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

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 coefficient 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.

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 frequency 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 frequency 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 coplanar 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

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

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). The wave equation is ∂2 ∂2 E = µ D = µ (ε E + P) (6.1) ∇×∇× − 0 ∂t2 − 0 ∂t2 0 The time-varying displacement can be separated into a part described by a constant average susceptibility or refractive index n, and a time-varying contribution described by ∆n(r, t)

2 D = ε0E + P =ε0 (n + ∆n(r, t)) E ε n2E +2ε n∆n(r, t)E, (6.2) ≈ 0 0 where we neglect potential higher-order terms in ∆n(r, t), i.e., ∆n(r, t) n. From Eqs. (6.1) and (6.2), we have ≪ 1 ∂2E 1 ∂2 ∆n(r, t) E+ = 2 E (6.3) ∇×∇× c2 ∂t2 − c2 ∂t2 n ! "

99 9 100 CHAPTER 6. ACOUSTO-OPTIC MODULATORS

2 2 with c =1/(ε0µ0n ). We consider a plane electromagnetic wave with wave vector ks in the x-z-plane, see Fig. 6.1. i direction of acoustic wave k-vector of diffracted wave

n large k-vector of acoustic wave

k-vector of incident wave

Figure 6.1: Diﬀraction grating in a medium generated by an acoustic wave.

The refractive index change generated by a wave with frequency ωs is proportional to the acoustic wave 10 ∆n(r, t)=∆nˆ cos (ω t k r) s − s · ∆nˆ j(ωst ks r) j(ωst ks r) = e − · + e− − · . (6.4) 2 ∆nˆ is the amplitude of the resulting! refractive index wave." From Gauss law for the electric field εE =ρ=ε E + E ε, (6.5) ∇· ∇ · · ∇ with ε = ε (n+∆n(r, t))2 ε n2 +2n∆n(r, t) . (6.6) 0 ≈ 0 If the electric field is polarized along the y#-direction, we obtain$ E ε =0. If there are in addition no charges ρ, then the divergence of E vanishes,· ∇ and the wave equation (6.3) simplifies to 1 ∂2E 1 ∂2 ∆n(r, t) ∆E =+2 E (6.7) − c2 ∂t2 c2 ∂t2 n % & The time-dependent refractive index multiplies with the field, which initially consists of only an incident field with index i and a diffracted wave with index d is generated j(ωit ki r) j(ωdt kd r) E = Eˆ ie − · + Eˆ de − · + c.c. (6.8) The product of incoming wave and index modulation generates a polarization, which is the source for the diffracted wave. Therefore, there is ω = ω + ω and k = k + k . d ± s i d ± s i 100 CHAPTER 6. ACOUSTO-OPTIC MODULATORS

2 2 with c =1/(ε0µ0n ). We consider a plane electromagnetic wave with wave vector ks in the x-z-plane, see Fig. 6.1.

direction of acoustic wave k-vector of diffracted wave

n large k-vector of acoustic wave

k-vector of incident wave

Figure 6.1: Diﬀraction grating in a medium generated by an acoustic wave.

The refractive index change generated by a wave with frequency ωs is proportional to the acoustic wave ∆n(r, t)=∆nˆ cos (ω t k r) s − s · ∆nˆ j(ωst ks r) j(ωst ks r) = e − · + e− − · . (6.4) 2 ∆nˆ is the amplitude of the resulting! refractive index wave." From Gauss law for the electric field εE =ρ=ε E + E ε, (6.5) ∇· ∇ · · ∇ with ε = ε (n+∆n(r, t))2 ε n2 +2n∆n(r, t) . (6.6) 0 ≈ 0 If the electric field is polarized along the y#-direction, we obtain$ E ε =0. If there are in addition no charges ρ, then the divergence of E vanishes,· ∇ and the wave equation (6.3) simplifies to 1 ∂2E 1 ∂2 ∆n(r, t) ∆E =+2 E (6.7) − c2 ∂t2 c2 ∂t2 n % & The time-dependent refractive index multiplies with the field, which initially consists of only an incident field with index i and a diffracted wave with index d is generated j(ωit ki r) j(ωdt kd r) E = Eˆ ie − · + Eˆ de − · + c.c. (6.8) The product of incoming wave and index modulation generates a polarization, which is the source for the diffracted wave. Therefore, there is

