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Electro-Optic Modulators

This is a continuation from the previous tutorial - Pockels Effect. The index changes induced by the Pockels effect can be utilized to construct a variety of electro-optic modulators, in either bulk or waveguide structures. An electro-optically induced rotation of principal axes is not required for the functioning of an electro-optic modulator though it often accompanies the index changes. However, the directions of the principal axes in the presence of an applied electric field, whether rotated or not, have to be taken into consideration in the design and operation of an electro-optic modulator. In this tutorial, we consider the operation...

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Pockels Effect

This is a continuation from the previous tutorial - Electro-Optic Effects. The majority of electro-optic devices are based on the Pockels effect. Structurally isotropic materials, including all gases, liquids, and amorphous solids such as glass, show no Pockels effect because they are centrosymmetric. Among the 32 point groups in the 7 crystal systems, 11 are centrosymmetric, and the remaining 21 are noncentrosymmetric. It is important to note that the linear optical property of a crystal is determined only by its crystal system, as mentioned in the propagation in an anisotropic medium tutorial and summarized in Table 2 in that tutorial,...

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Electro-Optic Effects

This is a continuation from the previous tutorial - Surface Input and Output Couplers. The optical property of a dielectric material can be changed through an electro-optic effect in the presence of a static or low-frequency electric field \(\pmb{E}_0\). The result is a field-dependent susceptibility and thus a field-dependent electric permittivity. \[\tag{6-1}\mathbf{P}(\omega,\pmb{E}_0)=\epsilon_0\boldsymbol{\chi}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)=\epsilon_0\boldsymbol{\chi}(\omega)\cdot\mathbf{E}(\omega)+\epsilon_0\Delta\boldsymbol{\chi}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)\] and \[\tag{6-2}\mathbf{D}(\omega,\pmb{E}_0)=\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)=\boldsymbol{\epsilon}(\omega)\cdot\mathbf{E}(\omega)+\Delta\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)\] where field-independent \(\boldsymbol{\chi}(\omega)=\boldsymbol{\chi}(\omega,\pmb{E}_0=0)\) and \(\boldsymbol{\epsilon}(\omega)=\boldsymbol{\epsilon}(\omega,\pmb{E}_0=0)\) represent the intrinsic linear response of the material at the optical frequency \(\omega\), while \(\Delta\boldsymbol{\chi}\) and \(\Delta\boldsymbol{\epsilon}\) represent changes induced by the low-frequency field \(\pmb{E}_0\). We can write \(\mathbf{D}(\omega,\pmb{E}_0)=\mathbf{D}(\omega)+\Delta\mathbf{P}(\omega,\pmb{E}_0)\), where \(\Delta\mathbf{P}(\omega,\pmb{E}_0)=\Delta\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)\). The total permittivity of the material in the presence of...

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Surface Input and Output Couplers

This is a continuation from the previous tutorial - Directional Couplers. In a system, it is always necessary to couple light from sources, such as lasers or light emitting diodes, to transmission components, which are usually dielectric waveguides or fibers, to various functional devices, such as optical switches, power dividers, amplifiers, and modulators, possible through transmission components again, and ultimately to photodetectors. The approaches to coupling light in and out of optical waveguides, including fibers, are basically classified into two categories: (1) surface coupling, also called longitudinal coupling, and (2) end coupling, also called end-fire coupling or transverse coupling. The...

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Directional Couplers

This is a continuation from the previous tutorial - grating waveguide couplers. Directional couplers are multiple-waveguide couplers used for codirectional coupling. They can be used in many different applications, including power splitters, optical switches, wavelength filters, and polarization selectors. We consider in this tutorial two-channel directional couplers, which consist of two parallel waveguides, as shown schematically in figure 4 below.  Figure 4. Schematic diagram of (a) a two-channel directional coupler and (b) its index profile assuming two step-index waveguides on the same substrate. The coupler is symmetric if \(n_a=n_b=n_1\) and \(d_a=d_b=d\). For simplicity, we consider only the case where each...

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