We present here a possible high-flux photon-pair source constructed by single lithium niobate optical superlattice (OSL) with a combined quasi-periodically and periodically poled structure, which is from the principle of electrically induced parametric down conversion (PDC) after second-harmonic generation (SHG), predicted by the united theory developed in this paper, in which SHG, PDC and electro-optic (EO) effect are comparably treated as two-order nonlinear effects. In the OSL, the e-polarized fundamental frequency photons are first converted to double frequency ones with the same polarization; then the PDC process is triggered by EO effect when the fundamental frequency photons are almost exhausted; finally, the double frequency photons are converted again to a series of two-photon pair of fundamental wave. It is demonstrated that at 100 °C, in a 20.2mm long OSL with a 30V / mm applied electric field, a 100MW/cm2, 1080 nm laser beam can be translated to a flux of high-purity two-photon pairs with a conversion efficiency close to 100%; and for a longer OSL the pump intensity can be further lowered. The device can also act as an ultra-low field electro-optic switch.
© 2007 Optical Society of America
Quantum entanglement plays a key role in quantum information science and has attracted lots of attention. The entanglement light is one of important resources in this realm, which is with potentially useful applications from quantum communications, including cryptography  and dense coding , to quantum teleportation , “entanglement swapping”  and quantum computation . Spontaneous [6–8] and seed injected [9, 10] PDC with type I  or type-II  phase matching are very useful techniques to generate entanglement and were widely used in experiments. It is a pity that spontaneous PDC is usually so weak, especially from the non-collinearly phase matching in which only small amount of photons in the output cone is collectible , being inconvenient for coupling the emitted photons into optical fibers , that it is unsuitable for long distance communications. Besides, to get entanglement beams in traditional methods, a double frequency pump laser beam [13–15] or two separated sets for SHG and PDC [16, 17] were required. However, a simple high-flux source of entangled photons, without two separated sets but with the same frequency of pump and signal beam, would be expected for some practical implementation of applications.
The optical parametric processes in periodically or/and quasi-periodically poled ferroelectric materials may be hopeful for our intention since it is with reasonably high efficiency and flexibility controllable. In fact, some efficient SHG [18–20], sum-frequency generation (SFG) , and optical parametric oscillation (OPO) [22, 23], notably third-harmonic generation [24–26 ] in these materials have been reported. Recently, cascaded frequency conversion and signal tuning in these materials received much interest [27–34]. A quasi-phase matched (QPM) crystal composed of an unpoled lithium niobate (LN) dispersion section sandwiched between two periodically poled LN (PPLN) sections was constructed, in which amplitude modulation and frequency conversion can be obtained simultaneously by using the EO effect to control the relative phase among mixing waves. The two functions were integrated in single OSL  subsequently. Very recently, a way different from those reported by Refs. and , using an applied electric field to control both polarization and magnitude of the second harmonic, was presented . Besides, QPM materials have also been used for efficient generation of photon pairs [13, 16, 35]. In this paper, we present a principle to integrate two functions including SHG and PDC into single LN OSL therefore yield squeezed light that could be turned into entangled light beams by using a 50/50 Y-type single mode fiber optic beam splitter . The basic idea of the principle is as follows: construct such an OSL with an applied electric field, in which the double frequency process is performed at first and then the PDC process is turned on when the fundamental frequency photons are almost converted to double frequency ones; finally, the double frequency photons are converted again to a series of two-photon pair of fundamental wave. The key step is utilizing EO effect to trigger the PDC process.
2. The united theory of SHG, PDC and EO effect
To discuss our principle we first develop a wave coupling theory describing the united effect of EO, SHG and PDC in a PPLN or/and quasi-periodically poled LN (QPPLN) OSL following the idea involved in Refs. [36, 37] instead of that including the use of refractive index ellipsoid theory in Ref. . In the united theory SHG, PDC and (EO) effect are comparably treated as two-order nonlinear effects. Figure 1 shows a schematic diagram of SHG and PDC in the OSL controlled by an applied electric field, where the propagation direction and the polarization of pump wave are along the x -axis and z -axis of the OSL, respectively. And the applied dc electric field E 0 is along the y -axis of Section 1 of the OSL; F is a filter, only allowing e-polarized fundamental wave to pass. Section 1 and 2 represent QPPLN and PPLN, respectively.
