Electro-optic effect in crystalline films of transverse planar octupolar symmetry

Mi-Yun Jeong; Sophie Brasselet; Bong Rae Cho; Tong-Kun Lim

doi:10.1364/OE.19.007979

1. Introduction

In recent years, molecular crystals of octupolar symmetry having large second order optical properties have been the center of a large interest, first with theoretical developments and then experimental demonstrations [1–4]. The major advantage of octupolar molecules with three-fold symmetry is their potential to lead to non-centrosymmetric crystallization due to the lack of ground-state dipole moment, avoiding the frequent head-to-tail centrosymmetric conformation found in one-dimensional dipolar molecules [5,6]. One of the most promising octupolar molecule crystallizing in macroscopic size crystals is TTB (1,3,5-tricyano-2, 4,6-tris (p-diethylaminostyryl) benzene), leading to a $D_{3 h}$ non-centrosymmetric structure with one of the largest Second Harmonic Generation (SHG) value reported among organic crystals [3]. This molecule has shown recently its ability to produce crystalline domains in polymer thin films with large multipolar SHG responses [4]. This achievement of octupolar crystalline thin films now brings the possibility to build electro-optic (EO) active devices from which large efficiencies are expected. While in the last decades, a large theoretical and experimental effort has been brought on EO integrated devices made of polymer thin films doped or grafted with immobilized dipolar molecules [7,8] with progressing performances [9], the octupolar structures based on crystalline thin films have been rarely studied [10]. Moreover, although many investigations have been carried-out in multipolar biaxial bulk crystals [11–13], the study of EO effect in three-fold symmetry structures has not been reported so far, to the best of our knowledge.

In this paper, we derive a theoretical expression of the electro-optic effect in a $D_{3 h}$ symmetry system, and apply it to the response of an octupolar crystalline thin film fabricated from the TTB molecule. In the geometry chosen here for practical reasons, the optical beam incident direction is set parallel to the molecular plane (Fig. 1 ), which is suitable to the fabricated crystalline TTB thin films where molecules naturally form stacks perpendicular to the sample substrate [4]. Although the resulting molecular-fields coupling geometry does not take full advantage of the polarization-independent efficiency possibilities of an illumination perpendicular to the molecular plane [1,3], it is still expected to benefit from a non-centrosymmetric arrangement, as well as from a high nonlinear efficiency.

Fig. 1 (a) Schematic drawing of a TTB molecule (3 branches) and its orientation by the Euler angles (θ,ϕ,ψ). The molecular coordinates are denoted $(U_{1}, U_{2}, U_{3})$ in the laboratory coordinate (X,Y,Z). The (x,y,z) notations are used as intermediate axes to define the orientation of the $(U_{1}, U_{2})$ molecular plane in the laboratory frame. (b) Geometry of the electro-optic modulation in the octupolar thin film. (c) Euler angles used in this geometry: the octupolar plane is defined by (θ = π/2, ϕ = 0), and ψ defines the octupoles orientation in the (Y,Z) plane. (d) The molecular structure of TTB.

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2. Electro-optic effect in an octupolar system: theory

The studied sample is a few micrometers thick molecular crystalline film sandwiched between two planar transparent electrodes, traversed by the incident optical beam and undergoing a modulated electric field perpendicular to the sample plane (Fig. 1). The molecular orientation in the macroscopic frame is defined by the Euler angles (θ,ϕ,ψ) in the laboratory frame (X,Y,Z), with the corresponding molecular frame denoted (U₁,U₂,U₃) (Fig. 1). From an optical point of view, the TTB crystal structure is uniaxial with its optical axis along U₃. The refractive index in the (100) plane, independent of the polarization orientation, is defined by the ordinary index n_o, whereas the extraordinary index n_e is associated to an incoming polarization along U₃. From the geometrical selection rules governed by the $D_{3 h}$ octupolar symmetry [1], the corresponding electro–optic tensor in the crystal unit cell frame (U₁,U₂,U₃) exhibits only three non-vanishing components: $r_{11} = - r_{21} = - r_{62}$ .

