Second-Order Nonlinear Interactions

Since the susceptibilitiesχ(n) are intrinsic properties of a material, their tensorial forms reflect the structural symmetry of the material. In all materials with inversion sym- metry, such as liquids, gases, amorphous solids, and many crystals, i.e. the so-called centrosymmetric materials, the χ(2n) _{tensor components must vanish. This vanishing}

χ(2n) tensor results from the odd inversion symmetry of the electric field vector E and the electric polarisation vectorP. However, there exists an important class of materials known as ferroelectrics, of which Lithium Niobate is a member. These materials possess a spontaneous electric dipole moment Ps in zero external field,1 which originates from

the shift of an ion from a symmetrical site.3 This symmetry-breaking results in nonzero

χ(2) tensor components,4 which are responsible for the nonlinear interactions we are interested in.

Several second-order nonlinear interactions which result from χ(2) are qualitatively described in this section. For linearly polarised light, which we restrict ourselves to, the scalar form of the nonlinear polarisation [Eq. (3.3)] for second-order nonlinear interac- tions is given by:

P(r, t) =ε0χ(2)E2(r, t). (3.5)

Let us consider an electric field that consists of two distinct frequency components, ω1

and ω2, which is represented in scalar form by:

E(r, t) =E1exp [i(k1·r−ω1t)] +E2exp [i(k2·r−ω2t)] + c.c., (3.6)

where _|kj| = ω_cjε(ωj) = ω_cjn2(ωj). With the above electric field, the second-order

nonlinear polarisation [Eq. (3.5)] can be rewritten as follows:

P(r, t) =ε0χ(2) n E₁2exp [2i(k1·r−ω1t)] +E22exp [2i(k2·r−ω2t)] + 2E1E2exp [i(k1+k2)·r−i(ω1+ω2)t] + 2E1E2∗exp [i(k1−k2)·r−i(ω1−ω2)t] + c.c. o + 2ε0χ(2) h |E1|2+|E2|2 i . (3.7) One readily identifies the different frequency components of the nonlinear polarisation

ω

ω

ω

=ω

+ω

χ(2)

(a) Sum-frequency generation

ω

ω

ω

=ω

-ω

χ(2)

ω

Figure 3.1: Illustration of geometries and energy-level descriptions for two different second-order nonlinear interactions.

as being due to the following interactions:

E₁2exp (−i2ω1t), E22exp (−i2ω2t) second-harmonic generation, (3.8a)

E1E2exp [−i(ω1+ω2)t] sum-frequency generation, (3.8b)

E1E2∗exp [−i(ω1−ω2)t] difference-frequency generation, (3.8c)

|E1|2+|E2|2 optical rectification. (3.8d)

We are only interested in the first three interactions, collectively known as parametric interactions. These interactions produce electromagnetic radiations at a new frequency, whilst the last one, optical rectification, does not. Instead, it creates a static electric field inside the nonlinear material.4 Although the three parametric interactions occur simultaneously (see Section 3.5), generally only one is preferred by the phase-matching condition in the nonlinear material to efficiently produce an output signal. Hence, we can consider them separately.

Let us first consider sum-frequency generation (SFG), whose geometry is illustrated in Fig. 3.1(a). The input electromagnetic field has two distinct frequencies ω1 and ω2

which interact with each other through the nonlinear material to produce an output wave at a frequency ω3 = ω1 +ω2. This interaction can be visualised in terms of

photon virtual energy levels, as depicted in Fig. 3.1(a). In this process, photons with energy~_{ω1 and}~_{ω2 are destroyed in the material to generate another one. The energy} 3.2 Second-Order Nonlinear Interactions

conservation dictates that the produced photon has an energy ~_{ω3 =} ~_ω1+~_{ω2. In} addition, the total momentum in the interaction must be conserved, i.e. k3 =k1+k2.

This requirement for conservation of momentum is also known as the phase-matching condition. For a collinear interaction, the conservation of momentum translates into

ω3n(ω3) = ω1n(ω1) +ω2n(ω2). However, this condition is generally prevented from

happening due to chromatic dispersion of the device, resulting in a phase-mismatch ∆k=k3−k1−k2. The phase-mismatch causes an alternation of power flow between

the three waves, reducing the conversion efficiency. Ways to achieve phase-matching condition is given in Section 3.5. When the input waves are at a degenerate frequen- cies ω1 = ω2, this interaction simply reduces to SHG. Second-harmonic generation is

analysed in more detail in the next sections. Both interactions are commonly used to generate electromagnetic waves at higher frequencies that are inaccessible by standard quantum transitions of atoms and/or molecules, such as frequencies in the ultraviolet.

The geometry of difference-frequency generation (DFG) is depicted in Fig. 3.1(b). The two input electromagnetic waves have distinct frequenciesω1andω2 which interact

to produce an output wave at a frequencyω3 =ω1−ω2. Both energy and momentum

must be conserved in this interaction, i.e. ~_{ω3 =}~_ω1−~_{ω2 and k3 =k1−k2. Super-} ficially, DFG looks similar to SFG. However, upon close inspection of the energy level diagrams in Fig. 3.1(b), not both input photons are destroyed. Only the photon at a higher frequency (ω1) is destroyed, whilst a second photon at the lower frequency (ω2) is

created in the interaction. Therefore, the input wave at a lower frequency is amplified. For this reason, this process is also known as optical parametric amplification (OPA).5 In the energy level diagram for DFG depicted in Fig. 3.1(b), the emission of a photon with energy ~_{ω3 after the excitation by a}~_{ω1 photon is stimulated by the presence of} a ~_{ω2 photon. However, spontaneous two photon emission (}~_{ω2 and} ~_{ω3) that follows} from the destruction of the ~_{ω1 photon can occur without the presence of the} ~_ω2 photon. This process is known as optical parametric fluorescence.6 If the nonlinear material is put inside a resonator such that the device is being used multiple times, the electromagnetic waves at ω2 and/or ω3 can build up to an extremely high value. This

device is known as an optical parametric oscillator (OPO).7