## Polarization-entangled photon pairs from a periodically poled crystalline waveguide |

Optics Express, Vol. 19, Issue 7, pp. 6724-6740 (2011)

http://dx.doi.org/10.1364/OE.19.006724

Acrobat PDF (1016 KB)

### Abstract

A proposal is made for the generation of polarization-entangled photon pairs from a periodically poled crystal allowing for high collection efficiency, high entanglement, and stable operation. The theory is formulated for colinear propagation for application to waveguides. The key feature of the theory is the use of type II phase matching using both the +1 and −1 diffraction orders of the poling structure. Although these conditions are fairly restrictive in terms of operating parameters, practical operating conditions can be found. For example, we find that a HeNe pump laser may be used for a periodically poled rubidium-doped potassium titanyl phosphate (Rb:KTP) waveguide to yield single mode polarization-entangled pairs. Fidelities of 0.98 are possible under practical conditions.

© 2011 OSA

## 1. Introduction

1. J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature **403**, 515–519 (2000). [CrossRef] [PubMed]

2. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature **390**, 575–579 (1997). [CrossRef]

3. M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. **11**, 045007 (2009). [CrossRef]

4. Z. Y. Ou and L. Mandel, “Violation of bell’s inequality and classical probability in a two-photon corrlelation experiment,” Phys. Rev. Lett. **61**, 50–53 (1988). [CrossRef] [PubMed]

5. P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. **75**, 4337–4341 (1995). [CrossRef] [PubMed]

6. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, 773–776 (1999). [CrossRef]

7. A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. **102**, 153902 (2009). [CrossRef] [PubMed]

8. J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. **95**, 260501 (2005). [CrossRef]

9. T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A **73**, 012316 (2006). [CrossRef]

5. P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. **75**, 4337–4341 (1995). [CrossRef] [PubMed]

6. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, 773–776 (1999). [CrossRef]

*et al.*[9

9. T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A **73**, 012316 (2006). [CrossRef]

10. R. W. Risk, “Fabrication and characterization of planar ion-exchanged ktiopo_{4} waveguides for frequency doubling,” Appl. Phys. Lett. **58**, 19–21 (1991). [CrossRef]

## 2. Polarization-entangled two-photon state in a crystalline waveguide

11. P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. **31**, 997–1008 (1995). [CrossRef]

*L*= 2ℓ is reversed in a pattern with period Λ, called the poling period, so that the usual wave vectors of phase matching are supplemented by wave vectors corresponding to 2

*π*/Λ times an integer

*m*[12

12. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: Tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631–2654 (1992). [CrossRef]

*mπ*/Λ but include only the integer pairs

*m*= ±1

*,*±3. To avoid an edge effect,

*L*should be in an integral multiple of Λ, although

*L*>> Λ is a sufficient condition. (Hereafter we will not distinguish between phase matching and quasi-phase matching.)

*β*(

*ω*,

_{s}*ω*) is a two-photon joint spectral amplitude, where

_{i}*ω*and

_{s}*ω*are the frequencies of the signal and idler fields,

_{i}*U*(

_{j}*r⃗*,

*ω*) is the spatial mode of the waveguide, for polarizations along crystal axes

*r⃗*with angular frequency

*ω*and polarization

*j*, and |0〉 is the photonic ground state. We use the convention that a bar over a state is unnormalized. We consider the case in which the confined modes at both the signal and idler frequencies are unique (

*i.e.,*single mode) for each of the two polarizations. The effect of the slightly different spatial modes for each polarization proves to be modest, so we defer its treatment to Appendix A. Neglecting the effect of spatial modes, Eq. (1) reduces to where the creation operators create photons in a mode with a fixed spatial dependence.

*ω*turns into a signal and idler photon pair. Energy is conserved, so

_{p}*ω*=

_{p}*ω*+

_{s}*ω*. If we consider parametric down conversion in a straight crystalline waveguide or a bulk crystal of length

_{i}*L*= 2ℓ, the joint spectral amplitude is given by Here,

*α*(

*ω*) is the pump spectrum, sinc[Δ

_{p}*k*(

_{m}*ω*,

_{s}*ω*)ℓ] is a phase matching function, and the

_{i}*c*are taken to be constants [13

_{m}13. Technically, the *c _{m}* are frequency-dependent, however, because their frequency dependence is scaled to the band gap of (3.6 eV for KTP) whereas that of

*β*(

*ω*,

_{s}*ω*) is scaled to the width of phase matching. For practical crystal lengths

_{i}*L*≳ 10 mm, this width is 1 meV or less, so the

*c*may be treated as constants.

