## Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures

Optics Express, Vol. 14, Issue 19, pp. 8866-8884 (2006)

http://dx.doi.org/10.1364/OE.14.008866

Acrobat PDF (640 KB)

### Abstract

We present new closed-form expressions for analysis of Teng-Man measurements of the electro-optic coefficients of poled polymer thin films. These expressions account for multiple reflection effects using a rigorous analysis of the multilayered structure for varying angles of incidence. The analysis based on plane waves is applicable to both transparent and absorptive films and takes into account the properties of the transparent conducting electrode layer. Methods for fitting data are presented and the error introduced by ignoring the transparent conducting layer and multiple reflections is discussed.

© 2006 Optical Society of America

## 1. Introduction

2. D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. **94**, 31–75 (1994). [CrossRef]

3. M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, and A. Dienes, “Second-harmonic generation and absorptionstudies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B **6**, 733–741 (1989). [CrossRef]

4. R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, “Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry,” J. Appl. Phys. **75**, 1869–1874 (1994). [CrossRef]

6. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. **83**, 1848–1853 (1998). [CrossRef]

7. H. Uchiki and T. Kobayashi, “New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer,” J. Appl. Phys. **64**, 2625–2629 (1988). [CrossRef]

8. D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, “Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation,” J. Opt. Soc. Am. B **10**, 1894–1900 (1993). [CrossRef]

9. L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, “Second-order nonlinear optical measurements in guest-host and side-chain polymers,” J. Appl. Phys. **68**, 456–465 (1990). [CrossRef]

10. S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, “Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films,” Appl. Phys. Lett. **69**, 275–277 (1996). [CrossRef]

11. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. **56**, 1734–1736 (1990). [CrossRef]

12. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. **29**, 2839–2841 (1990). [CrossRef] [PubMed]

*s*- and

*p*- polarized waves of a single laser beam reflected from the sample when a modulating voltage is applied across two parallel-plate electrodes. A transmission method was first mentioned by Schildkraut [12

12. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. **29**, 2839–2841 (1990). [CrossRef] [PubMed]

13. K. Clays and J. S. Schildkraut, “Dispersion of the complex electro-optic coefficient and electrochromic effects in poled polymer films,” J. Opt. Soc. Am. B **9**, 2274–2282 (1992). [CrossRef]

*et al*. [14

14. P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong“Electro-optic characterization of poled-polymer films in transmission,” App. Phys. Lett. **69**, 901–903 (1996). [CrossRef]

*et al*. [15

15. F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, “Measurement of the electro-optic properties of poled polymers at λ=1.55 µm by means of sandwich structures with zinc oxide transparent electrode,” Appl. Phys. Lett. **83**, 4477–4479 (2003). [CrossRef]

20. F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique,” Chem. Phys. **245**, 311–325 (1999). [CrossRef]

*et al*. [16] and Chollet,

*et al*. [17

17. P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, “Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers,” Thin Solid Films , **242**, 132–138 (1994). [CrossRef]

*et al*. [18

18. G. Khanarian, J. Sounik, D. Allen, S. F. Shu, C. Walton, H. Goldberg, and J. B. Stamatoff, “Electro-optic characterization of nonlinear-optical guest-host films and polymers,” J. Opt. Soc. Am. B **13**, 1927–1934 (1996) [CrossRef]

19. S. H. Han and J. W. Wu, “Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration,” J. Opt. Soc. Am. B **14**, 1131–1137 (1997). [CrossRef]

13. K. Clays and J. S. Schildkraut, “Dispersion of the complex electro-optic coefficient and electrochromic effects in poled polymer films,” J. Opt. Soc. Am. B **9**, 2274–2282 (1992). [CrossRef]

13. K. Clays and J. S. Schildkraut, “Dispersion of the complex electro-optic coefficient and electrochromic effects in poled polymer films,” J. Opt. Soc. Am. B **9**, 2274–2282 (1992). [CrossRef]

20. F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique,” Chem. Phys. **245**, 311–325 (1999). [CrossRef]

21. Y. Shuto and M. Amano, “Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films,” J. Appl. Phys. **77**, 4632–4638 (1995). [CrossRef]