ω = ω + ω and k = k + k . 11 d ± s i d ± s i 6.1. ACOUSTO-OPTIC INTERACTION 101

Usually, the frequency of sound is much less than the optical frequency, ω s ≪ ωi, and therefore kd ki . Fig. 6.1 shows the resulting k-diagram. We solve the wave equation| (6.7)| ≃ | approximately,| assuming that the amplitudes of the incoming and diﬀractive waves change slowly along the z-direction

j(ωit ki r) Ei = eyAi(z)e − · + c.c. (6.9) j(ωdt kd r) Ed = eyAd(z)e − · + c.c. (6.10)

If we substitute (6.7) into (6.8), and use the slowly varying envelope approximation (as in Chapter 3), we obtain

ω2 ω2 k2 d A (z) 2jk A (z) d ∆nAˆ (z)(6.11) − d − c2 d − d · ∇ d ≃−c2 i ! " ω2 ω2 k2 i A (z) 2jk A (z) i ∆nAˆ (z). (6.12) − i − c2 i − i · ∇ i ≃−c2 d ! " ω2 With k2 = d,i and k A (z)=k cos θ dAd , we obtain d,i c2 d · ∇ d d dz dA (z) ω ∆nˆ d j d A (z) (6.13) dz ≃−2c cos θ i dAi(z) ωi ∆nˆ j Ad(z). (6.14) dz ≃−2c cos θ 12 This set of equations describes coupled modes. However, the coupling coefficients ω ∆nˆ ω ∆nˆ d and i 2c cos θ 2c cos θ are not equal, because ωd = ωi. This results from the acoustic waves that excite and drive optical waves.̸ However, the difference is small, on the order of a millionth. Therefore, we can neglect the difference in Eqs. (6.13) and (6.14) and obtain with the coupling coefficient ω ∆nˆ ω ∆nˆ κ = d i (6.15) 2c cos θ ≃ 2c cos θ and initial conditions Ad(z)=0thesolution

A (z)=A (0) cos κ z (6.16) i i | | A (z)= jA (0) sin κ z. (6.17) d − i | | The incoming wave is depleted along the propagation and transformed into adiffracted wave. The diffracted wave is slightly shifted in frequency. If the interaction length is long enough, so that κ z>π/2, the diffracted wave is | | 6.1. ACOUSTO-OPTIC INTERACTION 101

j(ωit ki r) Ei = eyAi(z)e − · + c.c. (6.9) j(ωdt kd r) Ed = eyAd(z)e − · + c.c. (6.10)

If we substitute (6.7) into (6.8), and use the slowly varying envelope approximation (as in Chapter 3), we obtain

ω2 ω2 k2 d A (z) 2jk A (z) d ∆nAˆ (z)(6.11) − d − c2 d − d · ∇ d ≃−c2 i ! " ω2 ω2 k2 i A (z) 2jk A (z) i ∆nAˆ (z). (6.12) − i − c2 i − i · ∇ i ≃−c2 d ! " ω2 With k2 = d,i and k A (z)=k cos θ dAd , we obtain d,i c2 d · ∇ d d dz dA (z) ω ∆nˆ d j d A (z) (6.13) dz ≃−2c cos θ i dA (z) ω ∆nˆ i j i A (z). (6.14) dz ≃−2c cos θ d This set of equations describes coupled modes. However, the coupling coefficients ω ∆nˆ ω ∆nˆ d and i 2c cos θ 2c cos θ are not equal, because ωd = ωi. This results from the acoustic waves that excite and drive optical waves.̸ However, the difference is small, on the order of a millionth. Therefore, we can neglect the difference in Eqs. (6.13) and (6.14) and obtain with the coupling coefficient ω ∆nˆ ω ∆nˆ κ = d i (6.15) 2c cos θ ≃ 2c cos θ and initial conditions Ad(z)=0thesolution

transducer

optical wave

Applications: Ge crystal Deflection of light Frequency shifting

Figure 6.2: Typical k-diagram describing acousto-optic light diﬀraction at a standing acoustic wave.

