J. Eur. Opt. Society-Rapid Publ. 21, 3( 2025) 25
Figure
1.( a) Gaussian pump propagating through the chirped QPM PPLN crystal and splitting into signal and idler pulses during SPDC;( b) Schematic diagram of the down conversion for the pump, signal and idler pulses;( c) The real and imaginary parts for the response function of chirped QPM PPLN crystal;( d) The kernel of the biphoton state amplitude( BSA) function can be viewed as the product of pump function and phase matching function of the crystal.
versa. The spatial frequency of the crystal is K 0 + fz, wheref is the chirping parameter, similar to the cases in Refs. [ 6, 28, 30 ]. And K 0 is the initial value, which is the poled period in the entrance face of the crystal, matches with the incident pulse frequency 2p
K c to make phase compensation [ 28 ]. Following the procedures [ 26, 29, 31 ], the biphoton state reads
ZZ jbii
¼ T 0 dxdwTðx; wÞ^a y ðxÞ^a y ðwÞj 0i: ð1Þ
Here, ^a y is the creation operator of frequency x, andT 0 the normalization coefficient. Similar to the assumption in Ref. [ 28 ], the pump light can be treated as quasi-monochromatic, the frequency of the pump light can be approximately equal to the sum of the frequencies of the signal and the idler lights. As the Gaussian can be viewed as quasi-monochromatic laser if the bandwidth r p is rather narrower compared with the central frequency x p of pump light. And the relations between the frequencies for these three pulses associated with r p is shown in Figure 1( b). With the conditions of the collinear propagation of the three pulses without consideration for Poynting vector walk-off, we can set this case with noncritical type II QPM configuration [ 17 ] and the creation operator ^a y, which operated by the interaction between the pulses with the chirped QPM PPLN, is slowly varying with the distance, and the phase matching function U( x, w) can be expressed as follows [ 28 ]:
Z L
Uðx; wÞ ¼ exp ifz2
0 2 þ ikz dz
where Dk = k( x + w) � k( x) � k( w) � K 0 is the phase mismatching involves the phases of the three mixed waves and could be matched with the poled period of the crystal. k( x) is the wave-numbers of lights at frequency x, which is determined by the refractive indices from the Sellmeier equations by k( x)= nx / c.
The biphoton state amplitude( BSA) function T( x, w) = a( x, w) U( x, w) has two parts: the spectral envelope qffiffiffiffiffi
1 ð ax ð; wÞ ¼ exp � xþw�x pÞ 2
, and the phase-matching pr 2
2r 2 kernel [ 6 ]
exp ð 2if pffiffiffiffiffi 2f k x; wÞ2
Uðx; wÞ ¼ ð2Þ
fL þ k x; w erfi pffiffiffiffiffiffiffiffiffiffi
ð Þ � erfi k p ð ffiffiffiffiffiffiffiffiffiffi x; wÞ
�2if �2if ð3Þ
When the chirping parameter f is negligibly small, the nonlinear optical processes take place efficiently as some specific frequencies satisfying the conditions x + w = x p as well as Dk( x, w) = 0, then the phase-matching kernel becomes sinc functions, which is well-known in the Ref. [ 32 ]. However, in general cases, the phase matching kernel can be dramatically different due to the chirping parameter: