Frequency conversion; Optical parametric oscillators

Another use of lithium niobate is in optical parametric oscillators (OPOs). These devices allow frequency mixing, over a wide range of output wavelengths, and so are very useful for producing an easily tunable laser.

Frequency mixing is a nonlinear process, where two incident beams, with frequencies ω1 and ω2, in a suitable nonlinear material produce a third, with frequencyω3. This occurs due to the inability of dipoles in the material to respond to the electric field of light wave in a completely linear way. The deviation from the

linear response is small, but becomes significant when the intensity of light is high. This is discussed further in Chapter 6, on Surface Second Harmonic Generation. Second harmonic generation is a specific type of frequency mixing, where the two incident beams have the same frequency, and the output wave has twice the frequency (2ω). However, in the general case of frequency mixing, harmonics with frequencies of 2ω1, 2ω2,ω1+ω2andω1−ω2_{may be generated.}

Parametric oscillators use difference frequency generation, to convert a single incoming high frequency photon from a pump beam to two lower frequency photons, termed the signal and the idler. The difference with second harmonic generation is illustrated by an energy level diagram, shown in Figure 2.12.

pump signal idler 2nd harmonic a) b) pump pump

Figure 2.12:Energy level diagrams showing a) Second Harmonic Generation, and b) Difference Frequency Generation.

For successful difference frequency generation (and hence OPO operation), each of the three waves needs to be travelling in the same direction, at the same speed. This is a problem, as each of the frequencies will experience a different refractive index, due to the dispersion (increase of refractive index with frequency) of the material. The wavevector,kis defined as

k= ωn(ω)

c (2.17)

Due to dispersion there will be a wave vector mismatch

∆k=k3−k2−k1,0 (2.18)

If this is left uncorrected, as the beams propagate through the crystal their phase will become increasingly mis-matched, resulting in less efficient frequency conversion, until eventually the beams will start to destructively interfere. The coherence length is given by

lc= _∆π

To prevent destructive interference and maximise the conversion efficiency, the beams need to be ‘phase-matched’, with the ideal phase-matching condition defined as∆k=0.

2.5.3.1 Birefringent phase-matching

Phase-matching is possible using birefringent materials, such as lithium niobate, by making use of the two refractive indices. In this approach, the higher frequency pump wave is polarised along the direction of lowest refractive index, and the other waves are polarised at an angle to the first wave. Because they are travelling at an angle to the two optic axes, they will experience a refractive index that is a combination of the ordinary and extraordinary indices (Boyd, 2003), given by the relation 1 n2_(θ) = sin2θ n2_e + cos2θ n2_o (2.20)

Phase matching can be achieved by varyingθ, the angle between the optic axis and the direction of the signal/idler beam, to satisfy the phase matching condition. Birefringent phase-matching is not an ideal solution, because there are only a limited number of phase-matching angles, and parallel waves of perpendicular polarisations will walk-offfrom each other. One alternative is to use temperature tuning of the refractive index to achieve phase-matching. However, a more versatile technique is quasi-phase-matching, which has several advantages over birefringence phase-matching.

2.5.3.2 Quasi-phase-matching

Quasi-phase-matching is an alternative to birefringence phase matching, which uses the ferroelectric nature of lithium niobate to prevent a phase-mismatch from occurring during frequency conversion. It uses periodically poled lithium niobate (PPLN), where the direction of the ferroelectric domain is modified to alternate with a regular period,Λ, as shown in Figure 2.13.

Periodic domain inversion results in the sign of the nonlinear coefficient also being inverted with the same period, which compensates for the wavevector mismatch. The period of the alternately inverted domains is chosen to be twice that of the coherence length. Normally, when a generated wave travelling through the crystal goes further than a single coherence length, its amplitude will start to decrease.

L

Figure 2.13: Schematic representation of periodically poled lithium niobate (PPLN). The arrows represent the direction of the+zaxis, which alternates with periodΛ.

However, in PPLN, after a single coherence length the domain is inverted, so the wave experiences a nonlinear coefficient of opposite sign, so instead of decreasing, the amplitude of the wave continues to increase. Figure 2.14 shows the effect on field amplitude through a crystal in the case of ideal phase-matching, quasi-phase matching and no phase-matching.

Ideal phase-matching Quasi-phase-matching No phase-matching 0 1 2 3 4 5 6 7 8 9 10 11 12 0 z / lc FieldAmplitude

Figure 2.14: The field amplitude of a wave generated in nonlinear frequency conversion through a crystal, under conditions of ideal phase-matching, quasi- phase-matching (QPM) and no phase-matching.

In quasi-phase-matching, the equation for wavevector mismatch has to be modified with an extra term,∆kg, to take the grating period into consideration, (Myers et al., 1995): ∆k=kp−ks−ki−∆kg (2.21) where ∆kg= 2π Λ (2.22)

hence the optimum period of the PPLN is

Λ = 2π

(kp−ks−ki)

(2.23)

The use of quasi-phase-matching eliminates the dependence on birefringence, so can be used in a material which is not naturally birefringent, and does not require angle tuning which can cause problems with walk-offbetween the beams. It also allows access to the largest nonlinear coefficients, which may not be possible with birefringent phase-matching, due to the geometry of the crystal and the necessary polarisation of the beams.

For more detailed information on quasi-phase-matching and periodic poling, see, for instance, Houe and Townsend (1995).