Abstract

We numerically and analytically evaluate the delay of solitons propagating slowly, and without broadening, in an apodized Bragg grating. Simulations indicate that a 100 mm Bragg grating with Δn=10-3 can delay sub-nanosecond pulses by nearly 20 pulse widths without any change in the output pulse width. Delay tunability is achieved by simultaneously adjusting the launch power and detuning. A simple analytic model is developed to describe the monotonic dependence of delay on Δn and compared with simulations. As the intensity may be greatly enhanced due to a reduced velocity, a procedure for improving the delay while avoiding material damage is outlined.

© 2006 Optical Society of America

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  1. Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
    [CrossRef]
  2. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  19. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006).
    [CrossRef]
  20. R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shawand, and I. D. Aggarwal, "Large Raman Gain and Nonlinear Phase Shifts in High-Purity As2Se3 Chalcogenide Fibers," J. Opt. Soc. Am. B 21, 1146-1155 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2006

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Application of Slow Light in Telecommunications," Opt. Photon. News 17, 18-23 (2006).
[CrossRef]

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006).
[CrossRef]

2005

2004

2001

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical Delay Lines Based on Optical Filters," J. Quantum Electron. 37, 525-530 (2001).
[CrossRef]

2000

N. G. R. Broderick, D. J. Richardson, and M. Ibsen, "Nonlinear Switching in a 20-Cm-Long Fiber Bragg Grating," Opt. Lett. 25, 536-538 (2000).
[CrossRef]

M. De Sario, C. Conti, and G. Assanto, "Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback," IEEE J. Quant. Elect. 36, 931-943 (2000).
[CrossRef]

1999

1998

1996

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1991

C. M. de Sterke, K. R. Jackson, and B. D. Robert, "Nonlinear Coupled Mode Equations on a Finite Interval: A Numerical Procedure," J. Opt. Soc. Am. B 8, 403-412 (1991).
[CrossRef]

P. S. J. Russell, "Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
[CrossRef]

1989

D. N. Christodoulides and R. I. Joseph, "Slow Bragg Solitons in Nonlinear Periodic Structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

A. Aceves and S. Wabnitz, "Self-Induced Transparency Solitons in Nonlinear Refractive Periodic Media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Aceves, A.

A. Aceves and S. Wabnitz, "Self-Induced Transparency Solitons in Nonlinear Refractive Periodic Media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Aggarwal, I. D.

Andersen, K. N.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Assanto, G.

M. De Sario, C. Conti, and G. Assanto, "Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback," IEEE J. Quant. Elect. 36, 931-943 (2000).
[CrossRef]

Bjarklev, A.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Borel, P. I.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Application of Slow Light in Telecommunications," Opt. Photon. News 17, 18-23 (2006).
[CrossRef]

Broderick, N. G. R.

Chang-Hasnain, C. J.

Chen, X.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
[CrossRef]

Chenb, R. T.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, "Slow Bragg Solitons in Nonlinear Periodic Structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Conti, C.

M. De Sario, C. Conti, and G. Assanto, "Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback," IEEE J. Quant. Elect. 36, 931-943 (2000).
[CrossRef]

De Sario, M.

M. De Sario, C. Conti, and G. Assanto, "Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback," IEEE J. Quant. Elect. 36, 931-943 (2000).
[CrossRef]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory," J. Opt. Soc. Am. B 16, 587-599 (1999).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke, K. R. Jackson, and B. D. Robert, "Nonlinear Coupled Mode Equations on a Finite Interval: A Numerical Procedure," J. Opt. Soc. Am. B 8, 403-412 (1991).
[CrossRef]

Eggleton, B. J.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006).
[CrossRef]

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, D. J. Moss, M. Rochette, I. C. M. Littler, B. J. Eggleton, Y. Ruan, and B. Luther-Davies, "Integrated All-Optical Pulse Regenerator in Chalcogenide Waveguides," Opt. Lett. 30, 2900-2902 (2005).
[CrossRef] [PubMed]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical Delay Lines Based on Optical Filters," J. Quantum Electron. 37, 525-530 (2001).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory," J. Opt. Soc. Am. B 16, 587-599 (1999).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Fage-Pedersen, J.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Frandsen, L. H.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Fu, L.

Gaeta, A. L.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Application of Slow Light in Telecommunications," Opt. Photon. News 17, 18-23 (2006).
[CrossRef]

Gauthier, D. J.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Application of Slow Light in Telecommunications," Opt. Photon. News 17, 18-23 (2006).
[CrossRef]

Gu, L.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
[CrossRef]

Hansen, O.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Hodelin, J.

Ibsen, M.

Jackson, K. R.

Jacobsen, R. S.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Jiang, W.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
[CrossRef]

Jiang, Y.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, "Slow Bragg Solitons in Nonlinear Periodic Structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Khurgin, J. B.

Kristensen, M.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Ku, P.-C.

Laming, R. I.

Lavrinenko, A. V.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Lenz, G.

Littler, I. C. M.

Luther-Davies, B.

