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

Dong Hun Park, Chi H. Lee, and Warren N. Herman

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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

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Figures (9)
Fig. 1.
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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Fig. 2.
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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Fig. 3.
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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Fig 4.
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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Fig. 5.
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.
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.
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.
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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Fig. 9.
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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Tables (1)

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

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Equations (66)

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δ n ˜ μ = − 1 2 n ˜ μ 3 ( r μ 3 + i s μ 3 ) E 3 ,
I dc = I o 4 ∣ r s e i Ω − r p ∣ 2 = A + B sin 2 ( Ψ sp + Ω 2 ) ,
A = I o 4 ( ∣ r s ∣ − ∣ r p ∣ ) 2 , B = I o ∣ r s ∣ ∣ r p ∣ ,
r s = ∣ r s ∣ e i Ψ s , r p = ∣ r p ∣ e i Ψ p ,
Ψ sp = Ψ s − Ψ p .
I m = δ A + δ B sin 2 ( Ψ sp + Ω 2 ) + B 2 sin ( Ψ sp + Ω ) δ Ψ sp .
δ Ψ sp = I m ( π ⁄ 2 ) − I m ( 3 π ⁄ 2 ) 2 I c ,
δ B B = I m ( π ) − I m ( 0 ) 2 I c .
B ˜ ≡ I o r s r p * = B e i Ψ sp
δ B ˜ B ˜ = δ B B + i δ Ψ sp = δ r s r s + ( δ r p r p ) * .
δ B ˜ B ˜ = 1 r s ∂ r s ∂ n ˜ o δ n ˜ o + ( 1 r p ∂ r p ∂ n ˜ o δ n ˜ o + 1 r p ∂ r p ∂ n ˜ e δ n ˜ e ) * ≡ H r r 33 + i H s s 33 .
( δ Ψ sp δ B B ) = ( Im ( H r ) Re ( H s ) Re ( H r ) − Im ( H s ) ) ( r 33 s 33 ) .
r s = − e 2 i β s d , r p = − e 2 i β p d ,
β s = k o n ˜ s cos ( θ ˜ s ) , β p = k o n ˜ p cos ( θ ˜ p ) .
ψ sp = 2 ( β sr − β pr ) d ,
δ ψ sp = 2 ( δ β sr − δ β pr ) d
δ b = − 2 b ( δ β si + δ β pi ) d .
δ b ˜ b ˜ = δ b b + i δ ψ sp = 2 id [ ∂ β s ∂ n ˜ o δ n ˜ o − ( ∂ β p ∂ n ˜ o δ n ˜ o + ∂ β p ∂ n ˜ e δ n ˜ e ) * ] ≡ h r r 33 + i h s s 33 ,
( δ ψ sp δ b b ) = ( Im ( h r ) Re ( h s ) Re ( h r ) − Im ( h s ) ) ( r 33 s 33 ) .
