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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 13 — Jun. 25, 2007
  • pp: 8275–8283
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High-flux photon-pair source from electrically induced parametric down conversion after second-harmonic generation in single optical superlattice

Dong Huang and Weilong She  »View Author Affiliations


Optics Express, Vol. 15, Issue 13, pp. 8275-8283 (2007)
http://dx.doi.org/10.1364/OE.15.008275


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Abstract

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

1. Introduction

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 [1

1. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991). [CrossRef] [PubMed]

] and dense coding [2

2. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881 (1992). [CrossRef] [PubMed]

], to quantum teleportation [3

3. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993). [CrossRef] [PubMed]

], “entanglement swapping” [4

4. M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287 (1993). [CrossRef] [PubMed]

] and quantum computation [5

5. A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, “Conditional quantum dynamics and logic gates,” Phys. Rev. Lett. 74, 4083 (1995). [CrossRef] [PubMed]

]. Spontaneous [6–8

6. Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921 (1988). [CrossRef] [PubMed]

] and seed injected [9

9. F. De Martini, “Amplification of quantum entanglement,” Phys. Rev. Lett. 81, 2842 (1998). [CrossRef]

, 10

10. X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002). [CrossRef] [PubMed]

] PDC with type I [11

11. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773 (1999). [CrossRef]

] or type-II [12

12. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995). [CrossRef] [PubMed]

] 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 [13

13. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004). [CrossRef]

], being inconvenient for coupling the emitted photons into optical fibers [14

14. M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, “Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP,” Opt. Exp. 12, 3573 (2004). [CrossRef]

], that it is unsuitable for long distance communications. Besides, to get entanglement beams in traditional methods, a double frequency pump laser beam [13–15

13. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004). [CrossRef]

] or two separated sets for SHG and PDC [16

16. K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. 26, 1367(2001). [CrossRef]

, 17

17. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003). [CrossRef] [PubMed]

] 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

18. A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, “Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4,” Opt. Commun. 142, 265 (1997). [CrossRef]

], sum-frequency generation (SFG) [21

21. I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes,” J. Opt. Soc. Am. B 14, 3368 (1997). [CrossRef]

], and optical parametric oscillation (OPO) [22

22. L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663 (1997). [CrossRef]

, 23

23. K. El Hadi, M. Sundheimer, P. Aschieri, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, and F. Laurell, “Quasi-phase-matched parametric interactions in proton-exchanged lithium niobate waveguides,” J. Opt. Soc. Am. B 14, 3197 (1997). [CrossRef]

], notably third-harmonic generation [24

24. M. Fujimura, T. Suhara, and H. Nishihara, “Periodically domain-inverted LiNbO3 for waveguide auasi-phase-matched nonlinear optic wavelength conversion devices,” Bull. Mater. Sci. 22, 413 (1999). [CrossRef]

–26 ] in these materials have been reported. Recently, cascaded frequency conversion and signal tuning in these materials received much interest [27–34

27. C. J. K. Virmani ,Plasma Phys.15, 1039 (1973). [CrossRef]

]. 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[30

30. K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, “Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate,” Opt. Commun. 203, 163 (2002). [CrossRef]

]. The two functions were integrated in single OSL [31

31. Y. Chen, F. Fan, Y. Lin, Y. Huang, J. Shy, Y. Lan, and Y. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417 (2003). [CrossRef]

] subsequently. Very recently, a way different from those reported by Refs.[30

30. K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, “Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate,” Opt. Commun. 203, 163 (2002). [CrossRef]

] and [31

31. Y. Chen, F. Fan, Y. Lin, Y. Huang, J. Shy, Y. Lan, and Y. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417 (2003). [CrossRef]

], using an applied electric field to control both polarization and magnitude of the second harmonic, was presented [34

34. C. Huang, Q. Wang, and Y. Zhu, “Cascaded frequency doubling and electro-optic coupling in a single optical superlattice,” Appl. Phys. B 80, 741 (2005). [CrossRef]

]. Besides, QPM materials have also been used for efficient generation of photon pairs [13

13. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004). [CrossRef]

, 16

16. K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. 26, 1367(2001). [CrossRef]

, 35

35. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, “Highly efficient photo-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26 (2001). [CrossRef]

]. 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 [35

35. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, “Highly efficient photo-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26 (2001). [CrossRef]

]. 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

36. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195, 303 (2001). [CrossRef]

, 37

37. G. Zheng, H. Wang, and W. She, “Wave coupling theory of quasi-phase-matched linear electro-optic effect,” Opt. Exp. 14, 5535 (2006). [CrossRef]

] instead of that including the use of refractive index ellipsoid theory in Ref. [34

34. C. Huang, Q. Wang, and Y. Zhu, “Cascaded frequency doubling and electro-optic coupling in a single optical superlattice,” Appl. Phys. B 80, 741 (2005). [CrossRef]

]. 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.