## 2. Theory

*mm*(space group C

_{∞v}) [22] and has complex ordinary and extraordinary indices of refraction,

*ñ*

_{o}=

*n*

_{o}+

*iκ*

_{o}and

*ñ*

_{e}=

*n*

_{e}+

*iκ*

_{e}, respectively. Two independent complex electro-optic tensor elements

*r̃*

_{13}=

*r*

_{13}+

*is*

_{13}and

*r̃*=

*r*

_{33}+

*is*

_{33}determine the variations

*δñ*

_{o}and

*δñ*

_{e}of the complex refractive indices when an electric field

*E*

_{3}is applied to the film according to [13

**9**, 2274–2282 (1992). [CrossRef]

*µ*=1 or 3 (

*n*

_{1}=

*n*

_{o}and

*n*

_{3}=

*n*

_{e}). For a parallel plate structure,

*E*

_{3}=

*V*/

*d*, where

*V*is the peak voltage of the AC signal applied to the sample and

*d*is the thickness of the film. Usually it is argued that the real part,

*δn*

_{µ}, of

*δñ*

_{µ}depends only on

*r*

_{µ}

_{3}and the imaginary part,

*δκ*

_{µ}, depends only on

*s*

_{µ}

_{3}outside the polymer absorption band under the assumptions that

*n*≫

*κ*,

*nδκ*≫

*κδn*[13

**9**, 2274–2282 (1992). [CrossRef]

*κδκ*≪

*nδn*. A detailed description of the separation of Eq. (1) into real and imaginary parts is given in Appendix A, where we show that this simplification is not valid when the measurement wavelength is in the absorption band of the film.

### 2.1 General Expressions

11. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. **56**, 1734–1736 (1990). [CrossRef]

**9**, 2274–2282 (1992). [CrossRef]

*I*

_{dc}(Ω) and the modulated intensity

*I*

_{m}(

*V*, Ω) are collected. The optical bias curve is the intensity profile obtained by varying the retardation generated by the SBC with no application of voltage to the sample, while the modulation data set is obtained by applying an AC voltage

*V*sin(

*ωt*) to the sample and using a lock-in amplifier synced to the fundamental frequency of the applied voltage to record the resulting modulation of

*I*

_{dc}(Ω) for a given retardation.

*I*

_{dc}(Ω) can be expressed in terms of the complex reflection coefficients

*r*

_{s}and

*r*

_{p}of the

*s*- and

*p*- polarized waves, the intensity of the incident laser

*I*

_{o}, and the retardation in the form [15

15. F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, “Measurement of the electro-optic properties of poled polymers at λ=1.55 µm by means of sandwich structures with zinc oxide transparent electrode,” Appl. Phys. Lett. **83**, 4477–4479 (2003). [CrossRef]

17. P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, “Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers,” Thin Solid Films , **242**, 132–138 (1994). [CrossRef]

*V*, the modulated intensity

*I*

_{m}(

*V*, Ω) is obtained by differentiating Eq. (2) to get

*I*

_{dc}and modulation

*I*

_{m}are shown in Fig. 2. We note in general that

*I*

_{m}is not symmetric with respect to the horizontal axis and the modulated intensities at points 1 and 2 are not maximum and minimum, respectively. From Eq. (6), δΨ

_{sp}can be extracted by measuring the modulated intensities

*I*

_{m}

_{1}and

*I*

_{m}

_{2}at the two optical bias points corresponding to Ψ

_{sp}+Ω=

*π*/2, 3

*π*/2 as

*I*

_{c}=

*B*/2.

*δB*/

*B*can be obtained from the modulated intensities

*I*

_{m}

_{3}and

*I*

_{m}

_{4}at the two bias points corresponding to Ψ

_{sp}+Ω=0,

*π*according to

*I*

_{m}is measured only at points 1 and 2 as a function of voltage and the average of the difference in the slopes in plots of

*I*

_{m}versus

*V*is used in Eq. (7) to extract δΨ

_{sp}. This verifies that the applied voltage is low enough to stay in the linear regime, as required for the validity of Eq. (7). Usually,

*I*

_{m}is positive at point 1 and negative at point 2. But this is not always the case so care must be taken to note the phase of the reading on the lock-in amplifier, synced to the fundamental frequency of the applied voltage

*V*sin(

*ωt*) (in our case, 1 kHz). A similar technique can be applied to obtain

*δB*/

*B*from data at points 3 and 4 using Eq. (8).