h r = iV [ ( k o n ˜ o 3 n ˜ e n ˜ e 2 − N 2 ) * γ − k o n ˜ o 4 n ˜ o 2 − N 2 γ + ( k o n ˜ o n ˜ e N 2 n ˜ e 2 − N 2 ) * ]
h s = − iV [ k o n ˜ o 4 n ˜ o 2 − N 2 γ + ( k o n ˜ o 3 n ˜ e n ˜ e 2 − N 2 ) * γ + ( k o n ˜ o n ˜ e N 2 n ˜ e 2 − N 2 ) * ] ,
δ ψ sp = − ( n o 4 n o 2 − N 2 − n o 3 n e n e 2 − N 2 ) k o γ r 33 V + n o n e N 2 n e 2 − N 2 k o r 33 V = Im ( h r ) · r 33
δ b b = ( n o 4 n o 2 − N 2 + n o 3 n e n e 2 − N 2 ) k o γ s 33 V + n o n e N 2 n e 2 − N 2 k o s 33 V = − Im ( h s ) · s 33 ,
δ ψ sp = n 2 N 2 n 2 − N 2 ( 1 − γ ) r 33 k o V = Im ( h r ) · r 33
δ b b = ( 2 n 4 − n 2 N 2 n 2 − N 2 γ s 33 + n 2 N 2 n 2 − N 2 s 33 ) k 0 V = − Im ( h s ) · s 33 .
r = r 23 + r ̂ 34 e 2 i β 3 d 3 1 + r 23 r ̂ 34 e 2 i β 3 d 3 ⇐ r ̂ 34 = r 34 + r ̂ 45 e 2 i β 4 d 4 1 + r 34 r ̂ 45 e 2 i β 4 d 4 ⇐ r ̂ 45 = r 45 + r 56 e 2 i β 5 d 5 1 + r 45 r 56 e 2 i β 5 d 5 ,
r jk = Z k − Z j Z k + Z j
Z s = 1 n ˜ o 2 − N 2 , Z p = 1 n ˜ o 1 − ( N n ˜ e ) 2 .
( δ Ψ sp ( θ 1 ) ⋮ δ Ψ sp ( θ n ) δ B ⁄ B ( θ 1 ) ⋮ δ B ⁄ B ( θ n ) ) = ( Im [ H r ( θ 1 ) ] Re [ H s ( θ 1 ) ] ⋮ ⋮ Im [ H r ( θ n ) ] Re [ H s ( θ n ) ] Re [ H r ( θ 1 ) ] − Im [ H s ( θ 1 ) ] ⋮ ⋮ Re [ H r ( θ n ) ] − Im [ H s ( θ n ) ] ) ( r 33 s 33 ) ⇔ P = M · x .
M · x = P ⇒ QR · x = P ⇒ R · x = Q T P ≡ P ̂ .
R · x = P ̂ ⇒ ( R 2 × 2 R lower ) · x = ( P 2 × 1 P lower ) ⇒ x = R 2 × 2 − 1 · P 2 × 1 ,
Error = r 33 SM − r 33 r 33 .
Error = Im ( h s ) [ Im ( H r ) − Im ( h r ) ] + Re ( h s ) [ Re ( H r ) − Re ( h r ) ] Im ( h r ) Im ( h s ) + Re ( h r ) Re ( h s )
+ Im ( h s ) Re ( H s ) − Re ( h s ) Im ( H s ) Im ( h r ) Im ( h s ) + Re ( h r ) Re ( h s ) s 33 r 33 .
r = r 45 e 2 i β 3 d 3 e 2 i β 4 d 4 ,
f 234 q = e 2 i β q 3 d 3 , f 345 q = e 2 i β q 4 d 4 , g 345 q = 1 − ( r 45 q ) 2 e 2 i β q 4 d 4 , g 456 q = 1 .
H r ≡ − V ( n 4 ) 3 2 d 4 [ K 1 cos ( 2 β 4 d 4 ) − i K 2 sin ( 2 β 4 d 4 ) + K 3 ] − iV ( n 4 ) 3 K 4 ,
K 1 = [ 1 − ( r 45 s ) 2 ] γ r 45 s ∂ r 34 s ∂ n ˜ o + [ 1 − ( r 45 p * ) 2 ] ( γ r 45 p * ∂ r 34 p ∂ n o + 1 r 45 p * ∂ r 34 p ∂ n ˜ e )
K 2 = [ 1 + ( r 45 s ) 2 ] γ r 45 s ∂ r 34 s ∂ n ˜ o − [ 1 − ( r 45 p * ) 2 ] ( γ r 45 p * ∂ r 34 p ∂ n o + 1 r 45 p * ∂ r 34 p ∂ n e )
K 3 = γ r 45 s ∂ r 45 s ∂ n ˜ o + γ r 45 p * ∂ r 45 p * ∂ n o + 1 r 45 p * ∂ r 45 p * ∂ n ˜ e
K 4 = γ ∂ β 4 s ∂ n ˜ o − γ ∂ β 4 p ∂ n o − ∂ β 4 p ∂ n ˜ e ,
Error = 1 2 d 4 K 4 [ Im ( K 1 ) cos ( 2 β 4 d 4 ) − Re ( K 2 ) sin ( 2 β 4 d 4 ) + Im ( K 3 ) ] .
