## Spontaneous parametric down conversion in a nanophotonic waveguide

Optics Express, Vol. 15, Issue 14, pp. 8770-8780 (2007)

http://dx.doi.org/10.1364/OE.15.008770

Acrobat PDF (190 KB)

### Abstract

Recently, we verified that spontaneous parametric down conversion (SPDC) is enhanced in a waveguide, in agreement with theory showing an inverse dependence on mode confinement [^{9}/sec/nm/mW are predicted for periodically-poled KTP (PPKTP) nanophotonic waveguides. This results in an enhancement of the downconverted signal power greater than 45× that of low-index-contrast PPKTP waveguides and greater than 6500× that of bulk PPKTP crystals.

© 2007 Optical Society of America

## 1. Introduction

1. 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]

## 2. Theory of spontaneous parametric down conversion in a waveguide

1. 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]

*z*direction (quantization length

*L*) as:

*P*,

*S*,

*I*} describe the pump, signal, and idler, respectively. The classical pump power is

*P*, and we assume a single photon in the signal and idler modes. The mode angular frequencies are given by

_{P}*ω*

_{{P,S,I}}=

*ck*

_{{P,S,I}}where

*k*

_{{P,S,I}}= 2

*π*/λ

_{{P,S,I}}are the free-space wavenumbers, and the modes are represented by a modal effective refractive index

*n*

_{e,{P,S,I}}. The time dependent creation operator for the signal and idler photons is given by

*a*

^{†}

_{{S,I}}(

*t*). The transverse electric field distribution is contained in the term

*ϕ⃗*

_{{P,S,I}}(

*r⃗*), with the normalization:

_{T}*A*is the transverse integration area.

_{T}*χ*(2) material is given by:

*d*

_{eff}, and the integration is over the nonlinear interaction volume

*V*. Substituting the expressions for the electric fields of the pump, signal, and idler modes (Eq. 2) into the interaction Hamiltonian, and keeping only the terms which satisfy energy conservation (

*ω*=

_{P}*ω*+

_{S}*ω*), we obtain:

_{I}*β*= (

*n*-

_{e,p}k_{P}*n*-

_{e,S}k_{S}*n*).

_{e,I}k_{I}*R*=

*ρ*|〈

_{v}*H*〉|

_{I}^{2}/

*h¯*

^{2}, with the density of states in the signal detection bandwidth

*δv*= (

_{S}*c*/λ

_{S}

^{2})

*δλ*given by

_{S}*ρ*= (

_{v}*n*

_{e,S}n_{e,I}L^{2}/(

*c*λ

_{S}

^{2}))

*δλ*. Putting this all together, and integrating over the waveguide interaction length

_{S}*L*, we get the final expression for the signal generation rate:

_{C}*P*= (

_{signal}*hc*/λ

_{S})

*R*:

_{signal}*m*is an odd integer corresponding to the QPM poling order, and Λ is the QPM period. This expression is similar to that derived for a waveguide in Ref. [1

1. 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]

*n*waveguides conventionally fabricated in nonlinear crystals, but also for large index contrast waveguides and waveguiding structures consisting of both nonlinear and linear materials. Note that the expression for the effective area can be easily generalized for an arbitrary distribution of nonlinear material by including the nonlinear material coefficient distribution

*d*

_{eff}(

*r⃗*) in the transverse spatial integral for

_{T}*A*.

_{I}*L*

_{C}^{2}/ (

*n*). Therefore, for a fixed crystal length, the downconverted signal power can be increased by reducing the product of the effective mode indices and the nonlinear interaction area. High-confinement waveguides can often accomplish this feat because of two factors. First, for a higher index contrast between the core and cladding the effective index can be reduced over that of a small Δ

_{e,P}n_{e,S}n_{e,I}A_{I}*n*material (given that the core refractive index is the same and appropriate dimensions are chosen), as the confinement condition

*n*<

_{cladding}*n*<

_{e,wav}*n*allows waveguiding with a lower effective index for cases where

_{core}*n*is reduced. Secondly, the higher index contrast can result in a more tightly confined mode, reducing the interaction area. However, we must point out that this is counteracted in part by the reduced fraction of the optical mode which lies in the nonlinear core, although the fact that confinement can be increased approximately an order of magnitude with a less than 50% drop in nonlinear overlap indicates a large benefit occurs for high-contrast waveguides.

_{cladding}^{2}term in Eq. 8, which has a FWHM given by:

*L*.

