1. Introduction

Organic molecules have attracted great interest for their potential applications in nonlinear optical (NLO) devices. In particular, poled polymers based on second-order nonlinearity have been widely studied because they are a class of photonic material that may substitute for inorganic nonlinear crystals in high speed optical communication and signal processing [1].

Obtaining a macroscopic 2nd order NLO response requires orientation of the chromophores, which can be achieved by electric field poling [2]. Corona [3] and electrode contact poling techniques are usually used. Methods of measuring the linear electro-optic (EO) effect include Mach-Zehnder (MZ) interferometry [4–6], Fabry-Perot (FP) interferometry [7], attenuated total reflection [8] (ATR), waveguide method [9], and two slit interference method [10]. One of the most popular measurement techniques is a reflection method introduced by Teng and Man [11] (Teng-Man) as well as by Schildkraut [12]. The reflection measurement is generally easier to make compared to techniques such as MZ interferometry because it simply takes advantage of a relative phase-shift difference between the 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 [12,13], and later discussed in some detail by Lundquist, et al. [14] This method, which also measures a relative phase-shift, affords a somewhat simpler alignment compared to the reflection method, but requires two transparent electrodes.

Many researchers have been synthesizing chromophores for incorporation into polymeric materials to achieve high nonlinearity in a poled polymer thin film. The EO properties are often characterized using the Teng-Man reflection method because it is simple and quick. The vast majority of quoted values of the electro-optic coefficient that are obtained from Teng-Man measurements result from a simplified analysis of the data that assumes the transparent conducting oxide (TCO) is perfectly transparent and the gold is perfectly reflective. Consequently only the properties of the EO material [refractive indices (sometimes anisotropic), EO coefficients, and angle of incidence] are taken into account. However, in both Refs. 11 and 12, it was realized that experimental error can result from ignoring the reflection off the substrate-film interface and that an accurate determination of the EO coefficient could be achieved only by numerical calculation that applies anisotropic Fresnel equations to the stratified layers containing the nonlinear poled polymer. This requires knowledge of the refractive index and thickness of all the layers. In addition, the simple analysis is valid only when the absorption of the film is negligible. Furthermore, Michelotti, et al. [15] reported that the simple Teng-Man analysis can give unreliable measurements if the experimental setup is operated in a spectral region where the TCO layer is absorbing, and suggested that if it is only slightly absorbing it is still possible to evaluate the EO properties of the polymer films with reasonable precision.

The effect of electrochromism (variation of the imaginary part of the index of refraction under application of an electric field) and the complex EO coefficient was added in Ref. 13, followed by efforts by a few researchers to characterize other nonlinear properties, such as the complex quadratic EO coefficient, based on a simple analysis of Teng-Man data [20].

Levy, et al. [16] and Chollet, et al. [17] presented both simple and rigorous expressions for estimation of the complex EO coefficient. Numerical solutions to the rigorous expressions were obtained using the simplex method. They also pointed out that the modulated intensity at the maximum of the optical bias curve and some of the modulation dependence on angle of incidence are attributed to absorption of the film and to multiple reflections, respectively. Ignoring multiple reflections inside the polymer film, Khanarian, et al. [18] derived an approximate equation for the real part of the EO coefficient that takes into account the effects of reflectivity modulation at the glass/polymer and polymer/metal interfaces. Han and Wu [19] measured the modulated intensity as a function of optical bias, optical polarization, and angle of incidence in an effort to distinguish FP effects from EO phase modulation.

In this paper, we provide new closed-form expressions for analysis of Teng-Man data including absorption of both the film and TCO layers. The analysis also applies to any uniaxial nonlinear thin film with the c-axis perpendicular to the nonlinear film surface, such as thin z-cut Lithium Niobate. In general the modulation of both the real and imaginary parts of the complex refractive index under application of an electric field depends on both the real and imaginary parts of the complex EO coefficient, particularly inside the linear absorption band of the polymer [13]. Two models, both allowing for absorption of the nonlinear polymer layer, are analyzed and compared: a simple model that takes into account only the properties of the nonlinear polymer layer without multiple reflections and a rigorous model accounting for properties of the complete stratified layers of the test structure including multiple reflections. In both models, analytic expressions that are linearly dependent on the real and imaginary parts of the complex EO coefficient are presented so that standard linear least squares data analysis can be performed. In the case of the simple model, we obtain equations to characterize the imaginary part of the complex EO coefficients that are somewhat different from those reported earlier [13, 16, 20] as well as an identical equation to that usually used in the Teng-Man method for the real part [21] outside the absorption band of the polymer. The rigorous model has the same fundamental starting point as that of Ref. [16], namely, the description of the total reflectivity of the multilayer structure. But we provide explicit analytic expressions for all derivatives involved along with a new compact formalism that allows a matrix representation of the dependence of the real and imaginary parts of the complex EO coefficient on the reflectivity and phase modulation. The rigorous model analysis is used to show that simple model calculations, which do not include the effects of multiple reflections, can, in some cases, either grossly underestimate or overestimate the complex electro-optic coefficient. In addition, it is shown that the relative error in using the simple model can undergo a large cyclic variation, an asymptotic behavior, or an irregular FP effect with increasing film thickness depending on operating wavelength.

