Abstract

A thermal reflow technique is applied to high-index-contrast, sub-micron waveguides in As2S3 chalcogenide glass to reduce the sidewall roughness and associated optical scattering loss. We show that the reflow process effectively decreases sidewall roughness of chalcogenide glass waveguides. A kinetic model is presented to quantitatively explain the sidewall roughness evolution during thermal reflow. Further, we develop a technique to calculate waveguide optical loss using the roughness evolution model, and predict the ultimate low loss limit in reflowed high-index-contrast glass waveguides. Up to 50% optical loss reduction after reflow treatment is experimentally observed, and the practical loss limiting factors are discussed.

© 2010 OSA

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References

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2008

2007

2006

2005

A. L. Greer and N. Mathur, “Materials science: changing face of the chameleon,” Nature 437(7063), 1246–1247 (2005).
[CrossRef] [PubMed]

T. Barwicz and H. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005).
[CrossRef]

S. Ramachandran and S. Bishop, “Photoinduced integrated-optic devices in rapid thermally annealed chalcogenide glasses,” IEEE J. Sel. Top. Quantum Electron. 11(1), 260–270 (2005).
[CrossRef]

P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. St J Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
[CrossRef] [PubMed]

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

2002

1999

S. Ramachandran and S. Bishop, “Low loss photoinduced waveguides in rapid thermally annealed films of chalcogenide glasses,” Appl. Phys. Lett. 74(1), 13–15 (1999).
[CrossRef]

1996

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

1993

R. Syms and A. Holmes, “Reflow and Burial of Channel Waveguides Formed in Sol-Gel Glass on Si Substrates,” IEEE Photon. Technol. Lett. 5(9), 1077–1079 (1993).
[CrossRef]

1990

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137, 282–288 (1990).

1980

S. Dutta, H. Jackson, and J. Boyd, “Reduction of scattering from a glass thin-film optical waveguide by CO2 laser annealing,” Appl. Phys. Lett. 37(6), 512–514 (1980).
[CrossRef]

1971

1969

D. Marcuse, “Radiation Losses of Dielectric Waveguides in Terms of the Power Spectrum of the Wall Distortion Function,” Bell Syst. Tech. J. 48, 3233 (1969).

Agarwal, A.

Allen, P. J.

Anheier, N. C.

Antoine, K.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Baker, N. J.

Bar?, M.

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

Barwicz, T.

Birks, T.

Bishop, S.

S. Ramachandran and S. Bishop, “Photoinduced integrated-optic devices in rapid thermally annealed chalcogenide glasses,” IEEE J. Sel. Top. Quantum Electron. 11(1), 260–270 (2005).
[CrossRef]

S. Ramachandran and S. Bishop, “Low loss photoinduced waveguides in rapid thermally annealed films of chalcogenide glasses,” Appl. Phys. Lett. 74(1), 13–15 (1999).
[CrossRef]

Boyd, J.

S. Dutta, H. Jackson, and J. Boyd, “Reduction of scattering from a glass thin-film optical waveguide by CO2 laser annealing,” Appl. Phys. Lett. 37(6), 512–514 (1980).
[CrossRef]

Carlie, N.

Choi, D.

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

Choi, D. Y.

Couny, F.

Dutta, S.

S. Dutta, H. Jackson, and J. Boyd, “Reduction of scattering from a glass thin-film optical waveguide by CO2 laser annealing,” Appl. Phys. Lett. 37(6), 512–514 (1980).
[CrossRef]

Eggleton, B.

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

Eggleton, B. J.

Farr, L.

Feng, N. N.

Feng, N.-N.

Finsterbusch, K.

Fu, L.

Greer, A. L.

A. L. Greer and N. Mathur, “Materials science: changing face of the chameleon,” Nature 437(7063), 1246–1247 (2005).
[CrossRef] [PubMed]

Haus, H.

Hô, N.

Holmes, A.

R. Syms and A. Holmes, “Reflow and Burial of Channel Waveguides Formed in Sol-Gel Glass on Si Substrates,” IEEE Photon. Technol. Lett. 5(9), 1077–1079 (1993).
[CrossRef]

Hu, J.

Huang, W.-P.

Illekova, E.

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

Jackson, H.

S. Dutta, H. Jackson, and J. Boyd, “Reduction of scattering from a glass thin-film optical waveguide by CO2 laser annealing,” Appl. Phys. Lett. 37(6), 512–514 (1980).
[CrossRef]

Jain, H.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Kimerling, L.

Knight, J.

Krishnaswami, K.

Lacey, J.

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137, 282–288 (1990).

Lamont, M.

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

Lamont, M. R.

Li, W.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Lopez, C.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Luther-Davies, B.

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

V. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

R. Wang, S. Madden, C. Zha, A. Rode, and B. Luther-Davies, “Annealing induced phase transformation in amorphous As2S3 films,” J. Appl. Phys. 100(6), 063524 (2006).
[CrossRef]

Madden, S.

V. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

R. Wang, S. Madden, C. Zha, A. Rode, and B. Luther-Davies, “Annealing induced phase transformation in amorphous As2S3 films,” J. Appl. Phys. 100(6), 063524 (2006).
[CrossRef]

Mangan, B.

Marcuse, D.

D. Marcuse, “Radiation Losses of Dielectric Waveguides in Terms of the Power Spectrum of the Wall Distortion Function,” Bell Syst. Tech. J. 48, 3233 (1969).

Mason, M.

Mathur, N.

A. L. Greer and N. Mathur, “Materials science: changing face of the chameleon,” Nature 437(7063), 1246–1247 (2005).
[CrossRef] [PubMed]

Miller, A.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Moss, D. J.

Myers, T. L.

Myneni, S.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Nguyen, H. C.

Payne, F.

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137, 282–288 (1990).

Pelusi, M.

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

Petit, L.

Phillips, M. C.

Pope, A.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Poulain, M.

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

Qiao, H.

Ramachandran, S.

S. Ramachandran and S. Bishop, “Photoinduced integrated-optic devices in rapid thermally annealed chalcogenide glasses,” IEEE J. Sel. Top. Quantum Electron. 11(1), 260–270 (2005).
[CrossRef]

S. Ramachandran and S. Bishop, “Low loss photoinduced waveguides in rapid thermally annealed films of chalcogenide glasses,” Appl. Phys. Lett. 74(1), 13–15 (1999).
[CrossRef]

Richardson, K.

Riley, B. J.

Rivero, C.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Roberts, P.

Rode, A.

R. Wang, S. Madden, C. Zha, A. Rode, and B. Luther-Davies, “Annealing induced phase transformation in amorphous As2S3 films,” J. Appl. Phys. 100(6), 063524 (2006).
[CrossRef]

Sabert, H.

Schulte, A.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

Seal, S.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

St J Russell, P.

Surinach, S.

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

Syms, R.

R. Syms and A. Holmes, “Reflow and Burial of Channel Waveguides Formed in Sol-Gel Glass on Si Substrates,” IEEE Photon. Technol. Lett. 5(9), 1077–1079 (1993).
[CrossRef]

Ta’eed, V.

Ta'eed, V.

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

Tarasov, V.

Tien, P.

Tomlinson, A.

Wang, R.

R. Wang, S. Madden, C. Zha, A. Rode, and B. Luther-Davies, “Annealing induced phase transformation in amorphous As2S3 films,” J. Appl. Phys. 100(6), 063524 (2006).
[CrossRef]

Williams, D.

Xu, C.

Zha, C.

R. Wang, S. Madden, C. Zha, A. Rode, and B. Luther-Davies, “Annealing induced phase transformation in amorphous As2S3 films,” J. Appl. Phys. 100(6), 063524 (2006).
[CrossRef]

Zhang, G.

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

Zhou, G.-R.

Appl. Opt.

Appl. Phys. Lett.

S. Dutta, H. Jackson, and J. Boyd, “Reduction of scattering from a glass thin-film optical waveguide by CO2 laser annealing,” Appl. Phys. Lett. 37(6), 512–514 (1980).
[CrossRef]

S. Ramachandran and S. Bishop, “Low loss photoinduced waveguides in rapid thermally annealed films of chalcogenide glasses,” Appl. Phys. Lett. 74(1), 13–15 (1999).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, “Radiation Losses of Dielectric Waveguides in Terms of the Power Spectrum of the Wall Distortion Function,” Bell Syst. Tech. J. 48, 3233 (1969).

IEE Proc. J.

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137, 282–288 (1990).

IEEE J. Sel. Top. Quantum Electron.

