Electro-Optic Modulators and Switches

Lithium Niobate        Phase Modulators        Switching Speed vs. length

    Electro-optic modulators are normally referred to those devices that employ Kerr and Pockels effects in materials. In this case, the dielectric constant of the material changes as a function of the applied electric field, causing a change in the refractive index, n. The change in n due to an applied electric field is small and is conventionally expressed in terms of a power series as,

                (1)

Here, r and R are the linear and the quadratic electro-optic coefficients, respectively. Linear electro-optic effects are not observed in liquids or in crystals possessing inversion symmetry. In such materials, the reversal of the applied electric field (from E to -E) should not change the refractive index due to the absence of a preferred direction. The only way this requirement can be satisfied is for the linear electro-optic coefficient to vanish. In general, the applied electric field in above equation is a vector, the electro-optic coefficients are tensor elements, and the refractive index value depends on the polarization and direction of propagation of light.

Lithium Niobate

    Lithium niobate is one of the commonly used materials for electro-optic modulation. Linear electro-optic coefficients at room temperature at the wavelength 633 nm are given by (in units of pV/m):

r₁₃ = r₂₃ = 8.6, r₂₂ = - r₁₂ = - r₆₁ = 3.4, - r₃₃ = 30.8, r₅₁ = r₄₂ = 28.

Phase Modulators

    When light passes through an electro-optic material, an external electric field applied to the material causes a change in the speed of propagation of light that leads to a phase shift. In the presence of the linear electro-optic effect only and when the change in the refractive index is small, the phase shift Df is given by

  »               (2)

Here, l is the free-space wavelength and L is the length of the active region in the electro-optic material. The electric field required to cause 180° phase shift is called E_p and is obtained by setting Df = p:

                        (3)

There are direct trade-offs between the length of the modulator, the electro-optic coefficient, and E_p. Therefore, to keep the applied electric field at low amplitudes, the length of the modulator can be increased. However, if a higher operating frequency is required, the length need to be small (see Fig. 1 below).

Switching Speed of LiNO₃ modulator

Let us calculate the operating frequency of a lithium niobate modulator as a function of its length. The effective dielectric constant of this material is e_eff = (1 + e_r)/2 for er = 45. The length of the modulator has to be small compared with the wavelength of the electrical signal, L<<l_e. The relationship between l_e and frequency, f, in a dielectric medium is given by l_e = c/(fÖe_eff). If we assume L is one-tenth of l_e it can be shown that for L = 1 cm, f ~ 625 MHz. Fig. 1 displays the operating frequency of lithium niobate modulator as a function of its length.

Hz.                            (4)

Fig. 1. Operating frequency of a lithium niobate electro-optic modulator as a function of its thickness (semi-log plot).

Fig. 2. Extended x-axis view (log-log) of Fig. 1.

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