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 [1]. Here we investigate highly-confined nanophotonic waveguides designed to maximize the SPDC rate. A theory modified to include highly-confined waveguides is used to calculate the spectral width and pair generation rates in a sample system. Pair generation rates exceeding 109/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

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  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. 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]
  3. 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]
  4. 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]
  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 KTiOPO4 parametric downconverter,” Phys. Rev. A 69, 013807 (2004).
    [Crossref]
  6. D. A. Kleinman, “Theory of Optical Parametric Noise,” Phys. Rev. 174, 1027 (1968).
    [Crossref]
  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. 31, 769–781 (1995).
    [Crossref]
  8. H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optical coefficients of KiTiOPO4,” 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]
  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]
  11. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
    [Crossref]
  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]

2007 (1)

2005 (2)

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]

2004 (3)

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]

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]

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

2001 (1)

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]

1999 (1)

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]

1995 (2)

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]

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]

1992 (1)

1989 (1)

1984 (1)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[Crossref]

1968 (1)

D. A. Kleinman, “Theory of Optical Parametric Noise,” Phys. Rev. 174, 1027 (1968).
[Crossref]

Appelbaum, I.

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]

Assanto, G.

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]

Banaszek, K.

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]

Battle, P.

Beausoleil, R. G.

Bierlein, J. D.

Busacca, A. C.

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]

Chakmakjian, S. H.

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]

Cheung, E. C.

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]

Cino, A. C.

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]

Eberhard, P. H.

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]

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[Crossref]

Fiorentino, M.

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]

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

Gunter, P.

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]

Kawahara, K.

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]

Kleinman, D. A.

D. A. Kleinman, “Theory of Optical Parametric Noise,” Phys. Rev. 174, 1027 (1968).
[Crossref]

Koch, K.

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]

Kuga, T.

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]

Kuklewicz, C. E.

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

Kwiat, P. G.

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]

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]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[Crossref]

Liu, J. M.

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]

Mattle, K.

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]

Messin, G.

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

Moore, G. T.

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]

Munro, M. W.

Rabiei, P.

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

Ravaro, M.

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]

Riva-Sanseverino, S.

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]

Roberts, T. D.

Sanaka, K.

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]

Shapiro, J. H.

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

SIlberhorn, C.

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]

Spillane, S. M.

Steier, W. H.

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

U’Ren, A. B.

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]

Vanherzeele, H.

Waks, E.

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]

Walmsley, I. A.

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]

Weinfurther, H.

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]

Whitel, A. G.

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]

Wong, F. N. C.

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

Zeilinger, A.

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]

Appl. Phys. Lett. (2)

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]

Electron. Lett. (1)

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]

IEEE J. Quantum Electron. (1)

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]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[Crossref]

Phys. Rev. (1)

D. A. Kleinman, “Theory of Optical Parametric Noise,” Phys. Rev. 174, 1027 (1968).
[Crossref]

Phys. Rev. A (2)

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]

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 downconverter,” Phys. Rev. A 69, 013807 (2004).
[Crossref]

Phys. Rev. Lett. (3)

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]

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]

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]

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Figures (5)

Fig. 1.
Fig. 1.

Electric field profile for Type II SPDC in a low index contrast Rb ion-exchanged PPKTP waveguide for both the pump mode (λ= 405 nm, Y-polarized, left panel) and idler mode (λ = 810 nm, Z-polarized, right panel), after [1]. Inset shows the electric field profile for a highly-confined nanophotonic ridge PPKTP waveguide (w=450 nm, h=500 nm) for the same pump and signal wavelengths.

Fig. 2.
Fig. 2.

Effective index for pump mode (top left panel), signal mode (top center panel), and idler mode (top right panel), for various fixed waveguide widths {300,400,500,600} nm and various heights. The effective index increases as waveguide width and/or height increases, as expected. For comparison, the material index is {1.840,1.758,1.843} for the pump, signal, and idler, respectively. Bottom panels show (from left to right) the magnitude of the electric field distribution for the pump, signal, and idler modes for a 500 nm by 500 nm waveguide.

Fig. 3.
Fig. 3.

First order quasi-phase matching period for Type II SPDC in PPKTP versus waveguide height for waveguide widths of {300,400,500,600} nm.

Fig. 4.
Fig. 4.

