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Nonlinear Optics (Wise 2018/19) Lecture 6: November 18, 2018 5.2 Electro-Optic Amplitude Modulator 5.3 Electro-Optic Phase Modulator 5.4 Microwave Modulator

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Nonlinear Optics (Wise 2018/19) Lecture 6: November 18, 2018 5.2 Electro-Optic Amplitude Modulator 5.3 Electro-Optic Phase Modulator 5.4 Microwave Modulator Nonlinear Optics (WiSe 2018/19) Lecture 6: November 18, 2018 5.2 Electro-optic amplitude modulator 5.3 Electro-optic phase modulator 5.4 Microwave modulator 6 Acousto-optic modulators 6.1 Acousto-optic interaction 6.2 The acousto-optic amplitude modulator Chapter 7: Third-order nonlinear effects 7.1 Third-harmonic generation (THG) 7.2 The nonlinear refractive index 7.3 Molecular orientation and refractive index 7.4 Self-phase modulation (SPM) 7.5 Self-focusing 1 5.2 Electro-optic amplitude modulator How is phase retardation converted into a amplitude modulation? 5.2. ELECTRO-OPTIC AMPLITUDE MODULATOR 93 0 direction input polarizer wave plate output polarizer Figure 5.7: Transversal electro-optic amplitude modulator from LiNbO3. Here, ∆φWP is the phase retardation due to the field-independent birefrin- 2 gence or due to an additional wave plate as shown in Fig. 5.7, and a is a co- efficient describing the relationship between field-dependent phase retardation and applied voltage. Usually we use β =45◦ to achieve 100 % transmission Iout 1 = 1 cos [∆φWP + aV (t)] (5.43) Iin 2 { − } 1 = 1 cos ∆φ cos [aV (t)] + sin ∆φ sin [aV (t)] . 2 { − WP WP } There are various applications for modulators. If the transmission through the modulator should be linearly dependent on the applied voltage, we use a bias ∆φ = π/2andobtainforaV 1 (see also Fig. 5.8) WP ≪ I 1 out = [1 + aV (t)] . (5.44) Iin 2 For a sinusoidal voltage V (t)=V0 sin ωmt (5.45) and constant input intensity, we obtain a sinusoidally varying output intensity Iout 1 = (1 + aV0 sin ωmt) . (5.46) Iin 2 The constant a can easily be replaced by the half-wave voltage Vπ π 2πLn3r a = = 0 22 , (5.47) Vπ λ0d 94 CHAPTER 5. THE ELECTRO-OPTIC EFFECT AND MODULATORS retardation Figure 5.8: Transmission characteristic of the modulator shown in Fig. 5.7 as a function of phase retardation. where we used Eq. (5.33). With ∆φWP =0, the relationship between output intensity and applied voltage is I 1 out = [1 cos (aV (t))] , (5.48) Iin 2 − and, for small voltages aV (t), we obtain a quadratic dependence I 1 out = a2V 2 (t) . (5.49) Iin 4 3 5.3 Electro-optic phase modulator If the wave plate and polarizers in Fig. 5.7 are removed and only the ordinary or extraordinary wave is excited, then the electric field after the crystal is ω Ey (t)=E0 cos [ωt + φ (t)] , (5.50) with πn3r φ (t)= 0 22 V (t) . (5.51) λ0d With a sinusoidal voltage and from Eq. (5.45), we obtain ω Ey (t)=E0 cos (ωt + m sin ωmt) (5.52) = E [cos ωt cos (m sin ω t) sin ωt sin (m sin ω t)] , 0 m − m with modulation depth m πn3r m = 0 22 V . (5.53) λ d 0 ! 0 " 94 CHAPTER 5. THE ELECTRO-OPTIC EFFECT AND MODULATORS retardation Figure 5.8: Transmission characteristic of the modulator shown in Fig. 5.7 as a function of phase retardation. where we used Eq. (5.33). With ∆φWP =0, the relationship between output intensity and applied voltage is I 1 out = [1 cos (aV (t))] , (5.48) Iin 2 − and, for small voltages aV (t), we obtain a quadratic dependence I 1 out = a2V 2 (t) . (5.49) Iin 4 5.3 Electro-optic phase modulator 5.3 Electro-optic phase modulator If the wave plate and polarizers in Fig. 5.7 are removed and only the ordinary or extraordinary wave is excited, then the electric field after the crystal is ω Ey (t)=E0 cos [ωt + φ (t)] , (5.50) with πn3r φ (t)= 0 22 V (t) . (5.51) λ0d With a sinusoidal voltage and from Eq. (5.45), we obtain ω Ey (t)=E0 cos (ωt + m sin ωmt) (5.52) = E [cos ωt cos (m sin ω t) sin ωt sin (m sin ω t)] , 0 m − m with modulation depth m πn3r m = 0 22 V . (5.53) λ d 0 ! 0 " 4 5.3. ELECTRO-OPTIC PHASE MODULATOR 95 With the generating functions for the Bessel functions ∞ cos (m sin ωmt)=J0 (m)+2 J2k (m)cos(2kωmt) , (5.54) !k=1 ∞ sin (m sin ωmt)=2 J2k+1 (m)sin[(2k +1)ωmt] , (5.55) !k=0 and the addition theorem 2sinA sin B =cos(A B) cos (A + B) , (5.56) − − 2cosA cos B =cos(A B)+cos(A + B) , (5.57) − the spectrum of the output field is ω Ey (t)=E0 [J0 (m)cosωt (5.58) +J (m)cos(ω + ω ) t J (m)cos(ω ω ) t 1 m − 1 − m +J (m)cos(ω +2ω ) t + J (m)cos(ω 2ω ) t 2 m 2 − m +J (m)cos(ω +3ω ) t J (m)cos(ω 3ω ) t 3 m − 3 − m 5 + ...] . The spectrum consists of sidebands at multiples of the modulation fre- quency ωm. An example is shown in Fig. 5.9. sideband amplitude angular frequency Figure 5.9: Phase-modulated spectrum with a modulation depth of m = 1. 