*I*

_{dc}(point 3) and maximum

*I*

_{dc}(point 4), b) set the compensator to obtain the average of these two intensities to locate point 1 and measure

*I*

_{m}1, c) dial the compensator through the maximum to the same dc average on the other side to locate point 2 and measure

*I*

_{m}

_{2}, and finally d) use Eq. (7) to determine δΨ

_{sp}. This can be expected to give as accurate a value for δΨ

_{sp}as fitting a full modulation curve, provided multiple measurements are averaged. However, to measure

*δB*/

*B*using Eq. (8) requires precisely locating the maximum and minimum in the bias curve, but these are points where the slope of

*I*

_{dc}versus Ω is zero. One can show that for a small error ΔΩ in the compensator setting in locating points 3 and 4, the relative error in

*δB*/

*B*from using Eq. (8) is -[δΨ

_{sp}/

*δB*/

*B*)]ΔΩ. With moderate reflectivity modulation and (

*δB*/

*B*)/δΨ

_{sp}≈1 (higher ratios are possible), and if the error in determining the maximum

*I*

_{dc}is only 1% corresponding to ΔΩ≈0.2, then the relative error in determination of

*δB*/

*B*is already 20%. Better accuracy can be obtained by fitting the full

*I*

_{dc}versus Ω curve to reduce ΔΩ.

*I*

_{m}(

*V*, Ω) curves [15

15. F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, “Measurement of the electro-optic properties of poled polymers at λ=1.55 µm by means of sandwich structures with zinc oxide transparent electrode,” Appl. Phys. Lett. **83**, 4477–4479 (2003). [CrossRef]

20. F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique,” Chem. Phys. **245**, 311–325 (1999). [CrossRef]

*I*

_{dc}(Ω) as shown in Fig. 2 and then fit these data to Eqs. (2) and (6) to extract

*A*,

*B*, Ψ

_{sp},

*δA*,

*δB*, and δΨ

_{sp}. Acquiring the full modulation curve provides a visual check for consistency as well as improved statistics on the extracted parameters. Note that in fitting the optical bias curve we put Ψ

_{sp}+Ω→

*cx*+

*d*, where

*x*is the SBC setting, and obtain

*c*and d along with

*A*and

*B*from the fit. These are then used in fitting Eq. (6) to extract

*δA*,

*δB*, and δΨ

_{sp}.

*B̃*as

*δB̃*/

*B̃*.

*r̃*

_{13}=

*γr̃*

_{33}(

*γ*is generally assumed to be 1/3 for weak poling [2

2. D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. **94**, 31–75 (1994). [CrossRef]

*δñ*

_{o,e}, the complex quantity

*δB̃*/

*B̃*can be written as a linear function of

*r*

_{33}and

*s*

_{33}in the form

*r*

_{13}/

*r*

_{33}is not equal to

*δn*

_{o}/

*δn*

_{e}because of birefringence and that both δΨ

_{sp}and

*δB*/

*B*depend on both

*r*

_{33}and

*s*

_{33}. Once δΨ

_{sp}and

*δB*/

*B*are determined experimentally at any single angle of incidence, Eq. (12) can be inverted to solve for

*r*

_{33}and

*s*

_{33}provided

*H*

_{r}and

*H*

_{s}that determine the 2×2 matrix are known. This matrix depends on the linear properties of the multilayered structure and its form for the simple and rigorous models is discussed in the next two sections.

### 2.2 Simple Model

*s*- and

*p*- waves can be expressed as

*s*- and

*p*- waves normal to the film surface are defined as

*k*

_{o}=2

*π*/

*λ*is the wave vector in free space,

*ñ*

_{s}=

*ñ*

_{o},

*ñ*

_{p}is given in Eq. (B2), and

*and*θ ˜

_{s}*are the complex propagation angles inside the anisotropic nonlinear medium. Inserting Eq. (13) into the general expression Eq. (2) gives the optical bias curve. We introduce lowercase letters*θ ˜

_{p}*a*,

*b*, and

*ψ*

_{sp}instead of

*A*,

*B*, and Ψ

_{sp}to designate the simple model values. We emphasize the complex nature of the propagation constants

*β*

_{s,p}by writing

*β*

_{s}≡

*β*

_{sr}+

*iβ*

_{si}and

*β*

_{p}≡

*β*

_{pr}+

*iβ*

_{pi}. Then, the phase retardation inside the film is given by

*a*and

*b*depend only on the imaginary parts of the propagation constants, while ψ

_{sp}depends only on the real parts.

