Survey on Nonlinear Optical Effects

3.2.2.1 Second Order Processes Second Harmonic Generation

Assuming an electric field ˜E(t) =Eexp (−iωt) +c.c. incident on a non-centrosymmetric, loss- and dispersionless dielectric medium, Eq. (3.2) in scalar representation yields

P˜(t) =ε0χ⁽¹⁾(Eexp (−iωt) +c.c.) + 2ε0χ⁽²⁾EE^∗ +

ε0χ⁽²⁾E²exp (−i2ωt) +c.c. (3.7) with the asterisk denoting a complex conjugation. The first term on the right hand side of the equation is the previously discussed linear con-tribution to the polarization density re-emitting electromagnetic radia-tion at the same frequency. The second term is a non-oscillating field referred to as optical rectification,⁷ a steady polarization resulting from the anharmonic potential as shown in Fig. 3.1a. The third term is de-noted as second harmonic generation (SHG) as the polarization density oscillates at 2ω and is, therefore, generating electromagnetic radiation at twice the frequency of the incident field. Within the scope of this work, SHG is mainly used for the generation of photons with higher energies (E_ph =ω~+ω~ = 2ω~) and, thus, shorter wavelengths (λ=hc₀/E_ph), as illustrated by the energy-level diagram on the right side of Fig. 3.2.

7Caused by the noncentrosymmetric potential energy the electron moved by the incident electric field is, on average, not at its equilibrium position. This results in a non-zero average polarization denoted asoptical rectification.

Figure 3.2: Illustration of second harmonic generation (SHG). Left:

incident electric field with frequencyωis partially converted into an electric field at twice the intial frequency 2ω. Right: energy-level diagram of SHG illustrating the generation of a photon with energy Eph=ω~+ω~= 2ω~(adapted from [28]).

Figure 3.3: Illustration of sum-frequency generation (SFG). Left: in-cident electric fields with frequencies ω1 and ω2 generate a field at frequency ω3 =ω1+ω2. Right: energy-level diagram of SFG illus-trating the energy balanceEph(ω3) =Eph(ω1) +Eph(ω2) (adapted from [28]).

Figure 3.4: Illustration of difference-frequency generation (DFG).

Left: incident electric fields with frequenciesω1 and ω2 generate a field at frequencyω3=ω1−ω2. Right: energy-level diagram of DFG illustrating the generation of two photons at Eph(ω2) and Eph(ω3) from a single photon ofEph(ω1) yielding an amplification of the inci-dentω2-field (adapted from [28]).

Sum- & Difference-Frequency Generation

Applying a field with two frequencies ˜E(t) = E1exp (−iω1t) + E2

exp (−iω2t)+c.c.to the same medium as above gives (again with Eq. (3.2), but neglecting the linear susceptibility for clarity)

P˜⁽²⁾(t) =ε₀χ⁽²⁾h

E₁²e^−i2ω1t+E₂²e^−i2ω2t+ 2E₁E₂e^{−i(ω1+ω2)t} + 2E₁E^∗₂e^{−i(ω1−ω2)t}+c.c.i

+ 2ε₀χ⁽²⁾(E₁E₁^∗+E₂E₂^∗). (3.8)

Analogous to the previous case, the last term denotes optical rectifi-cation and the first two terms SHG of the ω1 and ω2 frequency, respec-tively. In addition to that, terms oscillating at frequencies (ω1+ω2) and (ω₁−ω₂), denoted as sum- (SFG) and difference-frequency generation (DFG) appear.

In SFG a higher frequency radiationω3is generated by the summation of the two incident frequenciesω1andω2 (see Fig. 3.3). Within the scope of this work, SFG is used to generate the third harmonic (third harmonic generation (THG)) of an initial frequency ω1 by a cascaded process of SHG (to generateω2= 2ω1) and SFG (to getω3=ω1+ω2= 3ω1).