Fig. 1. Schematic diagram of the SHG, PDC in the OSL controlled by an applied electric field. x , y and z represent three axes of the crystal. The arrows indicate the polarized directions. 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 [38

38. Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719 (2000). [CrossRef]

] and frequency conversions [39

39. C. Huang, Y. Wang, and Y. Zhu, “Effect of electro-optic modulation on coupled quasi-phase-matched frequency conversion,” Appl. Opt. 44, 4980 (2005). [CrossRef] [PubMed]

] appear simultaneously if there exits an external electric field. We take EO effect as the second-order nonlinear one as did Ref.[36

36. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195, 303 (2001). [CrossRef]

, 37

37. G. Zheng, H. Wang, and W. She, “Wave coupling theory of quasi-phase-matched linear electro-optic effect,” Opt. Exp. 14, 5535 (2006). [CrossRef]

]. 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) [36

36. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195, 303 (2001). [CrossRef]

] besides the well-known polarizations P (2) SHG and P (2) PDC relative to SHG and PDC[40

40. A. Yariv, Quantum Electronics (John Wiley & Sons, Inc., New York, 1975) Chaps.16-17.

], 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

36. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195, 303 (2001). [CrossRef]

, 37

37. G. Zheng, H. Wang, and W. She, “Wave coupling theory of quasi-phase-matched linear electro-optic effect,” Opt. Exp. 14, 5535 (2006). [CrossRef]

, 40

40. A. Yariv, Quantum Electronics (John Wiley & Sons, Inc., New York, 1975) Chaps.16-17.

] and noting r 32 = 0 and d 23, d 34 = 0 for LN, the coupling equations for present case (EO effect: ω1zω1y, ω2zω2y; SHG and/or PDC:ω1y + ω1yω2y, ω1y + ω1zω2y, ω1z + ω1zω2y, ω1y + ω1yω2z, ω1y + ω1zω2z, ω1z + ω1zω2z) can be deduced from Maxwell’s equations as follows:

dE1y(x)dx=id1(x)E1z(x)eiΔkaxid2(x)E1y(x)
+iω1cn1y(x)[d22(x)E1y*(x)E2y(x)eiΔkcx+d24(x)E1y*(x)E2z(x)eiΔkdx+d24(x)E1z*(x)E2y(x)eiΔkex],
(1)
dE1z(x)dx=id3(x)E1y(x)eiΔkaxid4(x)E1z(x)
+iω1cn1z(x)[d32(x)E1y*(x)E2y(x)eiΔkex+d33(x)E1z*(x)E2z(x)eiΔkfx],
(2)
dE2y(x)dx=2id5(x)E2z(x)eiΔkbx2id6(x)E2y(x)
+iω2cn2y(x)[12d22(x)E1y(x)E1y(x)eiΔkcx+d24(x)E1y(x)E1z(x)eiΔkex],
(3)
dE2z(x)dx=2id7(x)E2y(x)eiΔkbx2id8(x)E2z(x)
+iω2cn2z(x)[12d32(x)E1y(x)E1y(x)eiΔkdx+12d33(x)E1z(x)E1z(x)eiΔkfx],
(4)

where

Δka=k1yk1z=2πλ1(n1yn1z),Δkb=k2yk2z=4πλ1(n2yn2z),
(5a)
Δkc=2k1yk2y=4πλ1(n1yn2y),Δkd=2k1yk2z=4πλ1(n1yn2z),
(5b)
Δke=k1y+k1zk2y=2πλ1(n1y+n1z2n2y),Δkf=2k1zk2z=4πλ1(n1zn2z),
(5c)
d1(x)=k0n1yn1z2r422E0f(x),d2(x)=k0n1y3r222E0f(x),
(6a)
d3(x)=k0n1y2n1zr422E0f(x),d4(x)=k0n1z3r322E0f(x)=0,
(6b)
d5(x)=k0n2yn2z2r422E0f(x),d6(x)=k0n2y3r222E0f(x),
(6c)
d7(x)=k0n2y2n2zr422E0f(x),d8(x)=k0n2z3r322E0f(x),=0
(6d)
d22(x)=d22f(x),d24(x)=d24f(x),d32(x)=d32f(x),d33(x)=d33f(x),
(7)

where E , ω , k and n (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 [41

41. G. Luo, S. Zhu, J. He, Y. Zhu, H. Wang, Z. Liu, C. Zhang, and N. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006 (2001). [CrossRef]