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical Delay Lines Based on Optical Filters," J. Quantum Electron. 37, 525-530 (2001).
[CrossRef]

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006).
[CrossRef]

Moss, D. J.

Moulin, G.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Ou, H.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Peucheret, C.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Povinelli, M. L.

Richardson, D. J.

Robert, B. D.

Rochette, M.

Ruan, Y.

Russell, P. S. J.

P. S. J. Russell, "Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
[CrossRef]

Sanghera, J.

Shawand, L. B.

Shokooh-Saremi, M.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shawand, and I. D. Aggarwal, "Large Raman Gain and Nonlinear Phase Shifts in High-Purity As2Se3 Chalcogenide Fibers," J. Opt. Soc. Am. B 21, 1146-1155 (2004).
[CrossRef]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical Delay Lines Based on Optical Filters," J. Quantum Electron. 37, 525-530 (2001).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory," J. Opt. Soc. Am. B 16, 587-599 (1999).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Ta’eed, V. G.

Taverner, D.

Tucker, R. S.

Wabnitz, S.

A. Aceves and S. Wabnitz, "Self-Induced Transparency Solitons in Nonlinear Refractive Periodic Media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Zsigri, B.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005).
[CrossRef]

IEEE J. Quant. Elect.

M. De Sario, C. Conti, and G. Assanto, "Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback," IEEE J. Quant. Elect. 36, 931-943 (2000).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

P. S. J. Russell, "Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
[CrossRef]

J. Opt. Soc. Am. B

J. Quantum Electron.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical Delay Lines Based on Optical Filters," J. Quantum Electron. 37, 525-530 (2001).
[CrossRef]

Nature

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained Silicon as a New Electro-Optic Material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Nature Phys.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photon. News

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Application of Slow Light in Telecommunications," Opt. Photon. News 17, 18-23 (2006).
[CrossRef]

Phys. Lett. A

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Figures (4)

Fig. 1.
Fig. 1.

Illustration of (a) nonlinear bandgap shift, and the launching and evolution of (b) low- and (c) high-intensity pulses within the bandgap of a Bragg grating spanning 0<z<0.1 m. Contour lines representing the intensity are equally spaced on a logarithmic scale. At a sufficiently high launch intensity, the pulse propagates in the Bragg grating, as shown in (c), at a velocity less than c/n and emerges with a pulse delay.

Fig. 2.
Fig. 2.

(a) Definitions of κ, δ′ and δp . Solid line represents the Bragg grating transmission spectrum as a function of δ. Dotted line represents the pulse spectrum. (b) Typical effects of different launch peak powers P on the output pulse width and delay. P0 is the power that results in the same input and output pulse widths Δτ, and τd is the propagation delay under this condition. Output pulses (solid) are drawn to have the same pulse energy as the input (dashed).

Fig. 3.
Fig. 3.

(a) Delay and (b) launch power as a function of Δn in a 100 mm Bragg grating for 680 ps pulses for five detunings (1 kW corresponds to 3.65 GW/cm2). Data points (with error bars) in (a) show the simulated delay, whereas each dashed line is an analytical fit using Eqs. (5) and (6), with the fit parameters summarised in Table 1. Inset(top): Transmission vs delay at Δn=10-3. Inset(bottom): Incident (dashed) and transmitted (solid) pulses at Δn=10-3 with the detuning and launch peak power indicated by the grey box: (i) δ′=κ-1.2δp , P 0=0.393 kW (ii) δ′=κ-1.5δp , P0=0.463 kW (iii) δ′=κ-2.0δp , P 0=0.577 kW (iv) δ′=κ-2.5δp , P 0=0.691 kW and (v) δ′=κ-3.0δp , P 0=0.805 kW.

Fig. 4.
Fig. 4.

Results obtained from the gap soliton solutions for Δτ GS=680 ps in a L=100 mm Bragg grating.(a) Calculated delay vs Δn for different detunings, and (b) calculated delay (×) and total peak intensity (˚) vs detuning at Δn=10-3.

Tables (1)

Tables Icon

Table 1. Fitted Δδ used in Fig. 3(a) and calculated gap soliton intensity I tot for each of the five detunings for 680 ps pulses. δp =14.09 m-1 is the spectral width of the pulse.

Equations (10)

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λ B = 2 n Λ ,
n = n 0 + n 2 I ,
i A + z + i n c A + t + κ A + γ ( A + 2 + 2 A 2 A + ) = 0
i A z + i n c A t + κ A + + γ ( A + 2 + 2 A 2 A ) = 0
q 2 = δ 2 κ 2 ,
V g = c n [ 1 ( κ δ ) 2 ] 1 2 .
τ d τ d 0 = L ( 1 V g n c ) .
Δ τ GS = cosh 1 ( 2 + cos δ ~ ) γ ~ κ v ( c n ) sin δ ~ ,
δ γ ~ κ cos δ ~ .
I ± = ( λ 2 π n 2 ) κ Δ 2 sin 2 δ ~ ( 1 v 2 3 v 2 ) ( 2 1 + cos δ ~ ) ,

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