Quasi phase matching

Mathematical description

In nonlinear optics, the generation of other frequencies is the result of the nonlinear polarization response of the crystal due to fundamental pump frequency. When the crystal axis is flipped, the polarization wave is shifted by 180°, thus ensuring that there is a positive energy flow to the signal and idler beam. In the case of sum-frequency generation, polarization equation can be expressed by

where d is the nonlinear susceptibility coefficient, in which the sign of the coefficient is flipped when the crystal axis is flipped, and i represents the imaginary unit.

Development of signal amplitude

The following mathematical description assumes a constant pump amplitude. The signal wavelength can be expressed as a sum over the number of domains that exist in the crystal. In general the rate of change of the signal amplitude is

∂ A 2 ∂ z = A 1 2 χ e i Δ k z ,

where A 2 is the generated frequency amplitude and A 1 is the pump frequency amplitude and Δ k is the phase mismatch between the two optical waves. The χ refers to the nonlinear susceptibility of the crystal.

In the case of a periodically poled crystal the crystal axis is flipped by 180 degrees in every other domain, which changes the sign of χ . For the n t h domain χ can be expressed as

χ = χ 0 ( − 1 ) n

where n is the index of the poled domain. The total signal amplitude A 2 can be expressed as a sum

A 2 = A 1 2 χ 0 ∑ n = 0 N − 1 ( − 1 ) n ∫ Λ n Λ ( n + 1 ) e i Δ k z ∂ z

where Λ is the spacing between poles in the crystal. The above equation integrates to

A 2 = − i A 1 2 χ 0 Δ k ∑ n = 0 N − 1 ( − 1 ) n ( e i Δ k Λ ( n + 1 ) − e i Δ k Λ n )

and reduces to

A 2 = − i A 1 2 χ 0 e i Δ k Λ − 1 Δ k ∑ n = 0 N − 1 ( − 1 ) n e i Δ k Λ n

The summation yields

s = ∑ n = 0 N − 1 ( − 1 ) n e i Δ k Λ n = 1 − e i Δ k Λ + e i 2 Δ k Λ − e i 3 Δ k Λ + . . . + ( − 1 ) N e i Δ k Λ ( N − 2 ) − ( − 1 ) N e i Δ k Λ ( N − 1 ) .

Multiply above equation both sides by a factor of e i Δ k Λ

s e i Δ k Λ = e i Δ k Λ − e i 2 Δ k Λ + e i 3 Δ k Λ + . . . + ( − 1 ) N e i Δ k Λ ( N − 1 ) − ( − 1 ) N e i Δ k Λ N .

Adding both equation leads to the relation

s ( 1 + e i Δ k Λ ) = 1 − ( − 1 ) N e i Δ k Λ N .

Solving for s gives

s = 1 − ( − 1 ) N e i Δ k Λ N 1 + e i Δ k Λ ,

which leads to

A 2 = − i A 1 2 χ 0 ( e i Δ k Λ − 1 Δ k ) ( 1 − ( − 1 ) N e i Δ k Λ N e i Δ k Λ + 1 ) .

The total intensity can be expressed by

I 2 = A 2 A 2 ∗ = | A 1 | 4 χ 0 2 Λ 2 sinc 2 ( Δ k Λ / 2 ) ( 1 − ( − 1 ) N cos ⁡ ( Δ k Λ N ) 1 + cos ⁡ ( Δ k Λ ) ) .

For the case of Λ = π Δ k the right part of the above equation is undefined so the limit needs to be taken when Δ k Λ → π by invoking L'Hôpital's rule.

lim Δ k Λ → π 1 − ( − 1 ) N cos ⁡ ( Δ k Λ N ) 1 + cos ⁡ ( Δ k Λ ) = N 2

Which leads to the signal intensity

I 2 = 4 | A 1 | 4 χ 0 2 L 2 π 2 .

In order to allow different domain widths, i.e. Λ = m π Δ k , for m = 1 , 3 , 5 , . . . , the above equation becomes

I 2 = A 2 A 2 ∗ = | A 1 | 4 χ 0 2 Λ 2 sinc 2 ( m Δ k Λ / 2 ) ( 1 − ( − 1 ) N cos ⁡ ( m Δ k Λ N ) 1 + cos ⁡ ( m Δ k Λ ) ) .

With Λ = m π Δ k the intensity becomes

I 2 = 4 | A 1 | 4 χ 0 2 L 2 m 2 π 2 .

This allows quasi-phase-matching to exist at different domain widths Λ . From this equation it is apparent, however, that as the quasi-phase match order m increases, the efficiency decreases by m 2 . For example, for 3rd order quasi-phase matching only a third of the crystal is effectively used for the generation of signal frequency, as a consequence the amplitude of the signal wavelength only third of the amount of amplitude for same length crystal for 1st order quasi-phase match.

Calculation of domain width

The domain width is calculated through the use of Sellmeier equation and using wavevector relations. In the case of DFG this relationship holds true Δ k = k 1 − k 2 − k 3 , where k 1 , k 2 , and  k 3 are the pump, signal, and idler wavevectors, and k i = 2 π n ( λ i ) λ i . By calculating Δ k for the different frequencies, the domain width can be calculated from the relationship Λ = π Δ k .