]. 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

25. S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843 (1997). [CrossRef]

, 26

26. C. Zhang, H. Wei, Y. Zhu, H. Wang, S. Zhu, and N. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899 (2001). [CrossRef]

]. 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. [42

42. Keren Fradkin-Kashi and Ady Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649 (1999). [CrossRef]

], the reciprocal vector conditions of QPM in section 1 are kmn=2π(m+nτ')D=Δka=0.4262μm1 and km′n′=2π(m′+n′τ′)D=Δkf=0.8977μm1, 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 [42

42. Keren Fradkin-Kashi and Ady Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649 (1999). [CrossRef]

] 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 [39

39. C. Huang, Y. Wang, and Y. Zhu, “Effect of electro-optic modulation on coupled quasi-phase-matched frequency conversion,” Appl. Opt. 44, 4980 (2005). [CrossRef] [PubMed]

], 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/)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

dE1y(x)dx=ik0n1yn1z2r42E0fa2E1z(x)ik0n1y3r22E02f(x)E1y(x),
(8)
dE1z(x)dx=ik0n1y2n1zr42E0fa2E1y(x)+2fE1z*(x)E2z(x),
(9)
dE2z(x)dx=i12κ4fE1z(x)E1z(x),
(10)
κ1d=d24ω1cn1y,κ2f=d33ω1ffcn1z,κ4d=d32ω2cn2z,κ4f=d33ω2ffcn2z,
(11)
fa={f1,0forQPPLN0forPPLNandff={f1,1forQPPLNf1forPPLN,
(12)

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 [43

43. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. and Quant. Electron. 16, 373 (1984). [CrossRef]

] 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.

Fig. 2. Dependence 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 with different external electric field. The total length of the OSL is 20.2mm , and the pump intensity is 100MW / cm 2 . Red, green and brown lines represent the intensities of e-polarized pump wave, e-polarized second harmonic and o-polarized fundamental wave, respectively. And the normalized intensity of o-polarized fundamental wave is enlarged by 1000 times.

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 [44

44. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995) p.1071.

]

dAˆωdt=2igAˆω+Aˆ2ω
(13a)
dAˆ2ωdt=igAˆω2,
(13b)

which lead immediately to d n̑ω/dt = 0-2dnˆ2ω /dt , or d < n̑ω>/dt = -2d <nˆ2ω>/dt therefore

(d<n̑ω>dt)Δt=2(d<nˆ2ω>dt)Δt,
(14)

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

ħω1z+ħω1z=ħω2z.
(15)

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 [35

35. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, “Highly efficient photo-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26 (2001). [CrossRef]

]. 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 [35

35. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, “Highly efficient photo-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26 (2001). [CrossRef]

]. The coincidence rate can be obtained by using two one-channel analyzers [45

45. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart, ” Phys. Rev. Lett. 81, 3563 (1998). [CrossRef]

]. 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.

4. Conclusion

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) [38

38. Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719 (2000). [CrossRef]

]. It can be used as Q switch [46

46. Y. H. Chen and Y. C. Huang, “Actively Q-switched Nd:YVO4 laser using an electro-optic periodically poled lithium niobate crystal as a laser Q-switch,” Opt. Lett. 28, 1460 (2003). [CrossRef] [PubMed]

, 47

47. K. S. Abedin, T. Tsuritani, M. Sato, H. Ito, K. Shimamura, and T. Fukuda, “Integrated electro-optic Q switching in a domain-inverted Nd:LiTaO3,” Opt. Lett. 20, 1985 (1995). [CrossRef] [PubMed]

] or low-driving-power and high-speed optical switch in optical communication network or optical signal-processing system.

Acknowledgments

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).

References and links

1.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991). [CrossRef] [PubMed]

2.

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881 (1992). [CrossRef] [PubMed]

3.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993). [CrossRef] [PubMed]

4.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287 (1993). [CrossRef] [PubMed]

5.

A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, “Conditional quantum dynamics and logic gates,” Phys. Rev. Lett. 74, 4083 (1995). [CrossRef] [PubMed]

6.

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921 (1988). [CrossRef] [PubMed]

7.

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50 (1988). [CrossRef] [PubMed]

8.