_{C}## 3. Spontaneous parametric down conversion in a PPKTP waveguide

*d*

_{eff}= 2

*d*

_{{24,bulk}}/

*π*, where

*d*

_{{24,bulk}}= 3.92 pm/V [8

8. H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optical coefficients of KiTiOPO_{4},” Opt. Lett. **17**, 982–985 (1992). [CrossRef] [PubMed]

9. J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: properties and new applications,” J. Opt. Soc. Am. B **6**, 622–633 (1989). [CrossRef]

*n*denote the refractive indices in the

_{x,y,z}*x*,

*y*,

*z*directions, respectively, and λ is the optical wavelength expressed in microns. We neglect the refractive index dispersion of the surrounding dielectric, as for nearly all optically transparent cladding dielectrics the change is negligible (< 0.01) for the wavelengths of interest.

**15**, 7479–7488 (2007). [CrossRef] [PubMed]

*μ*m for the waveguide core geometry considered here. For comparison, the first order QPM period is ~ 8.36

*μ*m for a low-index waveguide [1

**15**, 7479–7488 (2007). [CrossRef] [PubMed]

*μ*m

^{2}with a variation of only ~ 10% over a waveguide width and height range of 500 – 600 nm and 450 – 550 nm, respectively. This effective area is ~ 35 × smaller than the low-index waveguide shown in Fig. 1 (16

*μ*m

^{2}).

^{9}photons/sec are achieved, with corresponding peak signal power spectral densities exceeding 1 nW/nm over a wide range of waveguide dimensions. The data trends primarily follow that of the effective area, due to its much stronger dependence on geometry. However, the counteracting effect of the effective indices shifts the location of the maximum in the rate and power plots towards smaller core geometry. The calculations show that as the waveguide width increases, the corresponding maximum shifts monotonically towards smaller waveguide heights. The highest signal photon generation rate occurs for a waveguide dimension of 500 nm by 500 nm, with a value of 6.55 × 10

^{9}photons/sec. The corresponding signal peak power spectral density is 1606 pW/nm. These high rates/powers are fairly tolerant to waveguide dimensions, with errors in waveguide dimensions of 100 nm leading to only slight < 10% changes for dimensions near the optimal value. For the optimal waveguide dimension of 500 nm by 500 nm, the spectral width determined by Eq. 11 is 0.47 nm, which is narrower than that calculated for the low-index waveguide shown in Fig. 1 (0.57 nm) due to the strong dispersion in the nanophotonic waveguide.

**15**, 7479–7488 (2007). [CrossRef] [PubMed]

*β*= 0). We see that the nanophotonic waveguides have much higher predicted pair generation rates than both low-index waveguides and bulk crystals. The enhancement factor is approximately 45 × that of low-index waveguides and more than 6500 × that of bulk crystals. This large enhancement over low-index waveguides is primarily attributed to the much smaller effective interaction area in a highly-confining nanophotonic waveguide (factor of ~35), with the remaining difference contributed by the slightly lower effective refractive indices for the pump, signal, and idler modes. The extremely large improvement over the bulk crystal is due to both the strong transverse confinement plus the increased interaction length in a waveguide.

**15**, 7479–7488 (2007). [CrossRef] [PubMed]

*n*≃ 0.02), over an area of a couple microns. The experimental measurements showed very good agreement with the predicted values, with an error < 5% for waveguides and < 20% for bulk crystals. We also note that the theory presented in this work is in agreement for a low-index contrast waveguide.

## 4. Heralded sources of single photons in a PPLN waveguide

10. A. B. U’Ren, C. SIlberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. **93**, 093601 (2004). [CrossRef] [PubMed]

*d*

_{33}= 31 pm/V nonlinear coefficient which is significantly larger than that possible in KTP. The calculations are performed as described above, with the waveguide core consisting of Z-cut PPLN surrounded by silica (

*n*= 1.45). The PPLN has a refractive index given by the Sellmeier equations [11]:

*n*denotes the extraordinary index (

_{e}*n*=

_{e}*n*),

_{z}*n*is the ordinary index (n

_{o}_{o}=

*n*=

_{x}*n*), and λ is the optical wavelength expressed in microns.

_{y}^{10}photons/sec/nm/mW are possible.

## 5. Conclusion

^{9}photons/sec/mW/nm are predicted over a wide range of waveguide core geometries, with only a slight dependence on geometry near the optimal dimensions of 500 nm by 500 nm. For the optimal waveguide core dimension the calculated spectral width (0.47 nm) is found to be narrower than that calculated for a low-index waveguide (0.57 nm) due to the additional dispersion caused by the waveguide. We have also shown that peak pair generation rates exceeding 10

^{10}photons/sec/nm/mW are possible in heralded sources based on Type I SPDC in a 10 mm long first-order QPM PPLN nanophotonic waveguide.