From the system geometry depicted in Fig. 1, the molecular crystal plane $(U_{1}, U_{2})$ lies in (Y,Z) macroscopic plane, while the electrode planes are parallel to (X,Y). The incident light is polarized in the (X,Y) plane at 45° relative to X. When a modulated electric field $E_{m}$ is applied along the Z direction, the electric field components in the crystal frame $(U_{1}, U_{2})$ are given by $E_{1} = - E_{m} \times \cos ψ$ and $E_{2} = E_{m} \times \sin ψ$ . In this configuration, the index ellipsoid is defined by the equation [12,13]:

(\frac{1}{n_{o}^{2}} - r_{11} E_{m} \cos ψ) U_{1}^{2} + (\frac{1}{n_{o}^{2}} + r_{11} E_{m} \cos ψ) U_{2}^{2} - 2 r_{11} E_{m} \sin ψ \cdot U_{1} U_{2} + \frac{U_{3}^{2}}{n_{e}^{2}} = 1

The system, initially uniaxial, becomes therefore biaxial upon the application of the electric field. Since the optical propagation is along the Z direction, the optical study can be performed in the

(U_{1}, U_{2})

plane. Defining

A = r_{11} E_{m} \cos ψ

and

B = r_{11} E_{m} \sin ψ

, and the initially ordinary and extraordinary indexes

n_{U_{1}} = n_{1} = n_{o}

and

n_{U_{3}} = n_{2} = n_{e}

, Eq. (1) can be written in a more synthetic formulation:

[\begin{matrix} U_{1} & U_{2} \end{matrix}] \cdot M \cdot [\begin{matrix} U_{1} \\ U_{2} \end{matrix}] = 1,
with
M = [\begin{array}{l} (\frac{1}{n_{o}^{2}} - A) -B \\ -B (\frac{1}{n_{o}^{2}} + A) \end{array}]

The resolution of the characteristic equation of this matrix

| M - μ \cdot I | = 0

(with I the identity matrix) leads to the corresponding characteristic values:

μ_{1} = \frac{1}{n_{o}^{2}} - r_{11} E_{m}

and

μ_{2} = \frac{1}{n_{o}^{2}} + r_{11} E_{m}

. In the new principal axes frame (

U_{1}^{'}, U_{2}^{'}

), the index ellipsoid takes the form:

\begin{array}{l} [\begin{matrix} U'_{1} & U'_{2} \end{matrix}] \cdot [\begin{matrix} μ_{1} & 0 \\ 0 & μ_{2} \end{matrix}] \cdot [\begin{matrix} U'_{1} \\ U'_{2} \end{matrix}] = 1
or \frac{U_{1}^{' 2}}{n_{1}^{'}} + \frac{U_{2}^{' 2}}{n_{2}^{'}} + \frac{U_{3}^{' 2}}{n_{3}^{'}} = 1 \\ with \frac{1}{n'_{1}^{2}} = \frac{1}{n_{o}^{2}} - r_{11} E_{m} and \frac{1}{n'_{2}^{2}} = \frac{1}{n_{o}^{2}} + r_{11} E_{m} \end{array}

After resolution of the eigenvector of the matrix M for the eigenvalues μ_1,2, the new optical axes

(U_{1}^{'}, U_{2}^{'})

of the system after application of the modulated field are seen to be related to the (

U_{1}^{}, U_{2}^{}

) frame by a rotation angle Θ with:

(\begin{matrix} U'_{1} \\ U'_{2} \end{matrix}) = (\begin{matrix} \cos Θ & \sin Θ \\ - \sin Θ & \cos Θ \end{matrix}) (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) and Θ = \frac{ψ}{2}

The modulation of an electric field in the Z direction therefore induces a rotation of the principal axes (

U_{1}^{}, U_{2}^{}

) by an angle Θ which is directly related to the Euler angle ψ, as depicted in Fig. 2 .

Fig. 2 Principal axes of the index ellipsoid ( $U_{1}^{}, U_{2}^{}$ ) in the absence of an applied field and principal axes intersection of the index ellipsoid ( $U_{1}^{'}, U_{2}^{'}$ ) in the presence of an applied field along the direction Z. The angles Θ, ψ and α are described in the text.