J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495 (1990). [CrossRef] [PubMed]

9.

F. De Martini, “Amplification of quantum entanglement,” Phys. Rev. Lett. 81, 2842 (1998). [CrossRef]

10.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002). [CrossRef] [PubMed]

11.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773 (1999). [CrossRef]

12.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995). [CrossRef] [PubMed]

13.

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004). [CrossRef]

14.

M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, “Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP,” Opt. Exp. 12, 3573 (2004). [CrossRef]

15.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801(R) (2004). [CrossRef]

16.

K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. 26, 1367(2001). [CrossRef]

17.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003). [CrossRef] [PubMed]

18.

A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, “Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4,” Opt. Commun. 142, 265 (1997). [CrossRef]

19.

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834 (1997). [CrossRef]

20.

S. Wang, V. Pasiskevicius, F. Laurell, and H. Karlsson, “Ultraviolet generation by first-order frequency doubling in periodically poled KTiOPO4,” Opt. Lett. 23, 1883 (1998). [CrossRef]

21.

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes,” J. Opt. Soc. Am. B 14, 3368 (1997). [CrossRef]

22.

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663 (1997). [CrossRef]

23.

K. El Hadi, M. Sundheimer, P. Aschieri, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, and F. Laurell, “Quasi-phase-matched parametric interactions in proton-exchanged lithium niobate waveguides,” J. Opt. Soc. Am. B 14, 3197 (1997). [CrossRef]

24.

M. Fujimura, T. Suhara, and H. Nishihara, “Periodically domain-inverted LiNbO3 for waveguide auasi-phase-matched nonlinear optic wavelength conversion devices,” Bull. Mater. Sci. 22, 413 (1999). [CrossRef]

25.

S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843 (1997). [CrossRef]

26.

C. Zhang, H. Wei, Y. Zhu, H. Wang, S. Zhu, and N. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899 (2001). [CrossRef]

27.

C. J. K. Virmani ,Plasma Phys.15, 1039 (1973). [CrossRef]

28.

G. Blau, M. Cairone, P. A. Chollet, and F. Kajzar, “Electro-optic modulation and second-harmonic generation through grating-induced resonant excitation of guided modes,” Proc. SPIE 2852, 237 (1996). [CrossRef]

29.

N. O’Brien, M. Missey, P. Powers, V. ominic, and K. L. Schepler, “Electro-optical spectral tuning in a continuous-wave, asymmetric-duty-cycle, periodically poled LiNbO3 optical parametric oscillator,” Opt. Lett. 24, 1750 (1999). [CrossRef]

30.

K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, “Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate,” Opt. Commun. 203, 163 (2002). [CrossRef]

31.

Y. Chen, F. Fan, Y. Lin, Y. Huang, J. Shy, Y. Lan, and Y. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417 (2003). [CrossRef]

32.

F. Xu, J. Liao, X. Zhang, J. He, H. Wang, and N. Ming, “Complete conversion of sum-frequency generation enhanced by controllable linear gratings induced by an electro-optic effect in a periodic optical superlattice,” Phys. Rev. A 68, 033808 (2003). [CrossRef]

33.

F. Xu, J. Liao, C. Guo, J. He, H. Wang, S. Zhu, Z. Wang, Y. Zhu, and N. Ming, “Highly efficient direct third-harmonic generation based on control of the electro-optic effect in quasi-periodic optical superlattices,” Opt. Lett. 28, 429 (2003). [CrossRef] [PubMed]

34.

C. Huang, Q. Wang, and Y. Zhu, “Cascaded frequency doubling and electro-optic coupling in a single optical superlattice,” Appl. Phys. B 80, 741 (2005). [CrossRef]

35.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, “Highly efficient photo-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26 (2001). [CrossRef]

36.

W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195, 303 (2001). [CrossRef]

37.

G. Zheng, H. Wang, and W. She, “Wave coupling theory of quasi-phase-matched linear electro-optic effect,” Opt. Exp. 14, 5535 (2006). [CrossRef]

38.

Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719 (2000). [CrossRef]

39.

C. Huang, Y. Wang, and Y. Zhu, “Effect of electro-optic modulation on coupled quasi-phase-matched frequency conversion,” Appl. Opt. 44, 4980 (2005). [CrossRef] [PubMed]

40.

A. Yariv, Quantum Electronics (John Wiley & Sons, Inc., New York, 1975) Chaps.16-17.

41.

G. Luo, S. Zhu, J. He, Y. Zhu, H. Wang, Z. Liu, C. Zhang, and N. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006 (2001). [CrossRef]

42.