14 diffracted into the initial wave. The obvious applications of this process are twofold. First, it can be used to deflect or steer light, and second, the frequency of the light is shifted by the sound frequency, see Fig. 6.2. Eqs. (6.16) and (6.17) also have a different meaning. If the frequency ωs =0,thentheopticalwaveinteractswithaconstantindexgrating.The diffracted wave has exactly the same frequency and the wave vectors must obey momentum conservation. Also this situation is described by the same coupled mode equations. The coupling coefficients are now identical and the total optical energy of both partial waves is conserved.

6.2 The acousto-optic amplitude modulator

In the previous section, we considered the interaction of a traveling acoustic wave with an optical wave. Then, the incoming wave is diffracted and the remainig wave is attenuated when it leaves the medium, but with constant amplitude. The diffracted wave is shifted in frequency and redirected. The intensity of the incoming wave is modulated, if it interacts with a standing acoustic wave, see Fig. 6.2. The acoustic waves in the crystal are typically generated with traveling- wave electrodes via the piezo-electric effect. The wave vectors of the incoming 102 CHAPTER 6. ACOUSTO-OPTIC MODULATORS

transducer

optical wave

Ge crystal

Figure 6.2: Typical k-diagram describing acousto-optic light diffraction at a standing acoustic wave. diffracted into the initial wave. The obvious applications of this process are twofold. First, it can be used to deflect or steer light, and second, the frequency of the light is shifted by the sound frequency, see Fig. 6.2. Eqs. (6.16) and (6.17) also have a different meaning. If the frequency ωs =0,thentheopticalwaveinteractswithaconstantindexgrating.The diffracted wave has exactly the same frequency and the wave vectors must obey momentum conservation. Also this situation is described by the same coupled mode equations. The coupling coefficients are now identical and the total optical energy of both partial waves is conserved.

6.2 The acousto-optic amplitude modulator

and diﬀracted waves are matched by the acoustic wave vector, see Fig. 6.3,

kdkd== ksk+ski+. ki. (6.18) (6.18)

Momentum conservation is impossible for the reversed wave. The index mod- Momentum6.2 conservationTheulation acousto is given is by impossible-optic amplitude for the reversed modulator wave. The index mod- ∆n ∆n(r,t)=∆n sinω t cos (k r)= exp [j (ω t k r)] ulation is given by s s 4j { s − s The intensity is modulated,+exp[ if thej (ω t interaction+ k r)] exp [ j (ω ist kwithr)] a (6.19)standing wave. s s − − s − s exp [ jω t + k r] . − − s s } ∆n ∆n(r,t)=Before, the∆ interactionn sinω wasst onlycos with (k a singlesr)= plane acoustic wave.exp Now, [j ( weωst ksr)] have two waves, and four terms, that change the frequency4j { of the incoming − wave to lower and higher values, while momentum is conserved (absorption of a phonon+exp[ by the forwardj (ω travelingst + waveksr and)] emissionexp of a [ phononj (ω bys thet ksr)] (6.19) backward wave gives the same momentum contribution).− Again,− since the − frequency of theexp sound [wavejωωs is mucht + lowerk thanr] the. frequency of the optical − − s s } electro-acoustic medium Before, the interaction was only with a single plane acoustic wave. Now, we have two waves, and four terms, that change the frequency of the incoming wave to lower and higher values, while momentumdiffracted is conserved (absorption wave of a phonon by the forward traveling wave and emission of a phonon by the backward wave gives the same momentum contribution). Again, since the frequency of the sound wave ωs is much lower than the frequency of the optical

incident electro-acoustic medium wave

16 Figure 6.3: Momentum conservation in the acousto-optical amplitude modulator.

diffracted wave

incident wave

Figure 6.3: Momentum conservation in the acousto-optical amplitude modulator. 104 CHAPTER 6. ACOUSTO-OPTIC MODULATORS

wave, the waves with frequencies ωd = ωi mωs stay in phase for a long time for even rather large m values. The simplest± case for treating these problems comes about for the case where the traveling time of the optical wave through the crystal is much less than the period of the sound wave. With the adiabatic approximation, i.e., the sound wave is assumed to be static, Eqs. (6.16) and (6.17) can be solved toω gives << ωi ωd = ωi + mωs