 

Fig. 1. Schematic of the experimental Teng-Man setup. L is laser, P polarizer, A aperture, S slit, SBC Soleil-Babinet Compensator, and PD photodetector.

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2. Theory

A poled organic thin film prepared by spin coating belongs to the point-group symmetry ∞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]

where 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] and 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

As shown in Fig. 1, light polarized at 45° is incident at an angle on the multilayered sample structure containing the nonlinear poled thin film. The polymer material is usually spin-coated on a glass substrate coated with a TCO, commonly indium tin oxide (ITO), and poled electrically to generate the second order nonlinearity. The first reflection off the air-glass interface and subsequent beams resulting from reflection of the first pass on its way out at the glass-air interface back into the polymer and out again are blocked [11–13]. The remaining light reflected off the sample experiences an additional controllable phase retardation Ω introduced by a Soleil-Babinet Compensator (SBC). The intensity of the light is detected after passing through the analyzer using a lock-in amplifier. At each angle of incidence, two different data sets, the optical bias curve 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 Vsin(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 [15–17]

 

Fig. 2. The optical bias curve and modulated intensity obtained as a function of SBC retardation setting x for a representative set of experimental data on a film with r ₃₃=42 pm/V and s ₃₃=0.3 pm/V using a peak voltage of 4 V. Points 1, 2, 3, and 4 correspond to compensator settings such that Ψ _sp +Ω=π/2, 3π/2, 0, and π.

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where

and

To first order in V, the modulated intensity I_m (V, Ω) is obtained by differentiating Eq. (2) to get

Representative data curves for optical bias 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, 3π/2 as

where 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

Typically, in the Teng-Man method, 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 Vsin(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).

To measure modulated intensities at only points 1 and 2 in Fig. 2, an often employed quick measurement technique is to a) vary the compensator to determine the minimum 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.

After verifying voltage values to be in the linear regime, we prefer to measure the full I_m (V, Ω) curves [15,20] as well as 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 .

To facilitate relating these parameters to the complex EO coefficients, we define a new complex parameter B̃ as

so that

Thus, the modulated reflectivity and phase are just the real and imaginary parts, respectively, of δB̃/B̃.

 

Fig. 3. Multilayered structures in a simple model (a) and a rigorous model (b). For simplicity, subscripts s and p in the reflection coefficients are omitted in (b).

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From Eq. (10), assuming r̃ ₁₃=γr̃ ₃₃(γ is generally assumed to be 1/3 for weak poling [2]), assuming no piezoelectric contribution and making a Taylor expansion to first order in δñ_o,e , the complex quantity δB̃/B̃ can be written as a linear function of r ₃₃ and s ₃₃ in the form

Separating into real and imaginary parts, we have

We note in general that the ratio 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

The simple model ignores the properties of the TCO layer and simplifies the multilayered structure of the sample to three layers, air-film-PEC (Perfect Electric Conductor), as shown in Fig. 3(a). By also ignoring the reflectance at the air-film interface, the reflection coefficients of the s- and p- waves can be expressed as

where the propagation constants of the s- and p- waves normal to the film surface are defined as

Here, k_o =2π/λ is the wave vector in free space, ñ_s =ñ_o , ñ_p is given in Eq. (B2), and $\tilde{\theta }$_s and $\tilde{\theta }$_p 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 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

which is purely real. In this simple model, the coefficients a and b depend only on the imaginary parts of the propagation constants, while ψ _sp depends only on the real parts.

From Eqs. (2)–(6) together with Eq. (15), we have

and

Inserting Eq. (13) into the general expression Eq. (11), we have

and separating into real and imaginary parts gives

The functions h_r and h_s are complex quantities given by

and

where N=sinθ 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 [12,13,16,20,21]. Then, the 2×2 matrix in Eq. (19) is diagonalized because the real parts of h_r and h_s vanish, and we get

and

For n_o ≈n_e ≡n, Eqs. (22) and (23) further reduce to

and

Equations (22) and (24) are identical with those reported previously [12,21], but our Eqs. (23) and (25) describe the electrochromic effect somewhat differently from that reported in Refs. 13, 16, and 20. In Refs. 13 and 20, the ellipsometric parameter tanΦ (=|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

In this section, we use rigorous expressions for the s- and p- reflectance [16, 17, 23]. In the case of a thin film on a TCO-glass substrate, the first reflection off the glass is blocked from reaching the detector [12], as is a second-pass beam resulting from reflection of the first pass on its way out at the glass-air interface back into the polymer and out again. This eliminates reflection fluctuations from the glass layer. Greater care is required to block these beams at high and low angles of incidence, because the transverse separation between reflections becomes narrow at these angles of incidence [24]. If the glass is sufficiently thick, it is possible to block the gross multiple reflections efficiently, i.e., the first reflection off the glass and the second, third, etc passes resulting from back-reflection off the glass-air interface, but it is not generally possible to avoid the multiple reflections inside a thin polymer film. These multiple reflections inside the film are accounted for in the rigorous model but not in the simple model.