_{m}*σ*is the width. We have chosen the constants in Eq. (4) so that for the CW case We only consider the CW case in this paper.

_{p}*c*

_{1}≈

*c*

_{–}_{1}in Eq. (3) and that all the other

*c*may be neglected. (The first assumption is justified in Appendix B. The terms

_{m}*c*with |

_{m}*m*| > 1 may be neglected because the effective nonlinear coefficient

*d*

_{eff}= 2

*d*

_{bulk}

*/*(|

*m*|

*π*), so the

*m*= ±1 terms are the largest; moreover, the other terms lead to phase matched frequencies which are different from those of

*m*= ±1 and are removed by spectral filtering of the light, as shown in Fig. 1(c).) Our goal is to find conditions so that both the

*m*= 1 and

*m*= −1 terms support parametric down conversion simultaneously, with these two processes being the heart of our entangled photons source. The case using type-0 SPDC with

*m*= 0 and type-II SPDC with |

*m*| = 1 has been implemented experimentally [14

14. J. Chen, A. J. Pearlmand, A. Ling, J. Fan, and A. Migdall, “A versitile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express **17**, 6727–6740 (2009). [CrossRef] [PubMed]

*Y*and

*Z*representing directions of polarization in the crystal. The wave vectors are defined by

*k*(

_{j}*ω*) =

*ωn*(

_{j}*ω*)/

*c*, where

*n*(

_{j}*ω*) is the index of refraction for the polarization indexed by

*j*, and

*c*is the speed of light.

*β*(

*ω*,

_{s}*ω*) will be largest when

_{i}*ω*is centered on angular frequency

_{p}*ξ*and

_{s}*ξ*as well as eliminating the variables

_{i}### Fidelity of entanglement

*β*(

*ω*,

_{s}*ω*) = ±

_{i}*β*(

*ω*,

_{i}*ω*). By considering the relevant matrix element of creation and destruction operators, the orthogonality relation

_{s}*F*= Tr{

*ρ*|2〉〈2|} [15

15. R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. **41**, 2315–2323 (1994). [CrossRef]

*ρ*= |2

_{+}〉〈2

_{+}|. (Any other choice of

*ρ*leads to a lower figure for the fidelity assuming we are testing for the dominant state.) Substituting this form for

*ρ*into the fidelity formula leads to using Eq. (16), Eq. (17), and

*F*has reduced to the evaluation of

*I*and

*J*are evaluated in Appendix C, leading to the expression using the symbols defined near Eq. (42). Eq. (25) implies

*F*= 1 is achieved when Δ

*τ*

_{+}= Δ

*τ*. Restated, we obtain a high fidelity symmetric state when |

_{–}*τ*

_{1}

_{s}*– τ*

_{1}

*| ≈ |*

_{i}*τ*

_{−}_{1}

*−*

_{s}*τ*

_{−}_{1}

*|, or equivalently, when*

_{i}*i.e.,*a CW source),

*F*is independent of both

*L*= 2

*ℓ*. While the natural range of

*F*is 0 to 1, the assumption

*c*

_{1}=

*c*

_{–}_{1}limits the range of

*F*. Although we do not explore it here,

*c*

_{1}=

*− c*

_{−}_{1}, leads to

*〉.*

_{−}## 3. Fidelity of polarization-entangled states for our source

14. J. Chen, A. J. Pearlmand, A. Ling, J. Fan, and A. Migdall, “A versitile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express **17**, 6727–6740 (2009). [CrossRef] [PubMed]

16. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A **69**, 013807 (2004). [CrossRef]

17. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled ktp waveguides and bulk crystals,” Opt. Express **15**, 7479–7488 (2007). [CrossRef] [PubMed]

18. A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled csource of narrowband engtangled photons,” Opt. Express **15**, 15377–15386 (2007). [CrossRef] [PubMed]

19. T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled ktiopo_{4} waveguide,” Opt. Express **17**, 12019–12029 (2009). [CrossRef] [PubMed]

5. P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. **75**, 4337–4341 (1995). [CrossRef] [PubMed]

16. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A **69**, 013807 (2004). [CrossRef]

20. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A **50**, 5122–5133 (1994). [CrossRef] [PubMed]

24. F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled ktiopo_{4} with zero group-velocity mismatch,” Appl. Phys. Lett. **84**, 1644–1646 (2004). [CrossRef]

*et al.*for

25. K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo_{4},” Appl. Phys. Lett. **74**, 914–916 (1999). [CrossRef]

*j*=

*Y,Z*, we use the measurements of Cheng

*et al.*[26

26. L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo_{4} isomorphs from the self-fluxes,” J. Cryst. Growth **137**, 107 (1994). [CrossRef]

27. http://www.comsol.com. Mention of commercial products is for information only; it does not imply recommendation or endorsement by NIST.

*w*and the profile of the Rb dopants whose index profile decreases with depth into the crystal from the bulk RTP value to the bulk KTP value following the function erfc(

*−Z/Z*

_{0}) [28

28. J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in ktiopo_{4},” Appl. Phys. Lett. **50**, 1216 (1987). [CrossRef]

*Z*= 0, and

*Z =*0 in the waveguide and in the crystal. We used a simulation region consisting of two quarter-circles filled with KTP with a 20

*μ*m radius, a rectangular region of width

*w*between the quarter circles to represent the waveguide proper, and a rectangular region of (40

*μ*m +

*w*)

*×*4

*μ*m above to represent the air. A sketch of the simulation region is shown in Fig. 4 (top), along with the variation of the index of refraction in a particular case (bottom left). On the boundary, the fields were required to vanish. This approximation causes little error because the fields were exponentially confined. The simulation region was large enough that boundary effects were negligible as tested by comparing to a 30

*μ*m radius boundary in selected cases and by visual inspection of the confined modes. The parameter

*w*was chosen to ensure that the down converted photons are in a single mode for the signal and the idler.

**75**, 4337–4341 (1995). [CrossRef] [PubMed]

*λ*= 1247 nm and

_{s}*λ*= 1284 nm. The spread in the wave packet arises because photons converted near the exit face of the waveguide exit at almost the same time, whereas those converted near the entrance of the waveguide exit with a time difference given by (

_{i}*L/c*)[

*n*(

_{Z}*λ*) −

_{s}*n*(

_{Y}*λ*)] or (

_{s}*L/c*)[

*n*(

_{Z}*λ*) −

_{i}*n*(

_{Y}*λ*)], corresponding to 0.08069

_{i}*L/c*and 0.08083

*L/c*. For the bulk crystal, the corresponding figures are 0.08325

*L/c*and 0.08348

*L/c*. With a single length for the bulk crystal, we may compensate the average delay by choosing the bulk crystal length to be 0.9688/2 of the waveguide instead of the traditional 1/2. This deviates from the ideal compensation by ±2

*×*10

^{−5}

*L/c*, which is only 2.5

*×*10

^{−4}

*L/c*of the broadening by the waveguide. Hence, we conclude that bulk KTP may be used to compensate the timing of the broadening of two-photon wave packets in the KTP waveguides in analogy with the standard compensation scheme. For reference, if

*L*= 10 mm, then

*L/c*= 33 ps, the delay is about 2.6 ps, and the compensation error is about 0.7 fs.

*Y*↔ –

*Y*about the waveguide centerline and hence yield a vanishing triple mode integral, yielding no signal and idler photons. The first higher mode with even parity, the third mode overall, yields a triple mode integral of 39 % that of the fundamental mode integral at (

*λ*,

_{p}*λ*,

_{s}*λ*) = (625, 1250, 1250) nm, and this factor enters the production rate squared, i.e., 0.15. Coupling to the input pump beam is likely to be weak compared to coupling to the fundamental pump mode of the waveguide. Finally, the effective index of refraction of this higher pump mode is 1.871 vs. 1.786 for the fundamental mode at

_{i}*λ*= 625 nm, suggesting the higher mode’s signal and idler photons will be generated far from phase-matching conditions.

_{p}*w*= 3.5

*μ*m and

*Z*

_{0}= 4.5

*μ*m with the parameters defined in Table 2 is shown in Fig. 5. The poling periods required for type II phase matching are much longer than that needed for type 0 phase matching. This may be understood because the higher index of refraction for the

*Z*polarized wave partially cancels the higher index of the (short wavelength) pump. Varying

*w*or

*Z*

_{0}has a very modest effect on the results as shown in Table 2. This Table also shows it is possible to obtain phase-matched degenerate SPDC with a HeNe laser

*λ*= 632.8 nm for certain values of (

_{p}*w,Z*

_{0}) such as (3.4

*,*4.5)

*μ*m and (3.7

*,*4.5)

*μ*m.

*k*(

*ω*,

_{s}*ω*) =

_{i}*–*Δ

*k*(

*ω*,

_{i}*ω*), illustrated in Fig. 2 for the case of Λ = 1 mm. Because we are imposing the constraints of Eq. (7), once the material, geometry, and Λ are chosen, there are no additional degrees of freedom. Hence, it is fortunate that the wavelengths are reasonably convenient, and that, with carefully chosen parameters, operation with HeNe laser for the pump should be possible. The curves are given for room temperature. The curves for the KTP are shifted upwards by about 0.6 nm for a 20 C rise above room temperature. Curves for RTP and the waveguide are similarly affected. The temperature coefficients for the index of refraction were taken from Emanueli and Arie [29

_{s}29. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for ktiopo_{4} and ktioaso_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

*et al.*for RTP [30

30. T. Mikami, T. Okamoto, and K. Kato, “Sellmeier and thermo-optic dispersion formulas for rbtiopo_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef]

*λ*,

_{p}*λ*,

_{s}*λ*) = (632.6, 1246.8, 1284.1) nm. As the poling period becomes increasingly long, the signal and idler wavelengths become more similar, as shown in Fig. 6.

_{i}*Y*and

*Z*polarized photons leaving the waveguide. For example, for 1250 nm photons, the overlap integral the electric fields for the two polarizations, (omitting the orthogonality due to polarization) is 0.97, when the two fields are normalized to unit self-overlap integrals. A simple way to nearly eliminate the effect of the spatial mode structure is to use a single mode optical fiber to produce identical spatial modes, albeit at the price of some coupling loss. (See Fig. 1.) Maintaining high efficiency throughout the optical path will require considerable care, including the use of antireflective coatings which reduces the need for reliance in posst selection by reducing the fraction of broken pairs. We also note that these losses do not affect fidelity, unless the contributions of background light or detector dark count rates become significant. In a similar way, polarization and frequency-dependent absorption can ameliorate the effect of an imbalance (i.e., deviation from unity) in the

*c*

_{1}/

*c*

_{−1}ratio discussed in Appendix B.

## 4. Concluding remarks

*m*= ±1 of the periodically poled KTP waveguide A disadvantage of the scheme is that the available pump, signal, and idler wavelengths are highly constrained. However, we have shown that solutions are possible and even a common HeNe laser generating light at 632.8 nm may be used as a pump for certain geometries.

## A. Spatial mode effect

*β*(

*ω*,

_{s}*ω*) ≈

_{i}*β*(

*ω*,

_{i}*ω*) in this Appendix. In analogy with Eq. (20), the norm is

_{s}*dr⃗*|

*U*(

_{j}*r⃗*,

_{s}*ω*)|

_{s}^{2}= 1 for

*j*=

*Y,Z*. If we assume that the exchange integral is independent of frequency, then we have Thus, the net effect of the spatial mode structure on the fidelity is to make the exchange integral smaller by a factor of the spatial mode overlap integral in Eq. (24). For our waveguide, for

*λ*= 625 nm, the spatial mode integral factor reduces the fidelity by a factor of 0.9738. The spatial modes are show for this case in Fig. 4 (bottom middle and bottom right). The modes are seen to be very similar, although not identical. The factor is only a weak function of the wave vector mismatch. For example, for Λ = 0.1 mm, 0.5 mm, and 1 mm, we obtain values of 0.8906, 0.9693, and 0.9697.

## B. Generation rates of two output channels

*c*

_{1}and

*c*

_{−1}are proportional to matrix elements of the interaction Hamiltonian for parametric down conversion. This theory was developed for periodically poled waveguides by Fiorentino and coworkers [17

17. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled ktp waveguides and bulk crystals,” Opt. Express **15**, 7479–7488 (2007). [CrossRef] [PubMed]

11. P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. **31**, 997–1008 (1995). [CrossRef]

11. P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. **31**, 997–1008 (1995). [CrossRef]

*d*(

*x,z*) as a Fourier series which is Eq. (8) of Ref. [11

**31**, 997–1008 (1995). [CrossRef]

*d*(

_{n}*x*).” They then present a phase matching equation equivalent to our Eq. (6). Here, we consider more than one such term. Each yields the same form for the interaction matrix element which is given in Eq. (11) of Ref. [17

17. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled ktp waveguides and bulk crystals,” Opt. Express **15**, 7479–7488 (2007). [CrossRef] [PubMed]

*d*is the effective nonlinear coefficient (2/

*π*)

*d*[11

_{bulk}**31**, 997–1008 (1995). [CrossRef]

*is the pump power,*

_{p}*ε*

_{0}is the electric constant,

*c*is the speed of light, the

*n*are indices of refraction,

*L*is the crystal length, and

*A*is the cross-sectional area of the waveguide, and the various

*U*are the modes in the waveguide. In our case, we must remember that the

*n*and

*d*refer to the applicable polarization. We may write the ratio with where (

*ω*,

_{s}*ω*) is evaluated at

_{i}*c*

_{−1}/

*c*

_{1}is 1.2084, 1.0365, and 1.0125 for the three cases, respectively. The deviation from unity is almost entirely due to the triple mode integral. The factors involving the index of refraction in Eq. (33) differ from unity by less than 10

^{−4}for Λ ≥ 0.1 mm. Using the expression at the end of the previous paragraph, the fidelity needs to be reduced from that shown in Fig. 7 by a factor of 0.9823, 0.9994, and 0.9999, all of which are smaller than deviations from unity shown in the plot for the associated value of Λ. Hence, the approximation

*c*

_{1}≈

*c*

_{−1}is justified.