*h*

_{r}and

*h*

_{s}are complex quantities given by

*N*=sin

*θ*as defined in Eq. (B1) and we have used

*r*

_{13}=

*γr*

_{33}and

*s*

_{13}=

*γs*

_{33}. The functions

*h*

_{r}and

*h*

_{s}have nonzero real and imaginary parts inside the absorption band. Outside the absorption band, we make the approximations discussed in Appendix A to compare with formulas reported earlier [12

12. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. **29**, 2839–2841 (1990). [CrossRef] [PubMed]

**9**, 2274–2282 (1992). [CrossRef]

**245**, 311–325 (1999). [CrossRef]

21. Y. Shuto and M. Amano, “Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films,” J. Appl. Phys. **77**, 4632–4638 (1995). [CrossRef]

*h*

_{r}and

*h*

_{s}vanish, and we get

**29**, 2839–2841 (1990). [CrossRef] [PubMed]

21. Y. Shuto and M. Amano, “Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films,” J. Appl. Phys. **77**, 4632–4638 (1995). [CrossRef]

*r*

_{p}/

*r*

_{s}|) was used to calculate the electrochromic effect, but

*δ*|

*r*

_{s}|=0 was assumed. In Ref. 16, the variation of complex phase retardation was introduced to estimate the electrochromic effect in the simple model, but their formula for the electrochromic effect is equivalent to that in Refs. 13 and 20 where δ|rs|=0 is assumed.

### 2.3 Rigorous Model

*s*- and

*p*- reflectance [16, 17

17. P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, “Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers,” Thin Solid Films , **242**, 132–138 (1994). [CrossRef]

23. F. Wang, E. Furman, and G.H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. App. Phys. **78**, 9–15 (1995). [CrossRef]

**29**, 2839–2841 (1990). [CrossRef] [PubMed]

24. D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications,” J. Opt. Soc. Am. A **22**, 1577–1588 (2005). [CrossRef]

*r*

_{s}and

*r*

_{p}have the form

*s*and

*p*subscripts to prevent the notation from becoming unduly cumbersome. In Eq. (26) the

*s*or

*p*propagation constant

*β*

_{j}in each layer

*j*is defined in Eq. (14) and the corresponding reflection coefficient from layer

*j*to

*k*is given by

*t*

_{12}and

*t*

_{21}, at the air-glass interface [17

**242**, 132–138 (1994). [CrossRef]

_{sp}and

*δB*/

*B*in our analysis. Using Eq. (26), the detailed expressions for the function

*H*that appear in Eq. (11) are given in Appendix C.

### 2.4 Data Analysis

*r*

_{33}and

*s*

_{33}. Data at multiple angles can be analyzed by solving Eq. (12) for

*r*

_{33}and

*s*

_{33}at each angle and then calculating mean values for

*r*

_{33}and

*s*

_{33}. Alternatively, for experimental data at

*n*angles of incidence, we can construct a 2

*n*×2 matrix that is derived by stacking up Eq. (12) to form a large matrix equation

*r*

_{33}and

*s*

_{33}simultaneously using several matrix decomposition methods such as QR and singular value decomposition to implement least squares fitting [27]. Using QR decomposition, the matrix

*M*can be decomposed into

*Q*and

*R*which are 2

*n*×2

*n*orthogonal and 2

*n*×2 upper triangular matrices, respectively and we define

*P̂*as

*R*and the 2×1 upper matrix from

*P̂*allows one to obtain

*x*,

*R*

_{lower}is the (2

*n*-2)×2 null matrix and the magnitude of

*P*

_{lower}represents the goodness of least squares fitting. Because the linear parameters (including the refractive index and thickness of the film) that determine the

*H*functions have experimental errors associated with their measurement, it is possible to tweak their values within their experimental uncertainty range and recalculate Eqs. (29)–(31) to attempt to improve the goodness of fit of

*r*

_{33}and

*s*

_{33}. Ref. 17 followed a more complicated approach by finding numerical fits to the rigorous expressions using the simplex method to fit the modulated intensities at three bias points as a function of angle. The complex variations of refractive indices as shown in Eq. (1) were not taken into account although it should be included for a highly absorptive medium.

