Phase Matching

power dependent change of the absorption ensures a nonlinear interaction between the electromagnetic field and the medium. In analogy to the third order interaction inside the OFC crystal fiber, the two forward propagating photons atω₁ and ω₂ < ω₁ interact with an additional counter propagating photon at ω₁ (Figure 4.4a). Energy conservation of

(a) (b)

Figure 4.4: (a) Schematic of the four wave mixing process in the MTS setup. Three collinear photons get destroyed in favor of a fourth photon atω3=ω2. (b) Energy level diagram of the four wave mixing process, with dashed virtual levels (adapted from^[91]).

this third order process requires the generation of a fourth photon at frequency ω₃ =ω₂ (Figure 4.4b). This new photon is also counter propagating towards the initial two photons, which is the basis of an error signal for laser stabilization.

4.1 Phase Matching

The electromagnetic field generated by a nonlinear interaction is described by the wave Equation (4.9). If multiple electromagnetic waves at different frequencies are involved, then a variety of frequency combinations is imaginable for the newly generated field.

However, the involved electromagnetic waves at different frequencies have different phase velocities. Additionally, the relative phases of the interacting waves potentially differ based on the point in space. Only if the phase difference between the involved fields adds up to zero, nonlinear conversion takes place. Thus, usually, one frequency is dominating the generated field, which is defined by this phase matching condition. Consider the concrete example of SFG using plane waves. The spatial dependence of each electromagnetic field component is given by

Ej(z, t) = Eje^{i(kjz−ωjt)}+c.c., (4.14) with the wave number k_j = n_jω_j/c and propagation along the z direction. The initial waves are denoted with the indices j = 0,1 and the generated wave with j = 3. The resulting wave frequency is ω₃ =ω₁+ω₂ (Section 4.0.2), which leads with Equation 4.13 to the nonlinear polarization

P₃(z, t) =P⁽²⁾(ω₁+ω₂)[e^i(k1+k2)z+e^−i(k1+k2)z] = 4₀d_effE₁E₂e^{i[(k1+k2)z−ω3t]}+c.c., (4.15)

Chapter 4. Nonlinear Optics

Only the combination of the electric fields E₁(z, t) and E₂(z, t) responsible for SFG is considered. Additionally, the susceptibility χ⁽²⁾ has been replaced, for simplicity, with the effective susceptibility 2deff, which assumes a fixed polarization, a fixed propagation and the Kleinman symmetry condition^[91,95]. Inserting E₃(z, t) and P₃(z, t) into the wave Equation (4.9) and using the slowly varying amplitude approximation |^∂_∂z^2E23| |k₃^∂E_∂z³| leads to the differential equation^[91]

∂E₃

∂z = 2id_effω²₃

k₃c² E₁E₂e^i∆kz, (4.16) where ∆k=k₁+k₂−k₃ is the wave vector mismatch. This equation describes the spacial change of the generated wave amplitudeE₃ as a function of the wave amplitudes E₁ and E2. In general, these amplitudes also have a spatial variation. However, for now, they are assumed to be constant. The intensity after passing a nonlinear medium is calculated by integration of Equation (4.16) from z = 0 to z = L, with L denoting the length of the medium. Squaring the expression of the amplitude yields the intensity^[91]

I₃ ∝sinc²

with the coherent build up length Lcoh = 2/∆k. If the length of the nonlinear medium exceedsL_coh, then the phase mismatch between the driving waves and the generated wave becomes significant. After this distance, the generated wave gets converted back into the initial waves. Thus, maintaining the phase matching condition ∆k = 0 is crucial, with

∆k = n₁ω₁

c + n₂ω₂

c − n₃ω₃

c = 0. (4.18)

However, normal dispersion makes it challenging. In this case, the refractive index increases monotonically for waves at a higher frequency, so n₁ > n₂ > n₃. The phase matching condition of Equation (4.18) can be written as

(n₃−n₂)ω₃ = (n₁−n₂)ω₁. (4.19) This obviously has no solution, because the left-hand side is negative and the right-hand side is positive. In an isotropic medium, phase matching is only possible for anomalous dispersion, like near an absorption feature. One way around this limitation is to make use of the birefringence displayed by some crystals. The refractive index is now not only dependent on the frequency of the wave, but also on the direction of the polarization.

According to Equation (4.19) the refractive index n1 at highest frequency is supposed to be smaller than n₂. The polarization providing this condition is dependent on the orientation of the crystal.

4.1. Phase Matching

4.1.1 Angle Tuning

The following discussion is limited to a negative uniaxial crystal, which is characterized by two equivalent optical axes, labeled ˆaand ˆb, and a third optical axis, ˆc. Light polarized along the ˆa and ˆb axes has a refractive index given by n_o, and light polarized along the ˆ

c axis has a refractive index given by n_e < n_o. The axis ˆc defines the orientation of the crystal relative to the propagation vector of the light k. If the propagation k is along ˆ

c, then no birefringence is displayed, and no phase matching is possible. For a non-zero angle theta between kand ˆc, the polarization of the light determines the refractive index experienced by it. If the polarization is perpendicular to the plane containing k and ˆc, then the beam is denoted as ordinary, with a refractive index n_o. Conversely, a beam with polarization parallel to this plane is extraordinary with the refraction index n_e(θ), dependent of the angle θ. For a negative uniaxial crystal, the refractive index n_e(θ) is smaller thenn_o. In the case of SHG, with two incoming waves at frequencyω, the generated wave at higher frequency 2ω must experience the refractive index ne(θ). This relation is described by^[91]

1

n_e(θ)² = sin²(θ)

¯

n²_e + cos²(θ)

n²_o . (4.20)

The two limits of the refractive index arene(0°) =no andne(90°) = ¯ne, with the principal value ¯n_e. Therefore, the initial waves have ordinary polarizations and the generated wave extraordinary polarization (Figure 4.5). In general, the polarizations of the two waves

Figure 4.5:Schematic of SHG for an uniaxial crytal, with an angleθbetween the propagation vectorkand the optical axis ˆc(adapted from^[91]).

at lower frequencies define the phase matching type. If both low-frequency waves have the same polarization, it is denoted as type I phase matching^[95]. The phase matching condition is in this case

n_e(2ω, θ) =n_o(ω), (4.21)

with Equation (4.20) leaving at most one solution for θ. For an angle between 0° and 90°, the Poynting vector and the propagation vector are not parallel for waves experiencing the extraordinary refractive index. Thus, the Poynting vectors of the ordinary beam and the extraordinary beam are different. The angle θ controls the phase matching condition and the walk-off between the two beams, which is always non-zero between 0° and 90°. Only a finite range of angles supplies sufficient phase matching at an accepted walk-off angle;

Chapter 4. Nonlinear Optics

this is the critical phase matching condition. The exception of no walk-off is an angle of θ = 90°, denoting the non-critical phase matching condition. Thus, it is popular to fix the angle at θ = 90° and tune the temperature instead of the angle for phase matching without the walk-off effect. Practically, this necessitates a strongly temperature dependent birefringence of the crystal, like in lithium niobate.