In contrast to SFG, where one photon is created from two photons, in DFG two photons atE_ph(ω₂) andE_ph(ω₃) are produced from a sin-gle photon with Eph(ω1) (see energy-level diagram on the right side of Fig. 3.4). In other words, theω2-field is amplified by DFG⁸ causing the process of DFG also being calledoptical parametric amplification (OPA).

In the course of this work, DFG (or OPA) is used in anoptical parametric oscillator (OPO)that is pumped atλ1= 410 nm and generates a so-called signal (with λ2 > λ1; ω2 < ω1) and idler radiation (with λ3> λ2> λ1; ω₃ < ω₂ < ω₁). The term oscillator denotes that the signal radiation is resonantly oscillating in the cavity of the OPO, thereby stimulating the process of OPA.

8DFG might also take place withoutω2orω3being initially present. However, as this process is notstimulated butspontaneousthe generated fields are much lower.

3.2.2.2 Nonlinear Refractive Index

In centrosymmetric material (e.g. glasses or liquids), represented by the symmetric potential energy function shown in Fig. 3.1b, the χ⁽²⁾ term vanishes for symmetry reasons and the χ⁽³⁾ of Eq. (3.2) term becomes the first nonlinear contribution. Applying an electrical field ˜E(t) = Eexp (−iωt) +c.c.gives

P˜⁽³⁾(t) =ε₀χ⁽³⁾ E³e^−i3ωt+ 3EE^∗Ee^−iωt+c.c.

(3.9) for the third order polarization density. The first term oscillates at fre-quency 3ωand represents THG and the second term oscillates atω, thus, at the same frequency as the linear component of polarization density P˜⁽¹⁾(t) =ε0χ⁽¹⁾Eexp (−iωt) +c.c. (see Eqs. (3.5) and (3.6)). Thus, the entire polarization density oscillating atω is given by

P(ω) =P⁽¹⁾(ω) +P⁽³⁾(ω) =ε₀

χ⁽¹⁾+ 3χ⁽³⁾|E|²

E (3.10)

with ˜P^(j)(t) =P^(j)(ω) exp (−iωt). Recalling thatn²₀ =εr = 1 +χ⁽¹⁾ in the linear case (see Section 3.2.1) yields for the third-order nonlinear case n²= 1 +χ⁽¹⁾+ 3χ⁽³⁾|E|² or

n²(I) =n²₀+ 3χ⁽³⁾ I

2n₀ε₀c₀ (3.11)

with intensityI= 2n0ε0c0|E|². Introducing the nonlinear refractive index n₂ and approximating n²(I) = (n₀+n₂I)² ≈n₀+ 2n₀n₂I gives for the nonlinear refractive index

n2= 3 4n²₀ε0c0

χ⁽³⁾. (3.12)

This nonlinear optical effect, being referred to as Kerr effect, results in an instantaneous change of the refractive index that, in turn, affects the propagation of the radiation itself by e.g. self-phase modulation.⁹

9Also, in the non-degenerate case, one frequency radiation can affect the prop-agation of another frequency radiation that is, for example, too weak to introduce

As this effect is of tremendous importance for the generation of optical supercontinua and the formation of temporal solitons more details are given in Section 3.2.5.

3.2.2.3 Multi-Photon Absorption

The previously discussed nonlinear processes are so-called parametric pro-cesses that incorporate virtual energy levels of the material. In contrast to these, multi-photon absorption is a nonparametric process, thus, it in-duces changes to the material properties by e.g. transferring population from one real level to another.

In multi-photon absorption processes this is accomplished by simulta-neous interaction of two or more photons with a single electron. By simul-taneous transfer of their energies to the electron a transition to a higher energy state can be reached. Naturally, the likeliness for such transitions increases with the photon rate and, thus, the transition cross section for two-photon absorption depends on the incident intensity I according to σ=σβI. Consequently,¹⁰ the transition rate for two-photon absorption reads

T_trans= σβI²

hν (3.13)

and has a square-dependence on the intensityI.