## 3. Results

*r*

_{33}and

*s*

_{33}from the rigorous model are calculated from Eq. (12) together with Eqs. (C1) and (C2), while the simple model values

*s*

_{33}/

*r*

_{33}, but not on the absolute magnitude of either coefficient. Outside the absorption band of the polymer, Re(

*h*

_{r})=Re(

*h*

_{s})=0 for the simple model as discussed in Sec. 2.2.

*n*

_{2}=

*n*

_{3}=

*n*

_{4}=1.5 in the absence of an applied voltage. One might think that the simple model should apply exactly in this case, but the index matching condition is broken upon application of a voltage leading to some reflectivity modulation, which is ignored in the simple model. Using

*r*

_{23}=

*r*

_{34}=0 and

*r̂*

_{45}=

*r*

_{45}(valid for gold thicker than ~75 nm), the reflection coefficient in Eq. (26) simplifies to

*d*

_{4}of the polymer film changes, while the third term is due to modulation of the reflectivity of the polymer-gold interface and contributes an offset. The terms involving the complex quantities

*K*

_{1},

*K*

_{2}, and

*K*

_{3}are inversely proportional to the film thickness, but the last term involving the real quantity

*K*

_{4}is independent of film thickness and equal to

*h*

_{r}in the simple model shown in Eq. (24). With

*s*

_{33}=0, the error expression Eq. (33) takes the form

*s*

_{33}/

*r*

_{33}is assumed to be 1 and 2 at 0.8 µm, and to be 0.1 at wavelengths 1, 1.3, and 1.55 µm for the simulation. The complex index of refraction of ITO measured by ellipsometry and the complex index of refraction of a representative doped NLO polymer film are shown in Fig. 5. The optical properties of ITO are strongly dependent on the manufacturing process so there are wide variations in commercial ITO properties. Free carrier absorption is usually noticeable in the near infrared range, whereas interband transitions dominate in the visible range [28]. The gold metal layer is assumed to be thicker than ~75nm because we have found that for thicknesses greater than this the thickness of the gold layer can be ignored because no light is reflected back from the gold/air interface.

18. G. Khanarian, J. Sounik, D. Allen, S. F. Shu, C. Walton, H. Goldberg, and J. B. Stamatoff, “Electro-optic characterization of nonlinear-optical guest-host films and polymers,” J. Opt. Soc. Am. B **13**, 1927–1934 (1996) [CrossRef]

*r*

_{33}values as a function of angle of incidence does not guarantee the value is correct. When the ITO is less absorptive as shown in the inset of Fig. 8(b), the correct

*r*

_{33}is approached for wavelengths of 1 and 1.3 µm because these wavelengths are in a spectral region where both polymer and ITO are more transparent.

_{sp}and

*δB*/

*B*are plotted at wavelengths 1.3 and 1.55 µm using the rigorous model. We notice that the δΨ

_{sp}’s are quite different from those of the simple Teng-Man method [13

**9**, 2274–2282 (1992). [CrossRef]

**245**, 311–325 (1999). [CrossRef]

*δB*/

*B*at a wavelength of 1.55 µm passes through 0 at several angles of incidence. Thus, measuring

*δB*/

*B*=0 at a fixed angle of incidence does not mean that there is no electrochromic effect, contrary to the suggestion in Ref. 16. We observe that δΨ

_{sp}and

*δB*/

*B*fluctuate more at 1.55 µm than at 1.3 µm because of the more reflective ITO.