12. P. Rabiei and P. Gunter, “Optical and electro-optical properties of submicrometer lithium niobate slab waveguides prepared by crystal ion slicing and wafer bonding,” Appl. Phys. Lett. **85**, 4603–4605 (2004). [CrossRef]

13. P. Rabiei and W. H. Steier, “Lithium niobate ridge waveguides and modulators fabricated using smart guide,” Appl. Phys. Lett. **86**, 161115 (2005). [CrossRef]

14. A. C. Busacca, A. C. Cino, S. Riva-Sanseverino, M. Ravaro, and G. Assanto, “Silica masks for improved surface poling of lithium niobate,” Electron. Lett. **41**, 01393492 (2005). [CrossRef]

_{3}. Furthermore, fabrication imperfections (such as surface roughness and side-wall tilt) will have a stronger effect on the overall device performance, however these can be minimized through optimization of the fabrication procedure.

## References and links

1. | 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 |

2. | P. G. Kwiat, K. Mattle, H. Weinfurther, and A. Zeilinger, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. |

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

4. | K. Sanaka, K. Kawahara, and T. Kuga, “New High-Efficiency Source of Photon Pairs for Engineering Quantum Entanglement,” Phys. Rev. Lett. |

5. | 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 KTiOPO |

6. | D. A. Kleinman, “Theory of Optical Parametric Noise,” Phys. Rev. |

7. | K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in Parametric Fluorescence with a Pump Beam of Finite Cross Section,” IEEE J. Quantum Electron. |

8. | H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optical coefficients of KiTiOPO |

9. | J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: properties and new applications,” J. Opt. Soc. Am. B |

10. | A. B. U’Ren, C. SIlberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. |

11. | G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. |

12. | P. Rabiei and P. Gunter, “Optical and electro-optical properties of submicrometer lithium niobate slab waveguides prepared by crystal ion slicing and wafer bonding,” Appl. Phys. Lett. |

13. | P. Rabiei and W. H. Steier, “Lithium niobate ridge waveguides and modulators fabricated using smart guide,” Appl. Phys. Lett. |

14. | A. C. Busacca, A. C. Cino, S. Riva-Sanseverino, M. Ravaro, and G. Assanto, “Silica masks for improved surface poling of lithium niobate,” Electron. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 24, 2007

Revised Manuscript: June 26, 2007

Manuscript Accepted: June 27, 2007

Published: June 28, 2007

**Citation**

Sean M. Spillane, Marco Fiorentino, and Raymond G. Beausoleil, "Spontaneous parametric down conversion in a nanophotonic waveguide," Opt. Express **15**, 8770-8780 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8770

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### References

- 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]
- P. G. Kwiat, K. Mattle, H. Weinfurther and A. Zeilinger, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995). [CrossRef] [PubMed]
- P. G. Kwiat, E. Waks, A. G. Whitel, I. Appelbaum and P. H. Eberhard, "Ultrabright source of polarization entangled photons," Phys. Rev. A 60, R773 (1999). [CrossRef]
- K. Sanaka, K. Kawahara and T. Kuga, "New High-Efficiency Source of Photon Pairs for Engineering Quantum Entanglement," Phys. Rev. Lett. 86, 5620-5624 (2001). [CrossRef] [PubMed]
- 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]
- D. A. Kleinman, "Theory of Optical Parametric Noise," Phys. Rev. 174, 1027 (1968). [CrossRef]
- K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian and J. M. Liu, "Hot spots in parametric fluorescence with a pump beam of finite cross section," IEEE J. Quantum Electron. 31, 769-781 (1995). [CrossRef]
- H. Vanherzeele and J. D. Bierlein, "Magnitude of the nonlinear-optical coefficients of KiTiOPO4," Opt. Lett. 17, 982-985 (1992). [CrossRef] [PubMed]
- J. D. Bierlein and H. Vanherzeele, "Potassium titanyl phosphate: properties and new applications," J. Opt. Soc. Am. B 6, 622-633 (1989). [CrossRef]
- A. B. U’Ren, C. Silberhorn, K. Banaszek and I. A.Walmsley, "Efficient conditional preparation of high-fidelity single photon states for Fiber-Optic Quantum Networks," Phys. Rev. Lett. 93, 093601 (2004). [CrossRef] [PubMed]
- G. J. Edwards and M. Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. Quantum Electron. 16, 373-375 (1984). [CrossRef]
- P. Rabiei and P. Gunter, "Optical and electro-optical properties of submicrometer lithium niobate slab waveguides prepared by crystal ion slicing and wafer bonding," Appl. Phys. Lett. 85, 4603-4605 (2004). [CrossRef]
- P. Rabiei and W. H. Steier, "Lithium niobate ridge waveguides and modulators fabricated using smart guide," Appl. Phys. Lett. 86, 161115 (2005). [CrossRef]
- A. C. Busacca, A. C. Cino, S. Riva-Sanseverino, M. Ravaro and G. Assanto, "Silica masks for improved surface poling of lithium niobate," Electron. Lett. 41, 01393492 (2005). [CrossRef]

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