## C. Evaluation of direct and exchange integrals

*β*is given by Eq. (3). In an obvious notation, let There are large contributions where

*I*

_{++}, and large contributions when

*I*

_{−−}, but the mixed terms are nowhere large, assuming that

*I*

_{+−}=

*I*

_{−+}. Equation (

**??**) may be achieved in practice with periodically poled KTP.

*I*

_{++}as an integral over

*ξ*and

_{s}*ξ*as For

_{i}*I*

_{−−}, we interchange

*ω*and

_{s}*ω*,

_{i}*J*

_{++}and

*J*

_{−−}may be neglected because the arguments to the two sinc functions are never simultaneously small. However, for

*J*

_{+−}=

*J*

_{−}_{+}both arguments may be simultaneously small.

*δτ*

_{±}=

*τ*

_{±1}

*−*

_{s}*τ*

_{±1}

*, the integrals in Eq. (37) and Eq. (38) reduce to where Δ*

_{i}*τ*

_{±}= |

*δτ*

_{±}|. To obtain Eq. (42), we use the change of variables

*ξ*=

_{p}*ξ*+

_{s}*ξ*and

_{i}*ξ*= (

*ξ*−

_{s}*ξ*)/2 which has a unit Jacobian; also

_{i}*ξ*= −

_{s}*ξ*where the

_{i}*δ*function of Eq. (5) is non-zero. Similarly, from Eq. (41), We may use Δ

*τ*

_{±}instead of

*δτ*

_{±}because the sinc function is even.

## Acknowledgments

## References and links

1. | J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature |

2. | D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature |

3. | M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. |

4. | Z. Y. Ou and L. Mandel, “Violation of bell’s inequality and classical probability in a two-photon corrlelation experiment,” Phys. Rev. Lett. |

5. | P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. |

6. | P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A |

7. | A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. |

8. | J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. |

9. | T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A |

10. | R. W. Risk, “Fabrication and characterization of planar ion-exchanged ktiopo |

11. | P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. |

12. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: Tuning and tolerances,” IEEE J. Quantum Electron. |

13. | Technically, the β(ω, _{s}ω) is scaled to the width of phase matching. For practical crystal lengths _{i}L ≳ 10 mm, this width is 1 meV or less, so the c may be treated as constants._{m} |

14. | J. Chen, A. J. Pearlmand, A. Ling, J. Fan, and A. Migdall, “A versitile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express |

15. | R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. |

16. | C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A |

17. | M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled ktp waveguides and bulk crystals,” Opt. Express |

18. | A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled csource of narrowband engtangled photons,” Opt. Express |

19. | T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled ktiopo |

20. | M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A |

21. | M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. |

22. | G. K. Samanta, S. C. Mathew, C. Canalias, V. Pasiskevicius, F. Laurell, and M. Ebrahim-Zadeh, “High-power, continuous-wave, second-harmonic generation at 532 nm in periodically poled ktiopo |

23. | F. Torabi-Goudarzi and E. Riis, “Efficient cw high-power frequency doubling in periodically poled ktp,” Opt. Commun. |

24. | F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled ktiopo |

25. | K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo |

26. | L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo |

27. | http://www.comsol.com. Mention of commercial products is for information only; it does not imply recommendation or endorsement by NIST. |

28. | J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in ktiopo |

29. | S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for ktiopo |

30. | T. Mikami, T. Okamoto, and K. Kato, “Sellmeier and thermo-optic dispersion formulas for rbtiopo |

31. | S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of bell inequalities for mixed states,” Phys. Ref. Lett. |

32. | P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. |

33. | R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(270.1670) Quantum optics : Coherent optical effects

(190.4975) Nonlinear optics : Parametric processes

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: January 13, 2011

Manuscript Accepted: February 27, 2011

Published: March 24, 2011

**Citation**

Zachary H. Levine, Jingyun Fan, Jun Chen, and Alan L. Migdall, "Polarization-entangled photon pairs from a periodically poled crystalline waveguide," Opt. Express **19**, 6724-6740 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6724

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### References

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- R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964). [CrossRef]

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