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As the input optical field polarization lies in the (X,Y) plane, a refractive index modulation will take place for the Y component of the polarization while the X component (perpendicular to the molecular stacks) is associated to a constant index n_e. The calculation of the refractive index along Y, defined as n_Y, can be deduced from the interception of the new index ellipsoid with the Y direction. Using the physically realistic limit $r_{11} E_{m} < < 1$ in Eq. (3), the new principal values of the refractive index are then given by:

n'_{1} = n_{o} + \frac{1}{2} r_{11} \cdot E_{m} \cdot n_{o}^{3} and n'_{2} = n_{o} - \frac{1}{2} r_{11} \cdot E_{m} \cdot n_{o}^{3}

In the presence of a modulating electric field, the intersection of the index ellipsoid with the plane

X = U_{3} = 0

(Fig. 2) is therefore given by the equation:

\frac{\cos α^{2}}{{(n_{o} + \frac{1}{2} r_{11} E_{m} n_{o}^{3})}^{2}} + \frac{\sin α^{2}}{{(n_{o} - \frac{1}{2} r_{11} E_{m} n_{o}^{3})}^{2}} = \frac{1}{n^{2} (α)}

where α is the angle between the

U'_{1}

direction and the direction defining the current point on the ellipsoid (Fig. 2). Denoting

n_{m} = \frac{1}{2} r_{11} E_{m} n_{o}^{3}

, the corresponding index for this current point is then:

n (α) = \frac{n_{o}^{2} - n_{m}^{2}}{\sqrt{{(n_{o} - n_{m})}^{2} \cos α^{2} + {(n_{o} + n_{m})}^{2} \sin α^{2}}}

The index n_Y is equal to

n_{Y} = n (α = \frac{π}{2} - (ψ + Θ))

with

Θ = ψ / 2

, therefore n_Y is a function of ψ with:

n_{Y} (ψ) = \frac{n_{o}^{2} - n_{m}^{2}}{\sqrt{{(n_{o} - n_{m})}^{2} \sin {(\frac{3 ψ}{2})}^{2} + {(n_{o} + n_{m})}^{2} \cos {(\frac{3 ψ}{2})}^{2}}}

Figure 3 shows the influence of the octupoles orientation angle ψ on the induced refractive index change. The minimum index n_Y is obtained at $ψ = 0 °$ , with $n_{Y}^{\min} = n_{o} - n_{m}$ and the maximum for $ψ = 60 °$ with $n_{Y}^{\max} = n_{o} + n_{m}$ . As expected, the maximum index change from its initial value $| Δ n_{Y} (ψ) | = | n_{Y} (ψ) - n_{o} |$ occurs for ψ = 0° with a π/3 periodicity, corresponding to one octupolar branch lying along the modulating field direction Z. At $ψ = π / 6$ (with a π/3 periodicity), the index change is almost zero, corresponding to a situation where one octupolar branch is perpendicular to the modulating field. In the present geometry, an electro-optic experiment can access the EO efficiency of the sample by measuring the induced phase shift between the X and Y polarization components of a field propagating along Z and polarized at 45° with the X direction (Fig. 1), for a crystal length L. This phase shift is denoted $Γ (ψ) = | n_{X} - n_{Y} (ψ) | L \cdot 2 π / λ$ , with λ the incident wavelength and $n_{X} = n_{e}$ . Its maximum expected value is therefore:

Γ^{\max} = Γ (ψ = k \cdot π / 3) = \frac{2 π}{λ} | n_{e} - (n_{o} - n_{m}) | L

with

n_{m} = \frac{1}{2} r_{11} E_{m} n_{o}^{3}

, which is obtained for

ψ = 0 °

for instance. This phase shift can be written

Γ^{\max} = Γ_{0} + Γ_{m}^{\max}

, sum of a constant phase shift

Γ_{0} = \frac{2 π}{λ} (n_{e} - n_{o}) L = \frac{2 π}{λ} Δ n L

and a modulated contribution

Γ_{m}^{\max} = \frac{2 π}{λ} n_{m} L = \frac{π}{λ} r_{11} E_{m} n_{o}^{3} L = \frac{π}{λ} r_{11} V_{m} n_{o}^{3}

with the associated potential

V_{m} = E_{m} \cdot L

. A π modulated phase change in this configuration can thus be obtained with the half voltage

V_{π} = \frac{λ}{r_{11} \cdot n_{o}^{3}}

, from which

r_{11}

can be measured providing that

ψ ≅ 0 °

is known.

Fig. 3 Calculated ψ-dependence of $n_{Y} (ψ)$ and $| Δ n_{Y} (ψ) |$ supposing $n_{o} = 1.5$ and $r_{11} E_{m} = 0.01$ . The corresponding orientations of the octupolar molecules in their plane are given for situations giving a maximum or minimum index change (k is an integer).

