Kerr Effect Dn l KEa 2 Kerr coefficient

Kerr Effect Dn = l KEa 2 Kerr coefficient Kerr effect term Applied field An applied electric field, via the Kerr effect, induces birefringences in an otherwise optically isotropic material

Kerr Effect (a) An applied electric field, via the Kerr effect, induces birefringences in an otherwise optically isotropic material. (b) A Kerr cell phase modulator.

Electro-Optic Properties Pockels (r) and Kerr (K) coefficients in a few selected materials Values in parentheses for r values are at very high frequencies Material Li. Nb. O 3 Crystal Uniaxial Indices Pockels Coefficients K Comment no = 2. 286 ´ 10 -12 m/V r 13 = 9. 6 (8. 6); r 33 = 309 (30. 8) m/V 2 l 633 nm l 546 nm ne = 2. 200 Uniaxial no = 1. 512 r 22 = 6. 8 (3. 4); r 51 =32. 6 (28) r 41 = 8. 8; r 63 = 10. 3 (KH 2 PO 4) KD*P (KD 2 PO 4) Uniaxial ne = 1. 470 no = 1. 508 r 41 = 8. 8; r 63 = 26. 8 l 546 nm Ga. As Isotropic ne = 1. 468 no = 3. 6 r 41 = 1. 43 l 1. 15 mm Glass Nitrobenzene Isotropic no 1. 5 0 0 3× 10 -15 3× 10 -12 KDP

Kerr Effect Example: Kerr Effect Modulator Suppose that we have a glass rectangular block of thickness (d) 100 mm and length (L) 20 mm and we wish to use the Kerr effect to implement a phase modulator in a fashion depicted in Figure 6. 26. The input light has been polarized parallel to the applied field Ea direction, along the z-axis. What is the applied voltage that induces a phase change of p (half-wavelength)? Solution The phase change Df for the optical field Ez is For Df = p, V = Vl/2, Although the Kerr effect is fast, it comes at a costly price. Notice that K depends on the wavelength and so does V 1/2.

Integrated Optical Modulators The electro-optic effect takes place over the spatial overlap region between the applied field and the optical fields. The spatial overlap efficiency is represented by a coefficient G The phase shift is Df and depends on the voltage V through the Pockels effect

Integrated Optical Modulators Induced phase change Length of electrodes Applied voltage Spatial overlap efficiency = 0. 5 – 0. 7 Electrode separation Pockels coefficient Different for different crystal orientations

Integrated Optical Modulators: An Example Df depends on the product V×L When Df = p, then V×L = Vl/2 L Consider an x-cut Li. Nb. O 3 modulator with d 10 mm, operating at l = 1. 5 mm This will have Vl/2 L 35 V cm A modulator with L = 2 cm has Vl/2 = 17. 5 V By comparison, for a z-cut Li. Nb. O 3 plate, that is for light propagation along the y-direction and Ea along z, the relevant Pockels coefficients (r 13 and r 33) are much greater than r 22 so that V l/2 L 5 V cm

A Li. Nb. O 3 Phase Modulator A Li. Nb. O 3 based phase modulator for use from the visible spectrum to telecom wavelngths, with modulation speeds up to 5 GHz. This particular model has Vl/2 = 10 V at 1550 nm. (© JENOPTIK Optical System Gmb. H. )

A Li. Nb. O 3 Mach-Zehnder Modulator A Li. Nb. O 3 based Mach-Zehnder amplitude modulator for use from the visible spectrum to telecom wavelengths, with modulation frequencies up to 5 GHz. This particular model has Vl/2 = 5 V at 1550 nm. (© JENOPTIK Optical System Gmb. H. )

Integrated Mach-Zehnder Modulators An integrated Mach-Zehnder optical intensity modulator. The input light is split into two coherent waves A and B, which are phase shifted by the applied voltage, and then the two are combined again at the output.

Integrated Mach-Zehnder Modulators Approximate analysis Input C breaks into A and B experience opposite phase changes arising from the Pockels effect A and B interfere at D. Assume they have the same amplitude A But, they have opposite phases Eout Acos(wt + f) + Acos(wt f) = 2 Acosf cos(wt) Output power Pout Eout 2 Amplitude

Mach-Zehnder Modulator Courtesy of Thorlabs

Coupled Waveguide Modulators (a) Cross section of two closely spaced waveguides A and B (separated by d) embedded in a substrate. The evanescent field from A extends into B and vice versa. Note: n. A and n. B > ns (= substrate index). (b) Top view of the two guides A and B that are coupled along the z-direction. Light is fed into A at z = 0, and it is gradually transferred to B along z. At z = Lo, all the light has been transferred to B. Beyond this point, light begins to be transferred back to A in the same way.

Coupled Waveguide Modulators Lo = Transfer distance If A and B are identical, full transfer of power from A to B occur over a coupling distance Lo, called the transfer distance

Coupling Efficiency Db = b. A b. B = Mismatch between propagation constants b. A When the mismatch Db = p 3/Lo then, power transfer is prevented We can induce this mismatch by applying a voltage (Pockels effect) b. B

Coupled Waveguide Modulator Applied voltage Voltage induced mismatch Pockels coefficient

Coupled Waveguide Modulator Voltage needed to switch the light off in B

Modulated Directional Coupler An integrated directional coupler. The applied field Ea alters the refractive indices of the two guides (A and B) and therefore changes the strength of coupling.

Modulated Directional Coupler: Example: Modulated Directional Coupler Suppose that two optical guides embedded in a substrate such as Li. Nb. O 3 are coupled as in Figure 6. 31 to form a directional coupler, and the transmission length Lo = 10 mm. The coupling separation d is ~10 mm, G 0. 7, the operating wavelength is 1. 3 mm where Pockels coefficient r 10´ 10 -12 m/V and n 2. 20. What is the switching voltage for this directional coupler? Solution

Acousto-Optic Modulator Fiber-coupled acousto-optic modulator (Courtesy of Gooch & Housego)