Assuming that the incident light is a plane wave, the iteration of the Airy formula [25] gives the reflection coefficient in the whole multilayered sample as shown in Fig. 3(b). As given in Refs. 16 and 17, the resulting expressions for r_s and r_p have the form

where we have omitted the 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

with the s- and p- wave impedances [26] of each layer given by

In principle, these reflection coefficients should be multiplied by two transmittances, t ₁₂ and t ₂₁, at the air-glass interface [17], but in practice these terms do not contribute to the calculation of δΨ _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

At a single angle and with knowledge of the linear parameters of the sample structure, the matrix equation shown in Eq. (12) can simply be inverted to solve for 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 2n×2 matrix that is derived by stacking up Eq. (12) to form a large matrix equation

It is easy to calculate 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 2n×2n orthogonal and 2n×2 upper triangular matrices, respectively and we define P̂ as

Taking the 2×2 upper triangular matrix from R and the 2×1 upper matrix from P̂ allows one to obtain x,

where R_lower is the (2n-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

In this section, we estimate the relative error that results from using the simple model for a number of different cases. Assuming the rigorous model gives the correct value, the relative error is defined as

For a given modulated phase and reflectivity, 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 $r_{33}^{\mathit{\text{SM}}}$ and $s_{33}^{\mathit{\text{SM}}}$ are calculated from Eqs. (19)–(21). To evaluate the error we equate the right hand sides of Eqs. (12) and (19) and rearrange to obtain

The result depends on the ratio 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.

For the first case, we consider the example of a sample structure with a thick gold electrode where the index of the nonlinear film and the TCO are matched with the glass substrate under the assumption of no electrochromic effect and no optical loss in the polymer or TCO layer. That is, 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

and the f’s and g’s in Eq. (C7) are

Then, H_r in Eq. (C1) can be expressed as

where

and the derivatives in these expressions are given in Eqs. (C8)–(C10). The first two terms in Eq. (36) are due to modulation of the reflectivity at the TCO-polymer interface and contribute oscillation as the thickness 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

The error percentage is plotted as a function of film thickness in Fig. 4. Notable is that the error oscillates from overestimation to underestimation and the envelope decreases with increasing thickness.

As other cases, we consider the relative error for different refractive index and thickness combinations of the polymer film for selected values of complex index and thickness of ITO at four wavelengths, 0.8, 1, 1.3, and 1.55 µm and the ratio 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.

 

Fig 4. Error percentage plot for varying film thickness when the refractive indices of film and TCO are matched with glass (n=1.5) at 1.3 µm wavelength. The positive and negative envelopes are proportional to ±1/d ₄ with a negative offset.

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Figure 6 shows the error plots as a function of thickness of the polymer for polymer indices of refraction shown in Fig. 5 (ignoring birefringence). Noteworthy is that for a given refractive index, the error cycles from positive to negative (overestimation to underestimation) as a function of film thickness when the loss of the film is negligible as shown in Figs. 6(b)–6(d) and 6(f)–6(h). This is because increasing the thickness causes a periodic phase retardation. The error extremes also tend to increase as the wavelength increases because the ITO becomes more reflective, which enhances FP effects inside the polymer layer. Thicker ITO also tends to increase the reflectivity. When the operating wavelength is in the absorption band of the polymer film as shown in Figs. 6(a) and 6(e), the error initially fluctuates irregularly when the film is thin and finally converges to a constant negative value as the film thickness increases. This is because propagation loss in the film has a dual effect of reducing multiple reflections as well as reducing the relative amount of phase modulation from the polymer bulk compared to the phase modulation of the light reflected from ITO/polymer interface, which is not accounted for in the simple model. Within the absorption band, a thick film results in a constant error, but the correct value is not predictable without the rigorous analysis.

 

Fig. 5. Optical properties (n+iκ) of (solid) a representative ITO (Abrisa Corporation) measured by ellipsometry and (dashed) a representative polymer film selected for the simulation. (real part : black, imaginary part : red).