Effective nonlinear interaction area for a nanophotonic KTP ridge waveguide surrounded by a linear dielectric (index 1.45) for Type II SPDC (405 nm pump Y-polarized, 810 nm signal Y-polarized, 810 nm idler Z-polarized). The area is shown for waveguide widths of {300,400,500,600} nm versus waveguide heights ranging from 200 to 550 nm. The calculations show that the interaction area has a minimum value of approximately 0.4 μm2 for waveguide widths around 500 – 600 nm and waveguide heights of 450 – 550 nm.

Fig. 5.
Fig. 5.

Signal photon generation rate and peak power spectral density (Δβ = 0) for Type II SPDC (405 nm pump Y-polarized, 810 nm signal Y-polarized, 810 nm idler Z-polarized) in a nanophotonic KTP ridge waveguide surrounded by a linear dielectric (index 1.45). The calculations assume a first-order QPM crystal of length 10 mm, and a pump power of 1 mW, with a detection bandwidth of 1 nm. The calculations are shown for waveguide core widths of {300,400,500,600} nm versus waveguide heights ranging from 200 to 550 nm. The calculations show that the highest photon generation rates occur for a waveguide core dimension of 500 nm by 500 nm, with rates exceeding 6 × 109 photons/sec. The corresponding spectral density exceeds 1600 pW/nm.

Tables (2)

Tables Icon

Table 1. Comparison of nanophotonic waveguides, low-index waveguides, and bulk crystals of PPKTP for Type II SPDC from 405 nm to 810 nm. All calculations assume a crystal length of 10 mm. Nanophotonic waveguides are predicted to have efficiency improvements of 45× over low-index waveguides and more than 6500× that of bulk crystals.

Tables Icon

Table 2. Predicted peak signal photon rates, powers, and efficiencies for Type I SPDC from a 405 nm pump to {1550,548} nm and {1310,586} nm signal and idler pairs in a nanophotonic PPLN waveguide. All calculations assume a crystal length of 10 mm.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E P ( r , t ) = P P 2 ε 0 c n e , P ( ϕ P ( r T ) e i ( ω P t n e , P k P z ) + h . c . )
E { S , I } ( r , t ) = h ¯ ω { S , I } 2 ε 0 L n e , { S , I } 2 ( ϕ { S , I } ( r T ) a { S , I } ( t ) e i ( ω { S , I } t n e , { S , I } k { S , I } z ) + h . c . )
A T ϕ { P , S , I } ( r T ) 2 d 2 r T = 1
H I = 4 ε 0 d eff V ( E P E S E I ) d 3 r
H I = 8 P P h ̅ 2 c π 2 d eff 2 ε 0 n e , P n e , S 2 n e , I 2 L 2 λ S λ I ( V ϕ P ( r T ) ϕ S * ( r T ) ϕ I * ( r T ) a S a I e i Δ β z + h . c . )
R signal = 8 π 2 d eff 2 ε 0 n e , P n e , S n e , I λ S 3 λ I sin c 2 ( Δ β L C 2 ) L C 2 P P A I δ λ S
A I ( A NL ϕ P ( r T ) ϕ S * ( r T ) ϕ I * ( r T ) d 2 r T ) 2
P signal = 8 hc π 2 d eff 2 ε 0 n e , P n e , S n e , I λ S 4 λ I sin c 2 ( Δ β L C 2 ) L C 2 P P A I δ λ S
d eff = 2 m π d eff , bulk
Δ β = 2 π ( n e , P λ P n e , S λ S n e , I λ I m Λ )
( Δ λ S ) FWHM = 0.886 λ S 2 L C ( n e , S n e , I ) λ S d n e , s d λ S + λ I d n e , I d λ I
n x 2 ( λ ) = 2.1146 + 0.89188 [ 1 ( 0.20861 λ ) 2 ] 0.01320 λ 2
n y 2 ( λ ) = 2.1518 + 0.87862 [ 1 ( 0.21801 λ ) 2 ] 0.01327 λ 2
n z 2 ( λ ) = 2.3136 + 1.00012 [ 1 ( 0.23831 λ ) 2 ] 0.01679 λ 2
n o 2 ( λ ) = 4.9048 + 0.1177 ( λ 2 0.0475 ) 0.0272 λ 2
n e 2 ( λ ) = 4.5820 + 0.0992 ( λ 2 0.0444 ) 0.0219 λ 2

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