δ n μ = − n μ 3 2 [ ( 1 − 3 κ μ 2 n μ 2 ) r μ 3 + ( κ μ 2 n μ 2 − 3 ) κ μ n μ s μ 3 ] E 3
δ κ μ = − n μ 3 2 [ ( 1 − 3 κ μ 2 n μ 2 ) s μ 3 + ( κ μ 2 n μ 2 − 3 ) κ μ n μ r μ 3 ] E 3 .
δ n μ ≈ − 1 2 n μ 3 ( r μ 3 − 3 κ μ n μ s μ 3 ) E 3
δ κ μ ≈ − 1 2 n μ 3 ( s μ 3 + 3 κ μ n μ r μ 3 ) E 3 .
δ n μ ≈ − 1 2 n μ 3 r μ 3 E 3 .
δ κ μ ≈ − 1 2 n μ 3 s μ 3 E 3 .
n ˜ s sin θ ˜ s = n ˜ p sin θ ˜ p = sin θ ≡ N ,
1 n ˜ p 2 = cos 2 θ ˜ p n ˜ o 2 + sin 2 θ ˜ p n ˜ e 2 .
β s = k o n ˜ o 2 − N 2
β p = k o n ˜ o n ˜ e n ˜ e 2 − N 2 .
δ β s = k o n ˜ o n ˜ o 2 − N 2 δ n ˜ o
δ β p = k o n ˜ e n ˜ e 2 − N 2 δ n ˜ o + k o n ˜ o N 2 n ˜ e 2 n e 2 − N 2 δ n ˜ e .
δ β s ≈ k o n o n o 2 − N 2 ( δ n o + i δ κ o )
δ β p ≈ k o n e n o 2 − N 2 ( δ n o + i δ κ o ) + k o N 2 n o n e 2 n e 2 − N 2 ( δ n e + i δ κ e ) .
H r = − V 2 d 4 [ γ r s ∂ r s ∂ n ˜ o n ˜ o 3 + γ ( 1 r p ∂ r p ∂ n ˜ o n ˜ o 3 ) * + ( 1 r p ∂ r p ∂ n ˜ e n ˜ e 3 ) * ] .
H s = − V 2 d 4 [ γ r s ∂ r s ∂ n ˜ o n ˜ o 3 − γ ( 1 r p ∂ r p ∂ n ˜ o n ˜ o 3 ) * − ( 1 r p ∂ r p ∂ n ˜ e n ˜ e 3 ) * ] .
1 r ∂ r ∂ n ˜ o , e = 1 r ∂ r ∂ r ̂ 34 ( ∂ r ̂ 34 ∂ r 34 ∂ r 34 ∂ n ˜ o , e + ∂ r ̂ 34 ∂ r ̂ 45 ∂ r ̂ 45 ∂ r 45 ∂ r 45 ∂ n ˜ o , e + ∂ r ̂ 34 ∂ β 4 ∂ β 4 ∂ n ˜ o , e ) ,
1 r s ∂ r s ∂ n ˜ o = f 234 s r s ( g 345 s ∂ r 34 s ∂ n ˜ o + f 345 s g 456 s ∂ r 45 s ∂ n ˜ o + 2 i d 4 r ̂ 45 s f 345 s ∂ β s 4 ∂ n ˜ o ) ,
1 r p ∂ r p ∂ n ˜ o = f 234 p r p ( g 345 p ∂ r 34 p ∂ n ˜ o + f 345 p g 456 p ∂ r 45 p ∂ n ˜ o + 2 i d 4 r ̂ 45 p f 345 p ∂ β p 4 ∂ n ˜ o ) ,
1 r p ∂ r p ∂ n ˜ e = f 234 p r p ( g 345 p ∂ r 34 p ∂ n ˜ e + f 345 p g 456 p ∂ r 45 p ∂ n ˜ e + 2 i d 4 r ̂ 45 p f 345 p ∂ β p 4 ∂ n ˜ e ) ,
f ijk q = [ 1 − ( r ij q ) 2 ] exp ( 2 i β qj d j ) [ 1 + r ij q r ̂ jk q exp ( 2 i β qj d j ) ] 2 , g ijk q = 1 − ( r ̂ jk q ) 2 exp ( 4 i β qj d j ) [ 1 + r ij q r ̂ jk q exp ( 2 i β qj d j ) ] 2
∂ r 34 s ∂ n ˜ o = − 2 n ˜ o Z 3 s ( Z 4 s ) 3 ( Z 3 s + Z 4 z ) 2 , ∂ r 34 p ∂ n ˜ o = − 2 Z 3 p Z 4 p n ˜ o ( Z 3 p + Z 4 p ) 2 , ∂ r 34 p ∂ n ˜ e = 2 Z 3 p N 2 n ˜ o 2 n ˜ e 3 Z 4 p ( Z 3 p + Z 4 p ) 2 ,
∂ r 45 s ∂ n ˜ o = 2 n ˜ o ( Z 4 s ) 3 Z 5 s ( Z 4 s + Z 5 s ) 2 , ∂ r 45 p ∂ n ˜ o = 2 Z 4 p Z 5 p n ˜ o ( Z 4 p + Z 5 p ) 2 , ∂ r 45 p ∂ n ˜ e = − 2 Z 5 p N 2 n ˜ o 2 n e 3 Z 4 p ( Z 4 p + Z 5 p ) 2
∂ β s 4 ∂ n ˜ o = k o Z 4 s n ˜ o , ∂ β p 4 ∂ n ˜ o = k o Z 4 p n ˜ o , ∂ β p 4 ∂ n ˜ e = k o 1 Z 4 p ( N 2 n ˜ e 3 ) .
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