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Fig. 6. Error plots by varying the thickness of the film for fixed indices of refraction of ITO and the film and two thicknesses of ITO, 100 nm (a–d) and 50 nm (e–h) at various wavelengths and 45° angle of incidence under assumption of γ=1/3. For plots (a) and (e), s ₃₃/r ₃₃=1 (black solid), and s ₃₃/r ₃₃=2 (red dashed). For the wavelengths other than 0.8 µm, values of s ₃₃/r ₃₃ between 0 and 0.1 produce curves that are indistinguishable on this scale. For (a) and (e), the errors approach -107% and -110%, respectively. For (b), (f), (c), (g), (d), and (h), the error extremes are -18% to 12%, -15% to 10%, -86% to 38%, -40% to 25%, -80% to 350%, and 37% to 63%, respectively.

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Fig. 7. Error contour plots at (a) λ=0.8 µm and s ₃₃/r ₃₃=1 (b) λ=1.3 µm and s ₃₃/r ₃₃=0.1 with thickness of ITO=100 nm, 45° angle of incidence, and γ=1/3. Each contour plot (a) and (b) shows asymptotic and cyclic behaviors with thickness of film irrespective of index of refraction of film, respectively.

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Fig. 8. EO coefficients r33 calculated by the simple model at various wavelengths and angles of incidence using thickness of film and ITO, 1.4 µm and 100 nm, respectively. EO coefficient r ₃₃=100 pm/V was used for the simulation. The ratios s ₃₃/r ₃₃ are 2 at 0.8 µm and 0.1 at the other wavelength. Insets in (a) and (b) show the optical properties of ITO selected for the simulation. The crossover points of n and κ of ITO are around 1.54 and 1.92 µm in (a) and (b), respectively.

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The error in using the simple model versus the film thickness shows qualitatively similar behavior irrespective of the index of refraction of the film as shown in Figs. 7(a) and 7(b). We also note, contrary to earlier suggestions [16,18], that the error from using the simple model is not in general reduced by using film thicknesses on the order of a wavelength or less.

 

Fig. 9. (a). δΨ _sp and (b) δB/B versus angle of incidence at wavelengths 1.3 and 1.55 µm. The ratio s33/r ₃₃=0.1 and film thickness=1.4 µm were used at both wavelengths for the ITO properties shown in Fig. 5.

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Figure 8 shows simple model r33 values as a function of angle of incidence for two different ITO samples where the ITO for Fig. 8(b) is less reflective than for Fig. 8(a). In Fig. 8(a), most EO coefficients calculated by the simple model with varying angle of incidence are far away from the correct r33. Note that although the simple model EO coefficient values seem to be flat at 1.3 µm for angles of incidence between 30° and 60°, they are nevertheless overestimated by about 50%. Thus, obtaining nearly constant simple model 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.

In Fig. 9(a) and 9(b), the simulated δΨ _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, 20]. Also, 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.

For those making Teng-Man measurements and desiring accuracies of only ±20%, we identify in Table 1 sets of ranges of the refractive indices and thicknesses of the layers that have been found to allow at least this accuracy using just the simple model and measurements at a 45° angle of incidence under assumption of no loss and no electrochromic effect in the film. As can be seen from Fig. 6, the error in using the simple model oscillates from positive to negative going through zero at isolated points. Thus, there can be particular combinations of parameters outside the table ranges that give an error better than ±20%, but the table has been constructed to keep the envelope of the error oscillation within ±20%. The rigorous model should be used when the parameters are outside these ranges, or when accuracy better than ±20% is desired.

4. Conclusions

In this paper, we have presented new mathematical expressions for analyzing Teng-Man data in both transparent and absorptive films that take into account the optical properties of the TCO layer. The formulas presented were derived using a rigorous model that includes the effect of multiple reflections inside the multilayered structure at any angle of incidence. In addition, a new expression has been derived for the electrochromic effect in a simple model. A curve fitting procedure has been described for the full profiles of angle dependent 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

Table 1. Parameter ranges to use the simple model with error less than ±20% at 45° angle of incidence.

View Table

obtain reliable EO coefficients for poled Teng-Man samples. In order to use the rigorous model, however, it is necessary to know detailed information about the sample because simultaneously fitting multiple parameters does not give a unique fit. We have shown that the relative error from using the simple model can experience cyclic, asymptotic, and irregular behaviors depending on the operating wavelength as the thickness of the polymer film increases. The error extremes can be reduced by choosing an operating wavelength where the absorption of the polymer and TCO are small simultaneously, not by decreasing the thickness of the polymer film on the order of a wavelength or less. We also note that the complex quadratic EO coefficient can be measured by detecting the signal at twice the fundamental frequency or by applying a DC offset to the polymer film with a sinusoidal bias as discussed in Ref. 20. Our formulas can easily be modified to include these effects. We also expect that our analysis can be applied to the interferometric types of measurement to estimate the relative error introduced by the TCO layer.