Quasi-phase-matching

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

${\displaystyle P_{3}=4dA_{1}A_{2}e^{i(k_{1}+k_{2})z},}$

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

${\displaystyle P_{3}=-4dA_{1}A_{2}e^{i(k_{1}+k_{2})z}=4dA_{1}A_{2}e^{i((k_{1}+k_{2})z+\pi )}.}$

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

${\displaystyle {\frac {\partial A_{2}}{\partial z}}=A_{1}^{2}\chi e^{i\Delta kz},}$

where ${\displaystyle A_{2}}$ is the generated frequency amplitude and ${\displaystyle A_{1}}$ is the pump frequency amplitude and ${\displaystyle \Delta k}$ is the phase mismatch between the two optical waves. The ${\displaystyle \chi }$ 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 ${\displaystyle \chi }$. For the ${\displaystyle n^{th}}$ domain ${\displaystyle \chi }$ can be expressed as

${\displaystyle \chi =\chi _{0}(-1)^{n}}$

where ${\displaystyle n}$ is the index of the poled domain. The total signal amplitude ${\displaystyle A_{2}}$ can be expressed as a sum

${\displaystyle A_{2}=A_{1}^{2}\chi _{0}\sum _{n=0}^{N-1}(-1)^{n}\int _{\Lambda n}^{\Lambda (n+1)}e^{i\Delta kz}\partial z}$

where ${\displaystyle \Lambda }$ is the spacing between poles in the crystal. The above equation integrates to

${\displaystyle A_{2}=-{\frac {iA_{1}^{2}\chi _{0}}{\Delta k}}\sum _{n=0}^{N-1}(-1)^{n}(e^{i\Delta k\Lambda (n+1)}-e^{i\Delta k\Lambda n})}$

and reduces to

${\displaystyle A_{2}=-iA_{1}^{2}\chi _{0}{\frac {e^{i\Delta k\Lambda }-1}{\Delta k}}\sum _{n=0}^{N-1}(-1)^{n}e^{i\Delta k\Lambda n}}$

The summation yields

${\displaystyle s=\sum _{n=0}^{N-1}(-1)^{n}e^{i\Delta k\Lambda n}=1-e^{i\Delta k\Lambda }+e^{i2\Delta k\Lambda }-e^{i3\Delta k\Lambda }+...+(-1)^{N}e^{i\Delta k\Lambda (N-2)}-(-1)^{N}e^{i\Delta k\Lambda (N-1)}.}$

Multiply above equation both sides by a factor of ${\displaystyle e^{i\Delta k\Lambda }}$

${\displaystyle se^{i\Delta k\Lambda }=e^{i\Delta k\Lambda }-e^{i2\Delta k\Lambda }+e^{i3\Delta k\Lambda }+...+(-1)^{N}e^{i\Delta k\Lambda (N-1)}-(-1)^{N}e^{i\Delta k\Lambda N}.}$

Adding both equation leads to the relation

${\displaystyle s(1+e^{i\Delta k\Lambda })=1-(-1)^{N}e^{i\Delta k\Lambda N}.}$

Solving for ${\displaystyle s}$ gives

${\displaystyle s={\frac {1-(-1)^{N}e^{i\Delta k\Lambda N}}{1+e^{i\Delta k\Lambda }}},}$

which leads to

${\displaystyle A_{2}=-iA_{1}^{2}\chi _{0}\left({\frac {e^{i\Delta k\Lambda }-1}{\Delta k}}\right)\left({\frac {1-(-1)^{N}e^{i\Delta k\Lambda N}}{e^{i\Delta k\Lambda }+1}}\right).}$

The total intensity can be expressed by

${\displaystyle I_{2}=A_{2}A_{2}^{*}=\left|A_{1}\right|^{4}\chi _{0}^{2}\Lambda ^{2}{\mbox{sinc}}^{2}(\Delta k\Lambda /2)\left({\frac {1-(-1)^{N}\cos(\Delta k\Lambda N)}{1+\cos(\Delta k\Lambda )}}\right).}$

For the case of ${\displaystyle \Lambda ={\frac {\pi }{\Delta k}}}$ the right part of the above equation is undefined so the limit needs to be taken when ${\displaystyle \Delta k\Lambda \rightarrow \pi }$ by invoking L'Hôpital's rule.

${\displaystyle \lim _{\Delta k\Lambda \to \pi }{\frac {1-(-1)^{N}\cos(\Delta k\Lambda N)}{1+\cos(\Delta k\Lambda )}}=N^{2}}$

Which leads to the signal intensity

${\displaystyle I_{2}={\frac {4\left|A_{1}\right|^{4}\chi _{0}^{2}L^{2}}{\pi ^{2}}}.}$

In order to allow different domain widths, i.e. ${\displaystyle \Lambda ={\frac {m\pi }{\Delta k}}}$, for ${\displaystyle m=1,3,5,...}$, the above equation becomes

${\displaystyle I_{2}=A_{2}A_{2}^{*}=\left|A_{1}\right|^{4}\chi _{0}^{2}\Lambda ^{2}{\mbox{sinc}}^{2}(m\Delta k\Lambda /2)\left({\frac {1-(-1)^{N}\cos(m\Delta k\Lambda N)}{1+\cos(m\Delta k\Lambda )}}\right).}$

With ${\displaystyle \Lambda ={\frac {m\pi }{\Delta k}}}$ the intensity becomes

${\displaystyle I_{2}={\frac {4\left|A_{1}\right|^{4}\chi _{0}^{2}L^{2}}{m^{2}\pi ^{2}}}.}$

This allows quasi-phase-matching to exist at different domain widths ${\displaystyle \Lambda }$. From this equation it is apparent, however, that as the quasi-phase match order ${\displaystyle m}$ increases, the efficiency decreases by ${\displaystyle m^{2}}$. 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.

The domain width is calculated through the use of Sellmeier equation and using wavevector relations. In the case of DFG this relationship holds true ${\displaystyle \Delta k=k_{1}-k_{2}-k_{3}}$, where ${\displaystyle k_{1},k_{2},{\mbox{and }}k_{3}}$ are the pump, signal, and idler wavevectors, and ${\displaystyle k_{i}={\frac {2\pi n(\lambda _{i})}{\lambda _{i}}}}$. By calculating ${\displaystyle \Delta k}$ for the different frequencies, the domain width can be calculated from the relationship ${\displaystyle \Lambda ={\frac {\pi }{\Delta k}}}$.

This method enables the generation of high-purity hyperentangled two-photon state. In orthogonal quasi-phase matching (OQPM),[10] a thin-layered crystal structure is combined with periodic poling along orthogonal directions. By combining periodic down- conversion of orthogonally polarized photons along with periodic poling that corrects the phase mismatch, the structure self corrects for longitudinal walkoff (delay) as it happens and before it accumulates. The superimposed spontaneous parametric downconversion (SPDC) radiation of the superlattice creates high-purity two-photon entangled state.