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For all other orientations ψ, the phase shift $Γ (ψ)$ is no more a simple expression as a function of $r_{11}$ and $E_{m}$ , in particular the dependence with respect to the modulating field amplitude is no more purely linear. For instance at $ψ ≅ 90 °$ where the minimum effect is expected, the calculation of the induced refractive index leads to $(ψ ≃ 90 °) = (n_{o}^{2} - n_{m}^{2}) / (n_{o} \sqrt{1 + \frac{n_{m}^{2}}{n_{o}^{2}}}) \approx n_{o} - \frac{3 n_{m}^{2}}{2 n_{o}^{2}}$ assuming $n_{m}^{4} < < 2 n_{o}^{3}$ . The corresponding phase shift is $Γ (ψ ≃ 90 °) = \frac{2 π}{λ} | n_{e} - (n_{o} - \frac{3 n_{m}^{2}}{2 n_{o}^{}}) | L = Γ_{0} + Γ_{m}^{\min}$ , with $Γ_{m}^{\min} = \frac{2 π}{λ} . \frac{3 n_{m}^{2}}{2 n_{0}} L = \frac{2 π}{λ} . \frac{3}{8 L} r_{11}^{2} V_{m}^{2} n_{o}^{5}$ , which is quadratically dependent on the electric field modulation amplitude. A π modulated phase change in this configuration is obtained with the half voltage $V_{π} = \sqrt{\frac{4 \cdot λ \cdot L}{3 \cdot r_{11}^{2} \cdot n_{o}^{5}}}$ , from which $r_{11}$ can be measured providing that $ψ ≃ 90 °$ is known.

3. Comparison with the electro-optic effect in a uni-dimensional symmetry system

In order to compare this situation to more traditional systems used in electro-optic devices based on organic materials, a similar calculation is undertaken in a system of non-centrosymmetric 1D symmetry. For comparison we consider that the molecular system lies along the U₁ axis, therefore only one electro-optic coefficient $r_{11}$ exists. This situation relates to crystals having a predominant nonlinear diagonal coefficient, or in a first approximation to $C_{\infty v}$ poled polymers neglecting the off-diagonal coefficient contributions [7]. In the case where the molecular symmetry axis (molecules main orientation direction) lies in the (Y, Z) plane, the applied electric modulation field $E_{m}$ can be either set in the transverse (Y) (Fig. 4(a) ) or longitudinal (Z) direction (Fig. 4(b)).

Fig. 4 Transverse (a) and longitudinal (b) electro-optic modulation geometries in 1D molecular systems. The angles and direction notations are the same as used for the octupolar systems.

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The transverse case (Fig. 4(a)) is naturally optimized if the molecules also lie along the Y direction, leading to an optimized modulated phase $Γ_{m} = \frac{π}{λ} r_{11} V_{m} n_{o}^{3} L$ . This value is the same expression as obtained previously for octupoles in a longitudinal configuration. This transverse geometry would however lead to more delicate electrodes fabrication and fields application [14].

The case of a longitudinal field application along Z (Fig. 4(b)) leads to a modified index ellipsoid which is reduced to:

(\frac{1}{n_{e}^{2}} + r_{11} E_{m} \cos ψ) U_{1}^{2} + \frac{1}{n_{o}^{2}} U_{2}^{2} = 1

with ψ the angle between Z and the molecular direction U₁. As the characteristic matrix of Eq. (10) is still diagonal, there is therefore no rotation of the index ellipsoid under a modulating field, the corresponding indexes being:

n_{1} (ψ) \approx n_{e} - \frac{1}{2} n_{o}^{3} r_{11} E_{m} \cos ψ

,

n_{2} = n_{o}

. The calculation of the n_Y index following the same procedure as above leads to:

n_{Y} = n (α = \frac{π}{2} - ψ) = \frac{n_{1} (ψ) n_{o}}{\sqrt{n_{o}^{2} \sin ψ^{2} + n_{1}^{2} (ψ) \cos ψ^{2}}}

The minimum index n_Y is obtained at

ψ = 0 °

(with a π periodicity), with

n_{Y}^{\min} = n_{o}

and the maximum for

ψ = 90 °

(with a π periodicity) with

n_{Y}^{\max} = n_{e}

. Such values are expected from situations where the molecules lie respectively along the Z and Y directions. The optimal case where the molecules lie along Y is however not favorable for electro-optic modulation since the corresponding index does not depend on the modulated field amplitude. The optimization of electro-optic coefficient coupling and field modulation are therefore antagonistic and cannot be combined in a common molecular orientation scheme. Obtaining an electro-optic modulation necessitates a compromise orientation such as

ψ = 45 °

, which leads to a loss of index change amplitude compared to the octupolar case.