## 4. Conclusions

*I*

_{dc}and

*I*

_{m}to extract more accurate complex EO coefficient values from the experimental data. Based on linear least squares fitting, this method is expected to facilitate the use of a rigorous model to

## Appendix A: Variation of index of refraction

*r*

_{µ}

_{3}and

*s*

_{µ}

_{3}coefficients. Separation of Eq. (1) into real and imaginary parts gives

*κ*=0.000024) so

*κ*/

*n*≪1 is a reasonable approximation. With these considerations in mind, for communication wavelengths in the 1300–1600 nm range, Eqs. (A1) and (A2) can be simplified to

*s*

_{µ}≫3(

*κ*

_{µ}/

*n*

_{µ})

*r*

_{µ}reduces Eq. (A4) to

*r*

_{µ}≫3(

*κ*

_{µ}/

*n*

_{µ})

*s*

_{µ}and

*s*

_{µ}≫3(

*κ*

_{µ}/

*n*

_{µ})

*r*

_{µ}are equivalent to those in Ref. [13

**9**, 2274–2282 (1992). [CrossRef]

*δn*and

*δκ*depend only on

*r*

_{33}and

*s*

_{33}, respectively. Within the absorption band of the polymer, however, this is not true and Eq. (1) or, equivalently, Eqs. (A1) and (A2) should be used.

## Appendix B: Variation of the propagation constants inside the nonlinear film

*s*- and

*p*-wave propagation constants. From Snell’s law we have

*ñ*

_{s}=

*ñ*

_{o}and

*N*, which is of course constant at a given angle of incidence, we can write the propagation constants in the forms

*β’*s in terms of

*N*is that we do not have to explicitly deal with changes in the internal angles

*θ*

_{s}and

*θ*

_{p}as the refractive index changes, they are automatically accounted for in this representation.

*β*

_{s,p}induced by the applied voltage are given by

## Appendix C: H functions in the rigorous model

*H*

_{r}and

*H*

_{s}that appear in Eq. (11) and describe the linear dependence of

*δB̃*/

*B̃*on

*r*

_{33}and

*s*

_{33}for the rigorous expression of reflectance. Also, because many software analysis packages can handle complex numbers efficiently, we retain the complex expressions for

*δñ*

_{o}and

*δñ*

_{e}as described in Eq. (1).

*r*

_{s}and

*r*

_{p}. Expanding Eq. (26), we see that the required derivatives in Eqs. (C1) and (C2) are off the form

*s*,

*p*designations to prevent the notation from becoming unduly cumbersome. The three derivatives with respect to

*ñ*

_{o}and

*ñ*

_{e}are given by

*f’*s and

*g’*s are defined by

*Z’*s are defined in Eq. (28).

*Z’*s in Eq. (28) and substitute from Eq. (C10) sequentially back to Eqs. (C1) and (C2) using also the definition of the total reflection coefficients

*r*

_{s}and

*r*

_{p}from Eq. (26). However, it is in general unduly cumbersome to express the functions

*H*

_{r}and

*H*

_{s}in Eqs. (C1) and (C2) as explicit functions of the linear parameters (complex refractive index and thickness) of the multilayer sample structure.

## Acknowledgments

## References and links

1. | G. A. Lindsay and K. D. Singer, eds., |

2. | D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. |

3. | M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, and A. Dienes, “Second-harmonic generation and absorptionstudies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B |

4. | R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, “Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry,” J. Appl. Phys. |

5. | Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, “Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer,” Appl. Phys. Lett. |

6. | M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. |

7. | H. Uchiki and T. Kobayashi, “New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer,” J. Appl. Phys. |

8. | D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, “Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation,” J. Opt. Soc. Am. B |

9. | L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, “Second-order nonlinear optical measurements in guest-host and side-chain polymers,” J. Appl. Phys. |

10. | S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, “Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films,” Appl. Phys. Lett. |

11. | C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. |

12. | J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. |

13. | K. Clays and J. S. Schildkraut, “Dispersion of the complex electro-optic coefficient and electrochromic effects in poled polymer films,” J. Opt. Soc. Am. B |

14. | P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong“Electro-optic characterization of poled-polymer films in transmission,” App. Phys. Lett. |

15. | F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, “Measurement of the electro-optic properties of poled polymers at λ=1.55 µm by means of sandwich structures with zinc oxide transparent electrode,” Appl. Phys. Lett. |

16. | Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, “Reflection method for electro-optical coefficient determination in stratified thin film structures,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B |