ω ∆n A (ℓ)=A (0) cos i ℓ (6.20) i i c 2cosθ ! " ω ∆n A (ℓ)= jA (0) sin i ℓ . (6.21) d − i c 2cosθ ! " Afterwards the amplitude of the wave is simply made time-dependent, i.e., ∆n(t)=∆n sinωst. A graphical construction of both amplitudes is attempted in Fig. 6.4. Again with the generating function of the Bessel functions, Eqs. (5.54) and (5.55),

jmωst cos (x sin ωst)= Jm (x) e , (6.22) m even # sin (x sin ω t)= j J (x) ejmωst, (6.23) s − m m#odd we obtain for the Fourier coeﬃcients of the incoming and diﬀracted waves at17 the output of the crystal

ω ∆n A (ℓ)=A (0) J i ℓ ejmωst, (6.24) i i m c 2cosθ m even # ! " ω ∆n A (ℓ)= A (0) J i ℓ ejmωst. (6.25) d − i m c 2cosθ m#odd ! " Two remarks need to be made. First, the incoming wave has only even sidebands and the diﬀractive wave only odd ones. I.e., the incoming wave is modulated with the frequency 2ωs. The incoming wave is extinct, if the modulation depth is adjusted such that the Bessel function of zeroth order shows azeroordisappears,i.e.,for ω ∆n i ℓ =2.405. (6.26) c 2cosθ 104 CHAPTER 6. ACOUSTO-OPTIC MODULATORS wave, the waves with frequencies ω = ω mω stay in phase for a long time d i ± s for even104 rather large m values.CHAPTER The simplest 6. ACOUSTO-OPTIC case for treating these MODULATORS problems comes about for the case where the traveling time of the optical wave through the crystal is much less than the period of the sound wave. With the adiabatic wave, the waves with frequencies ωd = ωi mωs stay in phase for a long time approximation,for even i.e., rather the large soundm values. wave is The assumed simplest± to case be for static, treating Eqs. these (6.16) problems and (6.17) cancomes be solved about for to the give case where the traveling time of the optical wave through the crystal is much less than the period of the sound wave. With the adiabatic approximation, i.e., the sound wave is assumed to be static, Eqs. (6.16) and ω ∆n (6.17) can beA solved(ℓ)= to giveA (0) cos i ℓ (6.20) i i c 2cosθ ! " ωi ∆n ωi ∆n Ad(ℓ)=A (ℓ)=jAi(0)A (0) sin cos ℓℓ . (6.21)(6.20) i − i c c2cos2cosθθ !! "" ω ∆n Afterwards the amplitudeA of(ℓ the)= wavejA is(0) simply sin i made time-dependent,ℓ . (6.21) i.e., d − i c 2cosθ ∆n(t)=∆n sinωst. A graphical construction of! both amplitudes" is attempted in Fig. 6.4.Afterwards the amplitude of the wave is simply made time-dependent, i.e., Again∆ withn(t)= the∆n generatingsinωst. A graphical function construction of the Bessel of bothfunctions, amplitudes Eqs. is (5.54) attempted and (5.55), in Fig. 6.4. Again with the generating function of the Bessel functions, Eqs. (5.54) and

(5.55), jmωst cos (x sin ωst)= Jm (x) e , (6.22) m even jmωst cos (x sin ωst)=# Jm (x) e , (6.22) sin (x sin ω t)= j m evenJ (x) ejmωst, (6.23) s − # m sin (x sin ω t)=m oddj J (x) ejmωst, (6.23) s #− m we obtain for the Fourier coefficients of the incomingm#odd and diffracted waves at the outputwe obtainof the for crystal the Fourier coefficients of the incoming and diffracted waves at the output of the crystal