This comparison shows overall that electro-optic modulation in octupolar systems is more flexible and can lead, with more amenable technology, to electro-optic modulation values as high (if not higher, depending on $r_{11}$ ) as in transverse 1D molecular systems.

4. Experimental results

The sample is an evaporated crystalline thin film of TTB (20 wt %)–PMMA(80 wt %) in chlorobenzene made of micrometric to millimetric size columnar cylindrical structures which naturally form on a polyimide (800 Å thickness) / ITO (PI/ITO) substrate [15]. The crystallinity of these structures could be confirmed by X-rd measurement (Fig. 5(a) ). Further confirmation of both the sample crystalline behavior and the nature of the molecular orientation have been obtained by measuring the birefringence of the columnar structures. Figure 5(b) shows the transmission of a sample region of with 1 mm diameter illuminated by a circularly polarized 633 nm beam at normal incidence. The sample is sandwiched between two crossed polarizers which are rotated simultaneously. In this situation, a π/4 periodic response is expected if the sample is strongly birefringent with a privileged direction, which is observed in the case of the TTB sample. Therefore, from the results of X-rd, birefringence measurements, and second harmonic generation measurements [15], we could confirm that the columnar cylindrical structures of the TTB molecules on PI/ITO were developed as a single crystal. At last, the birefringence and SHG measurements performed on large crystalline area support the evidence that this mono-crystalline behavior occurs at the optical-measurement spatial scale. Therefore, we suppose that the sample symmetry and orientation are constant over the whole illumination area. Any molecular orientation disorder would lead to an underestimation of the nonlinear coefficients in the sample.

Fig. 5 (a) X-rd measurement. (b) The birefringence measurement of the cylinder TTB crystal, performed by measuring the transmission of a TTB sample region of 1mm size between two crossed polarizers rotating simultaneously (the polar angle in the polar graph is the polarizers rotation angle).

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The electro-optic modulation experiment is performed following the geometry depicted in Fig. 6(a) . The sample of columnar cylindrical structures was sandwiched between polyimide (800 Å thickness)/ITO (PI/ITO) substrates [15]. The PI/ITO substrates were employed as transparent electrodes.

Fig. 6 Experimental electro-optic effect in a TTB thin crystalline film. (a) Setup. P: linear polarizer, $λ / 2$ : half wave plate, S: crystal sample, $λ / 4$ : quarter waver plate, A: analyzer. (b) Measured modulated light intensity vs. input beam polarization angle (dots) and theoretical fit to Eq. (12) (continuous line). (c) Modulated light intensity (dots) vs. fast axis angle of $λ / 4$ plate. The continuous line is the theoretical fit to Eq. (13). The experimental and fit curves are normalized to their maximum. The data of (b) and (c) were measured at a fixed applied electric field modulation of 61 $V_{P P}$ at 4 kHz.

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The orientation of the columnar structures is set such as the molecular stacking direction is along X, with both electric field modulation at AC 4 kHz and optical incidence lying along Z, as described below. The incident beam from a 633nm HeNe laser with a few $m W s$ power passes through an input linear polarizer, a $λ / 2$ plate used to rotate the polarization of the input field, the sample, a $λ / 4$ plate, and an analyzer (Fig. 6(a)). The polarizer and the analyzer are initially kept perpendicularly polarized at $\pm 45 °$ from the (X, Y) axes.

The optic field amplitude $E_{o u t}$ and the output intensity $I_{o u t}$ can be expressed using the Jones matrices of the each optical element.