Keren Fradkin-Kashi and Ady Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649 (1999). [CrossRef]

43.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. and Quant. Electron. 16, 373 (1984). [CrossRef]

44.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995) p.1071.

45.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart, ” Phys. Rev. Lett. 81, 3563 (1998). [CrossRef]

46.

Y. H. Chen and Y. C. Huang, “Actively Q-switched Nd:YVO4 laser using an electro-optic periodically poled lithium niobate crystal as a laser Q-switch,” Opt. Lett. 28, 1460 (2003). [CrossRef] [PubMed]

47.

K. S. Abedin, T. Tsuritani, M. Sato, H. Ito, K. Shimamura, and T. Fukuda, “Integrated electro-optic Q switching in a domain-inverted Nd:LiTaO3,” Opt. Lett. 20, 1985 (1995). [CrossRef] [PubMed]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(270.0270) Quantum optics : Quantum optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 19, 2007
Revised Manuscript: June 1, 2007
Manuscript Accepted: June 1, 2007
Published: June 18, 2007

Citation
Dong Huang and Weilong She, "High-flux photon-pair source from electrically induced parametric down conversion after second-harmonic generation in single optical superlattice," Opt. Express 15, 8275-8283 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8275


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References

  1. A. K. Ekert, "Quantum cryptography based on Bell’s theorem," Phys. Rev. Lett. 67, 661 (1991). [CrossRef] [PubMed]
  2. C. H. Bennett and S. J. Wiesner, "Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states," Phys. Rev. Lett. 69, 2881 (1992). [CrossRef] [PubMed]
  3. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels," Phys. Rev. Lett. 70, 1895 (1993). [CrossRef] [PubMed]
  4. M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, "Event-ready-detectors Bell experiment via entanglement swapping," Phys. Rev. Lett. 71, 4287 (1993). [CrossRef] [PubMed]
  5. A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, "Conditional quantum dynamics and logic gates," Phys. Rev. Lett. 74, 4083 (1995). [CrossRef] [PubMed]
  6. Y. H. Shih, C. O. Alley, "New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion," Phys. Rev. Lett. 61, 2921 (1988). [CrossRef] [PubMed]
  7. Z. Y. Ou, L. Mandel, "Violation of Bell’s inequality and classical probability in a two-photon correlation experiment," Phys. Rev. Lett. 61, 50 (1988). [CrossRef] [PubMed]
  8. J. G. Rarity, P. R. Tapster, "Experimental violation of Bell’s inequality based on phase and momentum," Phys. Rev. Lett. 64, 2495 (1990). [CrossRef] [PubMed]
  9. F. De Martini, "Amplification of quantum entanglement," Phys. Rev. Lett. 81, 2842 (1998). [CrossRef]
  10. X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, "Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam," Phys. Rev. Lett. 88, 047904 (2002). [CrossRef] [PubMed]
  11. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999). [CrossRef]
  12. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995). [CrossRef] [PubMed]
  13. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, "High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter," Phys. Rev. A 69, 013807 (2004). [CrossRef]
  14. M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, "Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP," Opt. Exp. 12, 3573 (2004). [CrossRef]
  15. M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, "Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints," Phys. Rev. A 69, 041801(R) (2004). [CrossRef]
  16. K. Banaszek, A. B. U’Ren, and I. A. Walmsley, "Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides," Opt. Lett. 26, 1367(2001). [CrossRef]
  17. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, K. Peng, "Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables," Phys. Rev. Lett. 90, 167903 (2003). [CrossRef] [PubMed]
  18. A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, "Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4," Opt. Commun. 142, 265 (1997). [CrossRef]
  19. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834 (1997). [CrossRef]
  20. S. Wang, V. Pasiskevicius, F. Laurell, and H. Karlsson, "Ultraviolet generation by first-order frequency doubling in periodically poled KTiOPO4," Opt. Lett. 23, 1883 (1998). [CrossRef]
  21. I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, "All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes," J. Opt. Soc. Am. B 14, 3368 (1997). [CrossRef]