When the pump wave is injected into the OSL, multi-effects including EO coupling  and frequency conversions  appear simultaneously if there exits an external electric field. We take EO effect as the second-order nonlinear one as did Ref.[36, 37]. So, corresponding to each monochromatic component of light ωi (i = 1,2), the total second-order polarization should involve a part describing the EO effect such as P (2) EO(ωi) = 2ε 0 χ (2)(ωi,0):E iy(x)E 0exp(ikiy x) + 2ε 0 χ (2)(ωi,0):E iz(x)E0exp(ikiz x)  besides the well-known polarizations P (2) SHG and P (2) PDC relative to SHG and PDC, where E iy(x) and E iz(x) denote two independent amplitude components of the monochromatic light field corresponding to wave vectors k iy and k iz (y,z represent the polarizations relative to o-ray and e-ray), respectively. By treating the total second-order polarization as a perturbation, following the way presented by Ref. [36, 37, 40] and noting r 32 = 0 and d 23, d 34 = 0 for LN, the coupling equations for present case (EO effect: ω1z ↔ ω1y, ω2z ↔ ω2y; SHG and/or PDC:ω1y + ω1y ↔ ω2y, ω1y + ω1z ↔ ω2y, ω1z + ω1z ↔ ω2y, ω1y + ω1y ↔ ω2z, ω1y + ω1z ↔ ω2z, ω1z + ω1z ↔ ω2z) can be deduced from Maxwell’s equations as follows:
where Ejμ , ωjμ , kjμ and njμ (j = 1,2 , referring to the fundamental wave and the second harmonic, respectively; μ = y,z) are the electric fields, the angular frequencies, the wave numbers and the refractive indices, respectively; k 0 is the wave number of the fundamental wave in vacuum; d 22 , d 24 , d 32 , d 33 ; r 22 , r 32 and r 42 are the double frequency and EO coefficients, respectively; c is the speed of light in vacuum; the asterisk denotes complex conjugation; and f(x) is the structure function that is +1 or -1 for the positive or negative domains of the OSL, respectively. The right side of each equation includes two parts: the bracketed stands for SHG and PDC; and the others refer to EO effect. When the external electric field is absent, the coupling equations (1)–(4) reduce to the familiar wave coupling equations describing SHG or PDC.
3. High-flux photon-pair source from electrically induced parametric down conversion after second harmonic generation in single OSL
We now use Eqs.(1)–(4) to find a OSL that satisfies our requirement. As well known, the key to QPM in OSL is to design a structure that provides a set of reciprocal vectors to compensate for the mismatches of wave vectors owing to the dispersion of refractive index. In general, the one-component periodic structure can provide one reciprocal vector to compensate for one mismatch of wave vector, but in special case, the first-order and third-order reciprocal vectors of the structure can simultaneously compensate for the phase mismatches of second-harmonic and sum-frequency effects, respectively . The two-component quasi-periodic OSL providing two reciprocal vectors has been proved useful to some coupled parametric processes such as the direct third-harmonic generation [25, 26]. One can see, from Eqs. (1)–(4), that in general, there exist six mismatches of wave vectors relative to SHG, PDC and EO effects. For our purpose, however, we need only to take account of two compensations for Δka (EO effect) and Δkf (SHG) in the first section of OSL and one compensation for Δkf (PDC) in the second section of OSL, respectively. So we can choose a hybrid OSL consisting of a two-component QPPLN (section 1) and a one-component PPLN (section 2). Suppose that a 1080nm extraordinary wave is used as a pump one and the temperature is at 100°c , then Δka = 0.4262μm -1 and |Δkf| = 0.8977μm -1. According to Ref. , the reciprocal vector conditions of QPM in section 1 are and , where D' =τ' ∙ lA+lB ; lA and lB are the lengths of the fundamental blocks A and B, respectively. By choosing m = 1, n = 0 , m' =1 and n' =1, the parameters τ' and D' can be determined immediately. They are τ' =1.1061 and D' =14.74μm , respectively. Therefore the structure of QPPLN can be determined  to be ABABABABABA…, in which block A or B is further composed of a positive ferroelectric domain and a negative one, respectively, i.e., lA = lA + +lA - and lB = lB + +lB -. We further choose lA =6.00μm and lA + =lB + = l = 3.00μm , then lB = D' -τ' ∙lA =8.10μm . Two Fourier coefficients corresponding to k 10 and k 11 can be determined , which are f 10 =0.1288, f 11 = 0.5324, respectively. For the PPLN, the fundamental block A’ is further composed of one positive and one negative ferroelectric domain, whose length is D = lA' + + lA' - . Then we expand f(x) as a Fourier series such as f(x) = Σm fm exp(-iGmx) + const. (m = ±1,±2,±3,⋯), in which the f m =(2/mπ)sin(mπl A' + /D) ; Gm=2πm/D . We choose D = 2π/(-Δkf) , then G 1 =-Δkf and Δkf can be compensated by G 1 in section 2. Under these conditions, D = 2π/(-Δkf) = 6.99μm . We further choose l A' + =3.50μm . Then the Fourier coefficient concerned is f 1 =0.6366 . Therefore, by ignoring those terms with nonzero mismatches of wave vectors, Eqs. (1)–(4) can be simplified as
We fix the pump intensity of e-polarized fundamental wave at 100MW/cm2 for calculation. It is found that a 3.4mm long QPPLN is more suitable. The numerical results reflecting the dependences of normalized intensities of e-polarized pump fundamental wave, e-polarized second harmonic and o-polarized fundamental wave respectively on the length of the crystal and electric field are shown in Fig. 2, in which (A)-(C) corresponds to electric fields 0, 30 and 100V/mm, respectively. In the calculations the Sellmeier equation  for LN and d 22 =3.0 ,d 24 =-5.0 ,d 32 =-5.0 ,d 33 = -33 ; r 22 = 3.0 , r 32 =0 , r 42 = 28 (in 10-12 m/V) are used.