ωi ∆n jmωst Ai(ℓ)=Ai(0) Jm ω ∆ℓn e , (6.24) A (ℓ)=A (0) Jc 2cosi θ ℓ ejmωst, (6.24) i mieven ! m c 2cos"θ # m even ! " # ωi ∆n jmωst Ad(ℓ)= Ai(0) Jm ωi ∆ℓn e jm.ωst (6.25) Ad(ℓ)=− Ai(0) Jmc 2cosθ ℓ e . (6.25) −m odd ! c 2cos"θ # m#odd ! " Two remarksTwo remarks need to need be made. to be made. First, First, the incoming the incoming wave wave has has only only even even side- side- bands andbands the and diﬀ theractive diﬀractive wave wave only only odd odd ones. ones. I.e., I.e., the the incoming incoming wave wave is is modulatedmodulated with the with frequency the frequency 2ωs. The 2ωs. incomingThe incoming wave wave is extinct, is extinct, if if the the mod- mod- ulation depthulation is depth adjusted is adjusted such that such the that Bessel the Bessel function function of zeroth of zeroth order order shows shows azeroordisappears,i.e.,forazeroordisappears,i.e.,for

ωi ∆n ωi ∆n ℓ =2.405. (6.26) c 2cosℓ =2θ .405. (6.26) c 2cosθ

18 6.2. THE ACOUSTO-OPTIC AMPLITUDE MODULATOR 105

19 Figure 6.4: Time dependence of amplitudes for incoming and diﬀracted waves. Chapter 7: Third-order nonlinear effects third-harmonic generation (THG) or frequency tripling self-phase modulation (SPM) due to the possible different permutations of the input fields:

if no resonances in between the fundamental and third harmonic 7.1 Third-harmonic generation (THG) THG possible in both centrosymmetric and non-centrosymmetric media, also possible in solids and liquids arguably the most interesting case: generation of UV and VUV in gases in low-conversion limit (almost always in THG), similar to SHG:

20 With stronger focusing, the Rayleigh range, over which the beam is focused, becomes smaller than the length of the conversion region effective interaction length ~ Rayleigh range, i.e., phase-matched case: detailed calculation for case of strong focusing for third-order processes:

21 in solids: in general difficult to achieve phase matching for THG solution: SHG + subsequent SFG in gases: by suitable mixing of different gases, the dispersion can be compensated, thus achieving phase matching, THG conversion efficiencies up to 10% achieved

7.2 The nonlinear refractive index SPM effects come along with an additional factor 3 compared to THG, (due to number of possible permutations of input frequencies) if only an electric field in x-direction:

22 linear polarization, definition via electric field: intensity

23 SPM XPM coherence term description via circular polarizations

independent, if the definition based on electric field or intensity

24 7.3 Molecular orientation and refractive index

7.3.1 The Lorenz-Lorentz law

25 spherical cavity inside a polarized isotropic medium

26 Clausius-Mossotti: local field enhanced: 27 7.3.2 Intensity-dependent refractive index

canonical ensemble of molecules (temperature T) exposed to external electric field

28 energy of dipole in electric field

29 7.4 Self-phase modulation (SPM) assume a purely linearly polarized or circularly polarized beam, ® polarization is conserved

ansatz

SPM coefficient

® |envelope|2 in time domain does NOT change

phase modified µ instantaneous intensity

30 instantaneous frequency

31 nonlinear phase shift

strong spectral broadening

32 always two instants during the pulse, which contribute to the same generated frequency

constructive/destructive interference depending on relative phase at these times ® maxima/minima in SPM spectrum zero points in spectrum for number of minima N on one side of the spectrum:

33 SPM for Gaussian pulse:

34 7.5 Self-focusing transverse beam profile becomes instable

intensity-dependent refractive index

for Dn2>0: •phase velocity in center reduced •phase fronts bend due to the induced lens (”Kerr lens”) •self-focusing of the beam

relevance: •Kerr-lens mode-locked laser oscillators •unwanted detrimental effect of ”hot spots”

35 simple physical consideration in 2D:

Snell’s law ® total internal reflection for

36 above this critical power, self-focusing exceeds diffraction.

Note the quadratic scaling with wavelength! in 2D (1 longitudinal, 1 transversal dimension): spatial solitons occur. in 3D (2 transversal dimensions): catastrophic self-focusing occurs, that eventually is balanced by other nonlinear effects, e.g., - saturation of the intensity-dependent refractive index - self-defocusing due to plasma formation by multi-photon ionization (”filamentation”)

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