1. Kerr Effect  n = KE a 2 Applied field Kerr effect term An applied electric field, via the Kerr effect, induces birefringences in an otherwise optically isotropic material Kerr coefficient

2. 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.

3. Electro-Optic Properties MaterialCrystalIndicesPockels Coefficients  10 -12 m/V K m/V 2 Comment LiNbO 3 Uniaxialn o = 2.286 n e = 2.200 r 13 = 9.6 (8.6); r 33 = 309 (30.8) r 22 = 6.8 (3.4); r 51 =32.6 (28)  633 nm KDP (KH 2 PO 4 ) Uniaxialn o = 1.512 n e = 1.470 r 41 = 8.8; r 63 = 10.3  546 nm KD*P (KD 2 PO 4 ) Uniaxialn o = 1.508 n e = 1.468 r 41 = 8.8; r 63 = 26.8  546 nm GaAsIsotropicn o = 3.6r 41 = 1.43   1.15  m GlassIsotropic n o  1.5 03×10 -15 NitrobenzeneIsotropic n o  1.5 03×10 -12 Pockels (r) and Kerr (K) coefficients in a few selected materials Values in parentheses for r values are at very high frequencies

4. Kerr Effect Example Example: Kerr Effect Modulator Suppose that we have a glass rectangular block of thickness (d) 100  m 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 E a direction, along the z-axis. What is the applied voltage that induces a phase change of  (half-wavelength)? Solution The phase change  for the optical field E z is For  = , V = V /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.

5. 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  The phase shift is  and depends on the voltage V through the Pockels effect

6. Integrated Optical Modulators Spatial overlap efficiency = 0.5 – 0.7 Pockels coefficient Different for different crystal orientations Electrode separation Length of electrodes Induced phase change Applied voltage

7. Integrated Optical Modulators: An Example Consider  an x-cut LiNbO 3 modulator with d  10  m, operating at  = 1.5  m This will have V /2 L  35 V  cm A modulator with L = 2 cm has V /2 = 17.5 V By comparison, for a z-cut LiNbO 3 plate, that is for light propagation along the y-direction and E a along z, the relevant Pockels coefficients (r 13 and r 33 ) are much greater than r 22 so that V  /2 L  5 V  cm  depends on the product V × L When  = , then V × L = V /2 L

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

9. A LiNbO 3 Mach-Zehnder Modulator A LiNbO 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 V /2 = 5 V at 1550 nm. (© JENOPTIK Optical System GmbH.)

10. 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.

11. Integrated Mach-Zehnder Modulators E out  Acos(  t +  ) + Acos(  t   ) = 2Acos  cos(  t) Approximate analysis Input C breaks into A and B 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 Output power P out  E out 2 Amplitude

12. Mach-Zehnder Modulator Courtesy of Thorlabs

13. 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 > n s (= 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 = L o, all the light has been transferred to B. Beyond this point, light begins to be transferred back to A in the same way.

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

15. Coupling Efficiency  =  3/L o  =  A   B = Mismatch between propagation constants AA BB When the mismatch then, power transfer is prevented We can induce this mismatch by applying a voltage (Pockels effect)

16. Coupled Waveguide Modulator Pockels coefficient Applied voltage Voltage induced mismatch

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

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

19. Modulated Directional Coupler: Example Example: Modulated Directional Coupler Suppose that two optical guides embedded in a substrate such as LiNbO 3 are coupled as in Figure 6.31 to form a directional coupler, and the transmission length L o = 10 mm. The coupling separation d is ~10  m,   0.7, the operating wavelength is 1.3  m where Pockels coefficient r  10  10  m/V and n  2.20. What is the switching voltage for this directional coupler? Solution

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