17. | P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, “Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers,” Thin Solid Films , |

18. | G. Khanarian, J. Sounik, D. Allen, S. F. Shu, C. Walton, H. Goldberg, and J. B. Stamatoff, “Electro-optic characterization of nonlinear-optical guest-host films and polymers,” J. Opt. Soc. Am. B |

19. | S. H. Han and J. W. Wu, “Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration,” J. Opt. Soc. Am. B |

20. | F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique,” Chem. Phys. |

21. | Y. Shuto and M. Amano, “Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films,” J. Appl. Phys. |

22. | I. P. Kaminow, |

23. | F. Wang, E. Furman, and G.H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. App. Phys. |

24. | D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications,” J. Opt. Soc. Am. A |

25. | M. Born and E. Wolf, |

26. | Ramo, Whinnery, and Van Duzer, |

27. | M. T. Heath, |

28. | R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films, 313–314, 394–397 (1998). |

**OCIS Codes**

(190.4710) Nonlinear optics : Optical nonlinearities in organic materials

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Thin Films

**History**

Original Manuscript: July 5, 2006

Revised Manuscript: August 31, 2006

Manuscript Accepted: September 1, 2006

Published: September 18, 2006

**Citation**

Dong H. Park, Chi H. Lee, and Warren N. Herman, "Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures," Opt. Express **14**, 8866-8884 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8866

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### References

- G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics, (ACS Symposium Series 601, 1995).
- D. M. Burland, R. D. Miller, and C. A. Walsh, "Second-order nonlinearity in poled-polymer systems," Chem. Rev. 94, 31-75 (1994). [CrossRef]
- M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins,and A. Dienes, "Second-harmonic generation and absorptionstudies of polymer-dye films oriented by corona-onset poling at elevated temperatures," J. Opt. Soc. Am. B 6, 733-741 (1989). [CrossRef]
- R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994). [CrossRef]
- Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004). [CrossRef]
- M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998). [CrossRef]
- H. Uchiki and T. Kobayashi, "New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer," J. Appl. Phys. 64, 2625-2629 (1988). [CrossRef]
- D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, "Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation," J. Opt. Soc. Am. B 10, 1894-1900 (1993). [CrossRef]
- L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990). [CrossRef]
- S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996). [CrossRef]
- C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990). [CrossRef]
- J. S. Schildkraut, "Determination of the electro optic coefficient of a poled polymer film," Appl. Opt. 29, 2839-2841 (1990). [CrossRef] [PubMed]
- K. Clays and J. S. Schildkraut, "Dispersion of the complex electro-optic coefficient and electrochromic effects in poled polymer films," J. Opt. Soc. Am. B 9, 2274-2282 (1992). [CrossRef]
- P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996). [CrossRef]
- F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003). [CrossRef]
- Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).
- P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films, 242, 132-138 (1994). [CrossRef]
- G. Khanarian, J. Sounik, D. Allen, S. F. Shu, C. Walton, H. Goldberg and J. B. Stamatoff, "Electro-optic characterization of nonlinear-optical guest-host films and polymers," J. Opt. Soc. Am. B 13, 1927-1934 (1996) [CrossRef]
- S. H. Han and J. W. Wu, "Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration," J. Opt. Soc. Am. B 14, 1131-1137 (1997). [CrossRef]
- F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999). [CrossRef]
- Y. Shuto and M. Amano, "Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films," J. Appl. Phys. 77, 4632-4638 (1995). [CrossRef]
- I. P. Kaminow, An Introduction to Electrooptic Devices, (Academic Press, Inc., 1974).
- F. Wang, E. Furman, and G.H. Haertling, "Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry," J. Appl. Phys. 78, 9-15 (1995). [CrossRef]
- D. Guo, R. Lin, and W. Wang, "Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications," J. Opt. Soc. Am. A 22, 1577-1588 (2005). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
- For a discussion of wave impedances, see, for example, Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, (John Wiley and Sons, New York, 1965), Chap. 6.
- M. T. Heath, Scientific Computing, 2nd ed. (McGraw Hill, 2002).
- R. A. Synowicki, "Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants," Thin Solid Films, 313-314, 394-397 (1998).

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