  22. L. E. Myers and W. R. Bosenberg, "Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators," IEEE J. Quantum Electron. 33, 1663 (1997). [CrossRef]
  23. K. El Hadi, M. Sundheimer, P. Aschieri, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, F. Laurell, "Quasi-phase-matched parametric interactions in proton-exchanged lithium niobate waveguides," J. Opt. Soc. Am. B 14, 3197 (1997). [CrossRef]
  24. M. Fujimura, T. Suhara, and H. Nishihara, "Periodically domain-inverted LiNbO3 for waveguide auasi-phase-matched nonlinear optic wavelength conversion devices," Bull. Mater. Sci. 22, 413 (1999). [CrossRef]
  25. S. Zhu, Y. Zhu, and N. Ming, "Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice," Science 278, 843 (1997). [CrossRef]
  26. C. Zhang, H. Wei, Y. Zhu, H. Wang, S. Zhu and N. Ming, "Third-harmonic generation in a general two-component quasi-periodic optical superlattice," Opt. Lett. 26, 899 (2001). [CrossRef]
  27. C. J. K. Virmani, Plasma Phys. 15, 1039 (1973). [CrossRef]
  28. G. Blau, M. Cairone, P. A. Chollet, F. Kajzar, "Electro-optic modulation and second-harmonic generation through grating-induced resonant excitation of guided modes," Proc. SPIE 2852, 237 (1996). [CrossRef]
  29. N. O’Brien, M. Missey, P. Powers, V. Dominic, and K. L. Schepler, "Electro-optical spectral tuning in a continuous-wave, asymmetric-duty-cycle, periodically poled LiNbO3 optical parametric oscillator," Opt. Lett. 24, 1750 (1999). [CrossRef]
  30. K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, "Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate," Opt. Commun. 203, 163 (2002). [CrossRef]
  31. Y. Chen, F. Fan, Y. Lin, Y. Huang, J. Shy, Y. Lan, and Y. Chen, "Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate," Opt. Commun. 223, 417 (2003). [CrossRef]
  32. F. Xu, J. Liao, X. Zhang, J. He, H. Wang, N. Ming, "Complete conversion of sum-frequency generation enhanced by controllable linear gratings induced by an electro-optic effect in a periodic optical superlattice," Phys. Rev. A 68, 033808 (2003). [CrossRef]
  33. F. Xu, J. Liao, C. Guo, J. He, H. Wang, S. Zhu, Z. Wang, Y. Zhu, N. Ming, "Highly efficient direct third-harmonic generation based on control of the electro-optic effect in quasi-periodic optical superlattices," Opt. Lett. 28, 429 (2003). [CrossRef] [PubMed]
  34. C. Huang, Q. Wang, Y. Zhu, "Cascaded frequency doubling and electro-optic coupling in a single optical superlattice," Appl. Phys. B 80, 741 (2005). [CrossRef]
  35. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, "Highly efficient photo-pair source using periodically poled lithium niobate waveguide," Electron. Lett. 37, 26 (2001). [CrossRef]
  36. W. She, W. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303 (2001). [CrossRef]
  37. G. Zheng, H. Wang, W. She, "Wave coupling theory of quasi-phase-matched linear electro-optic effect," Opt. Exp. 14, 5535 (2006). [CrossRef]
  38. Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, "Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications," Appl. Phys. Lett. 77, 3719 (2000). [CrossRef]
  39. C. Huang, Y. Wang, Y. Zhu, "Effect of electro-optic modulation on coupled quasi-phase-matched frequency conversion," Appl. Opt. 44, 4980 (2005). [CrossRef] [PubMed]
  40. A. Yariv, Quantum Electronics (John Wiley & Sons, Inc., New York, 1975) Chaps.16-17.
  41. G. Luo, S. Zhu, J. He, Y. Zhu, H. Wang, Z. Liu, C. Zhang, and N. Ming, "Simultaneously efficient blue and red light generations in a periodically poled LiTaO3," Appl. Phys. Lett. 78, 3006 (2001). [CrossRef]
  42. Keren Fradkin-Kashi and Ady Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649 (1999). [CrossRef]
  43. G. J.  Edwards and M.  Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. and Quant. Electron. 16, 373 (1984). [CrossRef]
  44. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995) p.1071.
  45. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Violation of Bell inequalities by photons more than 10 km apart, " Phys. Rev. Lett. 81, 3563 (1998). [CrossRef]
  46. Y. H. Chen and Y. C. Huang, "Actively Q-switched Nd:YVO4 laser using an electro-optic periodically poled lithium niobate crystal as a laser Q-switch," Opt. Lett. 28, 1460 (2003). [CrossRef] [PubMed]
  47. K. S. Abedin, T. Tsuritani, M. Sato, H. Ito, K. Shimamura, and T. Fukuda, "Integrated electro-optic Q switching in a domain-inverted Nd:LiTaO3," Opt. Lett. 20, 1985 (1995). [CrossRef] [PubMed]

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