In a first situation the input polarization and the analyzer are rotated simultaneously from their initial position indicated in Fig. 6(a), by an angle β relative to X. The other optical elements are fixed. The final output optical field $E_{o u t}$ and output intensity passing through the analyzer is given by:

\begin{array}{l} E_{o u t} = [\begin{matrix} \sin β^{2} & - \cos β \cdot \sin β \\ - \sin β \cdot \cos β & \cos β^{2} \end{matrix}] \cdot [\begin{matrix} e^{- i \frac{Γ_{T}}{2}} & 0 \\ 0 & e^{i \frac{Γ_{T}}{2}} \end{matrix}] \cdot E_{i n} [\begin{matrix} \cos β \\ \sin β \end{matrix}] \\ I_{o u t} = E_{o u t} \cdot E_{o u t}^{*} = 2 I_{i n} \cdot \sin β^{2} \cos β^{2} (1 + \sin Γ) \end{array}

where

Γ_{T} = \frac{π}{2} + Γ

is the total phase retardation induced by the

λ / 4

plate (π/2) and the sample (Γ). As shown above, the sample phase shift Γ is the sum of a constant phase shift

Γ_{0} = \frac{2 π}{λ} Δ n

and the modulated contribution

Γ_{m}

. Figure 6(b) shows the experimental dependence of the modulated light intensity vs. the input beam polarization angle β, which is seen to be in good agreement with Eq. (12).

In a second situation the $λ / 4$ plate is rotated by an angle γ relative to X and the other optical elements are fixed, such as the input polarization is at $+ 45^{\circ}$ relative to X and the analyzer is crossed at $- 45^{\circ}$ . The optical field and the output intensity become:

\begin{array}{l} E_{o u t} = \frac{1}{2} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}] \cdot \frac{1}{\sqrt{2}} [\begin{matrix} 1 + i (\sin γ^{2} - \cos γ^{2}) & - 2 i \sin γ \cdot \cos γ \\ - 2 i \sin γ \cdot \cos γ & 1 - i (\sin γ^{2} - \cos γ^{2}) \end{matrix}] \cdot [\begin{matrix} e^{- i \frac{Γ}{2}} & 0 \\ 0 & e^{i \frac{Γ}{2}} \end{matrix}] \cdot E_{i n} \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ 1 \end{matrix}] \\ I_{o u t} = I_{i n} \cdot [\frac{1}{2} [- \sin \frac{Γ}{2} + \cos \frac{Γ}{2} (\sin γ^{2} - \cos γ^{2})]^{2} + 2 \sin {\frac{Γ}{2}}^{2} \sin γ^{2} \cos γ^{2}] \end{array}

Figure 6(c) shows the experimental dependence of the modulated light intensity vs. the fast axis γ angle of the plate, which is also seen to be in good agreement with Eq. (13).

The experiments corresponding to the two described situations have been used to determine the electro-optic efficiency of the TTB crystalline samples. When the direction of the input polarizer is $+ 45^{\circ}$ relative to X ( $β = 45^{\circ}$ ), and the slow-axis of the quarter wave plate is horizontal ( $γ = 0^{\circ}$ ), the direction of analyzer remaining $- 45^{\circ}$ relative to X, then the above two expressions become a same equation:

I_{o u t} (E_{m}, ψ) = I_{i n} \frac{1}{2} [1 + \sin Γ (E_{m}, ψ)]

As mentioned above,

Γ (E_{m}, ψ) = | n_{X} - n_{Y} (E_{m}, ψ) | L \cdot 2 π / λ = Γ_{0} + Γ_{m} (E_{m}, ψ)

is a complex function of the orientation of the octupolar molecules in their plane, with

n_{Y} (E_{m}, ψ)

given in Eq. (8).

Γ_{m} (E_{m}, ψ)

exhibits in particular either a linear or a nonlinear dependence relative to the electric field modulation amplitude

E_{m}

, depending on the molecular orientation ψ. Examples of

I_{o u t} (E_{m})

dependences are shown in Fig. 7(a) . While the case

ψ = 0^{\circ}

shows a traditional sinusoidal behavior representative of a

Γ_{m} (E_{m}, ψ = 0 °) \propto E_{m}

linear dependence, the case

ψ ≅ 90^{\circ}

shows a modified behavior since in this situation

Γ_{m} (E_{m}, ψ ≅ 90^{o}) \propto E_{m}^{2}

, as described above. As can be observed in Fig. 7(b), this behavior has also been observed experimentally, where different responses can be measured in samples where different angles ψ are expected. This situation is primarily due to the complex fields-matter coupling occurring for not optimal molecular orientations.