One can see from Fig. 2 that the applied electric field can effectively control the parametric processes in OSL. For the case of zero electric field, OSL as a whole plays a role of frequency doubler and only the SHG takes place since in this case three coupling Eqs. (8) – (10) reduce to two ones describing SHG (please note that the coupling equations describing PDC are formally the same as those of SHG). The e-polarized fundamental frequency (pump) photons are only converted to second harmonic ones with the same polarization [Fig. 2(A)]. But for a properly weak electric field, for example, 30V/mm, due to the compensations of Δka and Δkf two processes including SHG and EO effect take place synchronously in the first section of OSL and the PDC process occurs in the second section of OSL after SHG and EO effect [Fig. 2(B)]. In the first section of OSL a large portion of e-polarized fundamental frequency photons are converted to double frequency ones with the same polarization and a very small portion of them are turned into o-polarized fundamental frequency photons by EO effect. At 10.4mm almost all the e-polarized fundamental frequency photons are converted to the double frequency ones and the remnant ratio is about 1.0987×10-8; while the o-polarized fundamental wave from EO effect is with a normalized intensity of 3.1692×10-5. In the first section of OSL there also exists another process: the o-polarized fundamental frequency photons are partially turned, again by EO effect, into a part of the seeds of e-polarized fundamental frequency photons for PDC, which is the key turning dE 1z(x)/dx from <0 to >0 therefore triggering PDC in the second section of OSL. At 20.2mm the double frequency photons are almost fully converted to a series of fundamental two-photon pairs still with e-polarization through PDC. Actually, in the OSL the second harmonic only plays a role of intermedium. When further increasing the electric field, for example, E 0 =100V/mm , the EO effect becomes stronger, which might obstruct the generation of pure two-photon pairs of e-polarized fundamental wave, because the e-polarized fundamental seeds for PDC is much more than that at lower electric field. For the process of degenerate PDC described above, the Heisenberg equations of motion are 
which lead immediately to d n̑ω/dt = 0-2dnˆ2ω /dt , or d < n̑ω>/dt = -2d <nˆ2ω>/dt therefore
where nˆ2ω=Aˆ+ 2ω Aˆ2ω, nˆω=Aˆ+ ω Aˆω, are the photon number operators of double frequency and fundamental lights, respectively. Besides, we have the frequency relation ω 1z + ω 1z = ω 2z and further the energy relation of photons
Equation (15) tells us that a pair of fundamental photons is consequentially created when a double frequency photon disappears. And Eq.(14) tells us that at any moment, the increase of photon number of fundamental light is always the double of the decrease of photon number of double frequency one. Since at the beginning of parametric downconversion, the seed of e-polarized fundamental one is so weak relative to the double frequency one that it can be ignored; the e-polarized double frequency photons and residual o-polarized fundamental ones can be conveniently removed by a band-polarizing filter F, the fundamental output will consist mainly of photon pairs. The output can be coupled to a 50/50 Y-type single mode fiber optic beam splitter to separate the twin photons . Two output beams of the splitter are then directed into two single mode fibers that are so long enough that their output beams become very weak, in which the photons do not overlap with each other. The two weak are introduced to passively quenched LN2 cooled germanium avalanche photodiodes (Ge-APDs) operating in Geiger mode for detecting . The coincidence rate can be obtained by using two one-channel analyzers . In a word, applying a properly weak electric field on an OSL with appropriate structure and length, pumped by a laser beam, we can get a source of high-flux and high-purity two-photon pairs. The further calculation indicates that if a longer OSL is used the intensity of pump beam can be lowered for performing the same function. For example, for an 42.5mm OSL, the intensity of pump can be as low as 20MW/cm2 with an external electric field of 30V / mm . One can see in fact that the device can also act as an ultra-low field electro-optic switch, which requires much lower field (~10V/mm) than that of traditional one with half-wave field (~102 -103 V/mm). The second harmonic also only plays a role of intermedium in the device.
In conclusion, a united wave coupling theory describing SHG, PDC and EO effect in PPLN or/and QPPLN OSL has been developed in this letter to the best of our knowledge for the first time, which shows a principle for constructing a high-flux photon-pair source from electrically induced PDC after second-harmonic generation in single OSL. It is demonstrated that at 100 °C, in a 20.2mm long OSL with appropriate structure and a 30V/mm applied electric field, a 100MW/cm2, 1080 nm laser beam can be translated to a flux of high-purity two-photon pairs with a conversion efficiency close to 100% ; and for a longer OSL the pump intensity can be further lowered. The device can also act as an ultra-low field electro-optic switch, which requires much lower field (~10V / mm) than that of traditional one with half-wave field (~102 -103 V / mm) . It can be used as Q switch [46, 47] or low-driving-power and high-speed optical switch in optical communication network or optical signal-processing system.
We thank Cheng-Ping Huang, Chao Zhang, and Hong Wei for their helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 10574167).
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