5.3. ELECTRO-OPTIC PHASE MODULATOR 95 With the generating functions for the Bessel functions ∞ cos (m sin ωmt)=J0 (m)+2 J2k (m)cos(2kωmt) , (5.54) !k=1 ∞ sin (m sin ωmt)=2 J2k+1 (m)sin[(2k +1)ωmt] , (5.55) !k=0 and the addition theorem 2sinA sin B =cos(A B) cos (A + B) , (5.56) − − 2cosA cos B =cos(A B)+cos(A + B) , (5.57) − the spectrum of the output field is ω Ey (t)=E0 [J0 (m)cosωt (5.58) +J (m)cos(ω + ω ) t J (m)cos(ω ω ) t 1 m − 1 − m +J (m)cos(ω +2ω ) t + J (m)cos(ω 2ω ) t 2 m 2 − m +J (m)cos(ω +3ω ) t J (m)cos(ω 3ω ) t 3 m − 3 − m + ...] . The spectrum consists of sidebands at multiples of the modulation fre- quency ωm. An example is shown in Fig. 5.9. sideband amplitude angular frequency Figure 5.9: Phase-modulated spectrum with a modulation depth of m = 1. 6 96 CHAPTER 5. THE ELECTRO-OPTIC EFFECT AND MODULATORS 5.4 Microwave5.4 Microwave modulator modulator High-speed or microwave signals are typically supplied by strip-lines or copla- nar lines, see Eq. 5.10. For efficient modulation, the index modulation must copropagate in phase with the optical signal (group velocity, however, here we neglect dispersion, thus group velocity is equal to phase velocity). coplanar strip electrode terminating resistor waveguide electro-optic substrate Figure 5.10: Electro-optic traveling wave modulator. Let’s assume the microwave signal ω n V (z,t)=V cos ω t m m z , (5.59) 0 m − c ! 0 " with phase velocity c0 c = . (5.60) 7 nm If we consider the time slot of the optical wave that enters the crystal at t =0,andcopropagatewiththattimeslotinthesignalinthewaveguidewith effective index n, this time slot will experience the applied voltage ω z V (z)=V cos m (n n ) (5.61) 0 c − m # 0 $ at time t = zn/c0. The refractive index change experienced along the waveguide is ∆n (z)=aV (z) , (5.62) and the total integral phase change from input to output is ℓ ω∆n (z) ∆φ = dz (5.63) c %0 0 aV ω ℓ ω = 0 cos mZ (n n ) dz, c c − m 0 %0 # 0 $ 96 CHAPTER 5. THE ELECTRO-OPTIC EFFECT AND MODULATORS 5.4 Microwave modulator High-speed or microwave signals are typically supplied by strip-lines or copla- nar lines, see Eq. 5.10. For efficient modulation, the index modulation must copropagate in phase with the optical signal (group velocity, however, here we neglect dispersion, thus group velocity is equal to phase velocity). coplanar strip electrode terminating resistor waveguide electro-optic substrate Figure 5.10: Electro-optic traveling wave modulator. Let’s assume the microwave signal ω n V (z,t)=V cos ω t m m z , (5.59) 0 m − c ! 0 " with phase velocity c c = 0 . (5.60) nm If we consider the time slot of the optical wave that enters the crystal at t =0,andcopropagatewiththattimeslotinthesignalinthewaveguidewith effective index n, this time slot will experience the applied voltage ω z V (z)=V cos m (n n ) (5.61) 0 c − m # 0 $ at time t = zn/c0. The refractive index change experienced along the waveguide is ∆n (z)=aV (z) , (5.62) and the total integral phase change from input to output is ℓ ω∆n (z) ∆φ = dz (5.63) c %0 0 aV ω ℓ ω = 0 cos mZ (n n ) dz, c c − m 0 %0 # 0 $ 8 Chapter 6 Acousto-optic modulators An optical wave, that propagates through a medium with temporally and spatially varying refractive index, generates modulation sidebands. As we have seen in the previous section, an acoustic wave can generate a refractive index variation via the acousto-optic effect. The interaction of the index modulation wave with the optical wave may lead to a deflection or modulation of the optical wave. The phase velocity of acoustic waves is typically six orders of magnitude slower than optical waves. Propagating or standing acoustic waves generate propagating or standing index modulation patterns. Optical waves scatter on layers with different refractive index. 6. Acousto-optic modulator 6.1 Acousto-optic6.1 Acousto interaction-optic interaction In a linear acousto-optic medium, the refractive index change is proportional to the applied voltage, see Eq. (5.8).
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