- Quasi-phase-matching
Quasi-phase-matching is a technique in
nonlinear optics which allows a positive net flow of energy from the pump frequency to the signal and idler frequencies by creating a periodic structure in the nonlinear medium. Momentum is conserved, as is necessary for phase-matching, through an additional momentum contribution corresponding to thewavevector of the periodic structure. Consequently, in principle any two-wave mixing process that satisfies energy conservation can be phase-matched. For example, all the optical frequencies involved can be collinear, can have the same polarization, and travel through the medium in arbitrary directions. This allows one to use the largestnonlinear coefficient of the material in the nonlinear interaction.Quasi-phase-matching ensures that there is positive energy flow from the pump frequency to signal and idler frequencies even though all the frequencies involved are not phase locked with each other. Energy will always flow from pump to signal as long the phase between the two optical waves is less than 180 degrees. Beyond 180 degrees, energy flows back from the signal to the pump frequencies. The
coherence length is the length of the medium in which the phase of pump and the sum of idler and signal frequencies are 180 degrees from each other. At each coherence length the crystal axes are flipped which allows the energy to continue to positively flow from the pump to the signal and idler frequencies.The most commonly used technique for creating quasi-phase-matched crystals is
periodic poling .Paschotta, Rüdiger. " [http://www.rp-photonics.com/quasi_phase_matching.html Quasi-phase matching] ." Encyclopedia of Laser Physics and Technology. Retrieved April 30, 2006]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 degrees thus insuring that there is a positive energy flow to the signal and idler beam. In the case of
Difference frequency generation polarization equation can be expressed by.P_3=4d A_1 A_2 e^{i(k_1+k_2)z}
Where d is the nonlinear susceptibility coefficient in which the sign of the coefficient is flipped when the crystal axis is flipped. Where i represents the
imaginary unit .P_3=-4d A_1 A_2 e^{i(k_1+k_2)z}=4d A_1 A_2 e^{i((k_1+k_2)z+pi)}
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 exists in the crystal. In general the rate for which the energy from the pump wavelength to the signal wavelength is.
frac{partial A_2}{partial z}=A_1 chi e^{i Delta k z}
Where A_2 is the generated frequency amplitude and A_1 is the pump frequency amplitude and Delta k is the phase mismatch between the two optical waves. The chi refers to the nonlinear susceptibility of the crystal.
In the case of a periodically poled crystal the optic axis is flipped by 180 degrees which changes the sign of chi. For the n^{th} domain chi can be express by this function.
chi=chi_0 (-1)^n
Where n is the index of the poled domain. The signal amplitude A_2 can be express as a sum.
A_2=A_1 chi_0 sum^{N-1}_{n=0} (-1)^n int^{Lambda (n+1)}_{Lambda n} e^{i Delta k z} partial z
Which integrates to.
A_2=-frac{i A_1 chi_0}{Delta k} sum^{N-1}_{n=0} (-1)^n (e^{i Delta k Lambda (n+1)}-e^{i Delta k Lambda n})
Which reduce to.
A_2=-i A_1 chi_0 frac{e^{i Delta k Lambda}-1}{Delta k} sum^{N-1}_{n=0} (-1)^n e^{i Delta k Lambda n}
The summation s can be expressed as a series.
s=sum^{N-1}_{n=0} (-1)^n e^{i Delta k Lambda n}=1-e^{i Delta k Lambda}+e^{i 2 Delta k Lambda n}-e^{i 3 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 e^{i Delta k Lambda}.
s e^{i Delta k Lambda}=e^{i Delta k Lambda} -e^{i 2 Delta k Lambda n}+e^{i 3 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 this relation.
s(1+e^{i Delta k Lambda})=1-(-1)^N e^{i Delta k Lambda N}
Solving for s.
s=frac{1-(-1)^N e^{i Delta k Lambda N} }{1+e^{i Delta k Lambda
Which leads to A_2 to equal.
A_2=-i A_1 chi_0 left( frac{e^{i Delta k Lambda}-1}{Delta k} ight)left(frac{1-(-1)^N e^{i Delta k Lambda N{e^{i Delta k Lambda}+1} ight)
The total intensity can be expressed by following equation.
I_2=A_2 A_2^*=A_1^2 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)} ight)
For the case of Lambda=frac{pi}{Delta k} the right part of the above equation is undefined so the limit needs to be taken when Delta k Lambda ightarrow pi by invoking
L'Hôpital's rule .lim_{Delta k Lambda opi}frac{1-(-1)^N cos(Delta k Lambda N)}{1+cos(Delta k Lambda)}=N^2
Which leads to intensity relation.
I_2=frac{4 A_1^2 chi_0^2 L^2}{pi^2}
In order to allow different domain widths ie. Lambda=frac{i pi}{Delta k}, for i=1,3,5,..., the above equation becomes.
I_2=A_2 A_2^*=A_1^2 chi_0^2 Lambda^2 mbox{sinc}^2(i Delta k Lambda/2) left(frac{1-(-1)^N cos(i Delta k Lambda N)}{1+cos(i Delta k Lambda)} ight)
With Lambda -> frac{pi}{Delta k} the intensity trivially becomes.
I_2=frac{4 A_1^2 chi_0^2 L^2}{i^2 pi^2}
From this equation it is apparent that as quasi-phase match order increases the efficiency decreases by that amount. 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. This allows quasi-phase-matching to exist at different domain widths Lambda.
Calculation of domain width
The domain width is calculated through the use of
Sellmeier equation and usingwavevector relations. In the case of DFG this relationship holds true Delta k=k_3-k_1-k_2 where k_1, k_2, mbox{and } k_3 are the pump, signal, and idler wavevectors and k_i=frac{2 pi n(lambda_i)}{lambda_i}. By calculating Delta k for the different frequencies the domain width can be calculated from the relationship Lambda=frac{pi}{Delta k}.References
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