption concerning poling birefringence is supported by birefringence measurements reported in Ref. 6. In Fig. 1 we show the percent deviation of r33 calculated using Eq. (10), which assumes an isotropic film from r33 calculated using Eq. (8), which includes the effect of poling birefringence as a function δ. For this calculation we have used n = 1.55, θ = 70°, λ = 632.8 nm, and r33 = 3rí3. This result is independent of the magnitude of r33 because of the linearity of both equations. We conclude that poling birefringence can result in the underestimation of r33 if not taken into consideration.

We now consider the affect of interfacial reflections on the calculation of the electrooptic coefficient. Equations (8) and (10) are of value because they provide a linear relation­ship between δΔ/δV and r. As mentioned earlier they strict­ly apply only to an isolated poled polymer film on a perfectly transmitting or reflecting substrate with no reflections oc­curring at the air-film interface. Experiments generally involve substrates that are neither perfectly transmitting nor reflecting and also include electrodes and sometimes addi­tional layers. Light will be partially transmitted and reflect­ed at all the interfaces, and overlapping beams will interfere. These effects will differ for p- and s-polarized light. For this reason we now compare δΔ/δV calculated using Eq. (10) to a numerical calculation, which includes these effects. This numerical calculation involves using the Fresnel factors that are derived in Ref. 4, which apply to the interface between an isotropic and uniaxial media to determine the reflectance or transmittance of a multilayer structure containing a poled polymer film. δΔ/δV is obtained by performing this calcula­tion at different values of V and using Eqs. (9a) and (9b) to relate V to δnt and δnn of the poled polymer film.

We specialize in a configuration of practical importance that is shown in Fig. 2: a poled polymer film on ITO covered glass on which a gold electrode has been evaporated. For the numerical calculation it is necessary to determine which beams are to be included. Since the variation in glass thick-

Fig. 3. Comparison of Ω (see text) vs the angle of incidence deter­mined numerically (solid line) and from Eq. (10) (dashed line). The parameters used in the calculation are: λ = 632.8 nm; nglass = 1.52; nITO

= 2.0; d ITO = 200 nm; np o l y m e r = 1.55, dpolymer = 1.5 μm; UAu

= 0.18 + 3.439i.

ness will normally be »λ , the phase relation between rays A and B in Fig. 2 will not generally be know, and, therefore, ray A must be blocked. Also, multiple reflections in the glass caused by ray C will be spatially separated, and it will often be unclear how many impinge on the metal electrode and are collected. Therefore, it is best to mask the multiple reflec­tions in the glass.

For the purpose of comparing Eq. (10) to the numerical calculation we define a dimensionless parameter

Ω is independent of r33 because, in general, δΔ/δV changes linearly with r33. Figure 3 shows Ω plotted as a function of the incident angle calculated numerically (ΩN) and from Eq. (10) (Ω10). The two calculations closely coincide for θ below 20°, but at higher angles ΩN follows Ω10 but alternately passes below and above it as θ increases. From these results we conclude that Eq. (10) represents a general approximation of the relationship of δΔ/δ V to r33. A more accurate determina­tion of r33 from δΔ/δV can be performed by numerically

2840 APPLIED OPTICS / Vol. 29, No. 19 / 1 July 1990

polarized light reflected from the metal electrode.

calculating δΔ/δVby applying anisotropic Fresnel equations to the multilayer containing the poled polymer film, but this requires that the refractive index and thickness of all the layers be accurately known. The affect of interfacial reflec­tions on the determination of r33 of a film cannot be eliminat­ed by comparing δΔ/δV with that of a film for which r33 is known unless both the standard and film with unknown r33 are situated in the same multilayer configuration and have identical optical properties and thickness.

References 1. D. Williams, "Polymers in Nonlinear Optics," ACS Adv. Chem.

Ser. 218, 297-330 (1988).

2. G. Khanarian et al., "Characterization of Polymeric Nonlinear Organic Materials," Proc. Soc. Photo-Opt. Instrum. Eng. 824, 72-78 (1988).

3. L. P. Mosteller, Jr., and F. Wooten, "Optical Properties and Reflectance of Uniaxial Absorbing Crystals," J. Opt. Soc. Am. 58, 511-518 (1968).

4. M. J. Dignam, M. Moskovits, and R. W. Stobie, "Specular Reflec­tance and Ellipsometric Spectroscopy of Oriented Molecular Layers," Trans. Faraday Soc. 67, 3306-3317 (1971).

5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 673.

6. W. H. G. Horsthuis and G. J. M. Krijnen, "Simple Measuring Method for Electro-Optic Coefficients in Poled Polymer Wave­guides," Appl. Phys. Lett. 55, 616-618 (1989).

1 July 1990 / Vol. 29, No. 19 / APPLIED OPTICS 2841