Fig. 7 Experimental electro-optic responses in a TTB thin crystalline film. (a) Theoretical I_out(V_m) dependence, assuming the parameters $L = 1 μ m$ , $n_{o} = 1.5825$ , $n_{e} = 1.59$ , $r_{11} = 300 p m / V$ ( $ψ = 0 °$ curve) and $r_{11} = 1100 p m / V$ ( $ψ = 90 °$ curve), $λ = 633 n m$ . (b,c) Experimental I_exp(V_m) dependence measured on the modulation intensity for two different samples and the data of Fig. 7(b) and 7(c) were renormalized with maximum value.

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The experimental measurement of the EO coefficient $r_{11}$ finally requires a preliminary knowledge of the molecular orientation ψ. Assuming an orientation $ψ = 0^{\circ}$ in the sample leads to an estimation of a the lower limit for r₁₁, since this situation leads to optimal coupling.

Experimental data on TTB thin crystalline films are treated by measuring the $V_{π}$ voltage necessary to obtain a phase shift $Γ_{m} = π$ , which corresponds to a cancellation of the modulated contribution of the output intensity. In the investigated samples, a modulated electric field of maximum amplitude 140V is applied between the electrodes distant by $L = 3.35 μ m$ , at a modulation frequency of 4 kHz. The sample ordinary index $n_{o} = 1.5825$ is known from a separate measurement. ψ can be in principle determined from a Second Harmonic Generation measurement where both polarization and incidence angle are tuned [3,15]. Typical values of $V_{π} \approx 230 V$ are measured in samples whose output intensity dependence $I_{o u t} (V_{m})$ resembles Fig. 6(b), for which one can consider that ψ is close to 0°. The knowledge of the exact molecules orientation in the crystal being however inaccurate, this $V_{π}$ measurement can nevertheless be used to estimate a range of possible values for the nonlinear coefficient $r_{11}$ of the measured crystal. The previous equations given in Section 2 lead to an estimation of $r_{11}$ from $r_{11} = \frac{λ}{V_{π} \cdot n_{o}^{3}}$ (estimated for $ψ = 0 °$ ) to $r_{11} = \sqrt{\frac{4 \cdot λ \cdot L}{3 \cdot V_{π}^{2} \cdot n_{o}^{5}}}$ (estimated for $ψ = 90 °$ ), which makes $r_{11}$ falls in the range $r_{11} \approx 695 - 2300 p m / V$ . This range of values is consistent with values of $d_{11} \approx 1000 - 1500 p m / V$ previously measured in similar samples by Second Harmonic Generation [3,15,16]. Overall this order of magnitude of EO efficiencies of TTB crystalline thin films is high compared to more traditional molecular materials [7,10,17,18]. Note that in the most general case, $r_{11}$ can also be deduced without the need to measure $V_{π}$ by further simplifying Eq. (14) using the preliminary assumption $Γ_{0}, Γ_{m} < < 1$ . Then $\sin Γ \approx Γ_{m} + Γ_{0}$ and $I_{o u t} = I_{0} + I_{m}$ is the sum between a constant intensity $I_{0} = I_{i n} / 2$ (measured without electric field modulation) and the modulated contribution $I_{m} \approx I_{0} \cdot Γ_{m}$ . Under this assumption, $r_{11}$ can be directly inferred from the measurement of $I_{m} / I_{0} = Γ_{m}$ .

5. Conclusion

The electro-optic effect in crystalline films of octupolar symmetry has been studied theoretically and experimentally. Simple relations have been found between the molecular orientation in the sample plane and the expected electro-optic response, which show that a control of the octupoles orientation in their plane is essential to optimize the macroscopic response. The derived longitudinal electro-optic response, compared to the one from traditional 1D systems, shows that octupolar films can provide a flexible and efficient solution for electro-optic modulation where the compromise molecular orientation versus fields-molecular coupling is more readily obtained. The experimental data fit are shown to be in good agreement with the expected modulation, and provide an estimation of the order of magnitude of the EO efficiency in TTB crystalline thin films.

Acknowledgments

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00348 and 2009-0068755), the National Research Foundation (NRF) grant (No. 2010-0018921), and Priority Research Centers Program through the NRF funded by the Ministry of Education, Science and Technology (2010-0020209).

References and links

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Electro-optic effect in crystalline films of transverse planar octupolar symmetry

Abstract

1. Introduction

2. Electro-optic effect in an octupolar system: theory

3. Comparison with the electro-optic effect in a uni-dimensional symmetry system

4. Experimental results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (14)

Optics Express