Phase matching

Matching the relative phases between the three fields is key to obtaining nonlinear conversion as shown in equation4.28. If there is no phase mismatch between the three fields (∆k=0) the

4.3 Nonlinear optics and generating squeezed states

amplitude of the generated field increases linearly with z. Hence the intensity, |A3|2, increases

quadratically along the length of the crystal.

Perfect phase matching is often impossible to achieve due to the normal dispersion in a non- linear medium, where the refractive index (normally) increases with increasing optical frequency. For beams propagating co-linearly through the nonlinear medium, the phase matching condition may be written in terms of the pump, signal and idler refractive indices,

n1ω1+n2ω2=n3ω3. (4.29)

In a crystal with nonzero phase mismatch, the phases of the pump and fundamental fields diverge as they propagate through the crystal. When the relative phases reach a mismatch ofπ/2 the nonlinear conversion reverses and energy is transferred to the opposite nonlinear process. In the case of the sum-frequency generation, light is converted from the desired pump field back into the signal/idler. The field exiting the crystal with a nonzero phase mismatch can be found by integrating equation4.27over the length of the nonlinear medium [22]:

A3= 4πiχ(2)ω₃2 k3c2 A1A2 Z L 0 ei∆kz dz (4.30) =4πiχ (2) ω₃2 k3c2 A1A2 ei∆kz₋₁ i∆k .

The intensity of theithfield is calculated from the time-averaged Poynting vector,

Ii=

nic

2π|Ai|

2_{. (4.31)}

For the generated field at the pump wavelength this is given by

I3= 16π(χ(2))2ω₃4|A1|2|A2|2n3) k2 3c3 ei∆kL₋₁ ∆k 2 . (4.32)

The modulus-squared term in this equation can be re-written as a sinc-squared function, and wavenumbers and angular frequencies rewritten in terms of wavelength, giving

I3= 256π5(χ(2))2I1I2 n1n2n3λ32c L2sinc2 ∆kL 2 . (4.33)

This is the full equation for the generated field intensity, and will be used in later chapters to calculate the nonlinearity of a crystal from measured powers.

There are several methods of achieving zero phase mis-match, or close to zero phase mismatch, in a nonlinear crystal. Two types of phase matching, and quasi-phase matching are summarised below.

Type I and type II (birefringent) phase matching

Type I and type II phase matching make use of the birefringence in an anisotropic crystal, that is a crystal with different refractive indices along the different crystal lattice directions. For a medium with normal dispersionω3is polarised in a direction with lower refractive index thanω1 andω2.

Type I phase matching implies thatω1andω2 are polarised along the same crystal axis, while in

type II phase matching they are polarised in different directions.

There are two methods to control the three refractive indices – angle tuning and temperature tuning. Tuning the angle of the crystal changes the refractive index seen by the input beam to

an angle-dependent combination of the refractive indices of the crystal axes. In this thesis we focus on temperature tuning, which exploits the temperature dependence of refractive index of the crystal.

Phase matching can be achieved by adjusting the temperature of the crystal such that the crystal refractive index for the pump and fundamental fields causes a zero relative phase shift. Experimentally accessible temperatures are limited, resulting in some crystal dimensions being inaccessible using this technique.

Quasi phase matching

Instead of modifying the crystal refractive indices, quasi phase matching (QPM) [7] flips the sign of the nonlinearity before the phase mismatch between the interacting fields becomesπ/2 and the desired nonlinear process reverses. This technique allows access to crystal axes with high non- linearity rendered unreachable using birefringent phase matching due to the extreme temperature requirements, since QPM does not tune the crystal index.

QPM is achieved using the periodic poling technique, where crystals are processed such that the direction of the effective nonlinearity is periodically flipped over the length of the crystal. The period of the periodic poling is set by rate of divergence of the phases of the interacting fields, which can be represented by the coherence length of the fields, given by [117]

Lc=

2π

|∆k|. (4.34)

mthorder periodic poling flips the crystal domains everymcoherence lengths. For a degenerate process (ω1=ω2), the length of poling regions is approximated by

Γ0=

mλ1

2×∆n, (4.35)

where∆n=|n1,2−n3|is the refractive index difference between the pump and fundamental.

In the case of quasi-phase matched periodically poled materials, the effective nonlinearity (de f f =χ(2)/2) is affected by deviation from perfect phase matching [84]. Formth order phase

matching, the modified effective nonlinearity is given by

de f f =

2

mπdi j, (4.36)

wheredi j is the unpoled nonlinearity along thei, jthcrystal axes.

The poling length is optimised for a particular difference in refractive index between pump and fundamental, which depends on temperature. The sinc-squared phase matching curve has been recorded for all crystals used in this thesis to establish the optimal operating temperature. To measure the phase matching curve the crystal is operated in a single-pass SHG experiment, and the conversion efficiency measured as a function of crystal temperature.