S. Ramachandran and S. Bishop, “Photoinduced integrated-optic devices in rapid thermally annealed chalcogenide glasses,” IEEE J. Sel. Top. Quantum Electron. 11(1), 260–270 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

R. Syms and A. Holmes, “Reflow and Burial of Channel Waveguides Formed in Sol-Gel Glass on Si Substrates,” IEEE Photon. Technol. Lett. 5(9), 1077–1079 (1993).
[CrossRef]

M. Pelusi, V. Ta'eed, M. Lamont, S. Madden, D. Choi, B. Luther-Davies, and B. Eggleton, “Ultra-High Nonlinear As2S3 Planar Waveguide for 160-Gb/s Optical Time-Division Demultiplexing by Four-Wave Mixing,” IEEE Photon. Technol. Lett. 19(19), 1496–1498 (2007).
[CrossRef]

J. Appl. Phys.

W. Li, S. Seal, C. Rivero, C. Lopez, K. Richardson, A. Pope, A. Schulte, S. Myneni, H. Jain, K. Antoine, and A. Miller, “Role of S/Se ratio in chemical bonding of As-S-Se glasses investigated by Raman, x-ray photoelectron, and extended x-ray absorption fine structure spectroscopies,” J. Appl. Phys. 98(5), 053503 (2005).
[CrossRef]

R. Wang, S. Madden, C. Zha, A. Rode, and B. Luther-Davies, “Annealing induced phase transformation in amorphous As2S3 films,” J. Appl. Phys. 100(6), 063524 (2006).
[CrossRef]

J. Lightwave Technol.

J. Mater. Res.

S. Surińach, E. Illekova, G. Zhang, M. Poulain, and M. Barό, “Optical fiber drawing temperature of fluorogallate glasses,” J. Mater. Res. 11(10), 2633–2640 (1996).
[CrossRef]

Nature

A. L. Greer and N. Mathur, “Materials science: changing face of the chameleon,” Nature 437(7063), 1246–1247 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Other

Patterned As2S3 films are highly susceptible to surface oxidation and thus need to be protected by polymer coatings (e.g. SU8) for long-term stability. Our XPS study has confirmed the presence of surface As2Ox oxides on as-patterned As2S3 waveguides exposed in ambient air for just a few hours.

http://www.amorphousmaterials.com/IR%20Fibers.htm

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic illustration of a glass surface with sinusoidal roughness characterized by a spatial period L; (b) calculated internal pressure due to surface tension in glass with a rough surface.

Fig. 2
Fig. 2

(a) Autocorrelation functions of waveguide sidewall roughness before and after reflow treatment predicted by the kinetic analysis: the black curve is the initial roughness in as-patterned waveguide (assume a correlation length Lc = 100 nm and roughness variance σ = 10 nm), the red curve corresponds to roughness after reflow treatment for a duration t0 = μ/T·50 nm, and the green curve is the numerical fit of the roughness autocorrelation after reflow using an exponential model; (b) scattering loss evolution as a function of reflow time (given in μ/T·nm) for TE polarization light in a 500 nm thick As2S3 slab waveguide (index contrast Δn = 0.92) with a rough top surface, calculated using the Payne-Lacey model and the autocorrelation functions predicted by the reflow kinetic theory. The ultimate loss is limited by equi-partition of energy in surface capillary wave modes. The inset gives the assumed initial roughness parameters and the slab waveguide configuration.

Fig. 3
Fig. 3

Surface morphology of As2S3 chalcogenide waveguides measured by AFM: (a) as-patterned; (b) reflowed at 230 °C for 15 s exhibiting reduced sidewall roughness; and (c) reflowed at 245 °C for 15 s showing significant cross-sectional geometry modification.

Fig. 4
Fig. 4

(a). A top-view SEM micrograph showing the presence of edge roughness on photo resist mask prior to lift-off; (b). waveguide roughness amplitude as a function of roughness wavelength for different reflow temperatures: the olive curve is the roughness amplitude after 15 s reflow at 220 °C predicted by the reflow kinetic theory based on as-patterned roughness values.

Fig. 5
Fig. 5

Measured As2S3 waveguide loss ( ± 10%) after 15 s reflow treatment at different temperatures. The optical loss values in as-patterned As2S3 waveguides are represented by the two horizontal lines.

Fig. 6
Fig. 6

Measured As2S3 waveguide loss after 15 s reflow at 245 °C: the increased optical loss at longer wavelength is characteristic of radiative loss mechanisms but is not expected by the crystalline precipitation scattering hypothesis.

Tables (1)

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Table 1 Simulated optical radiative loss at 1550 nm wavelength in as-patterned and reflowed As2S3 waveguides

Equations (20)

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H ' ( z ) = H L sin ( 2 π L z )
Continuity equation:   v z z + v x x = 0
Force  equilibrium:   μ 2 v x z 2 = p x
μ 2 v z x 2 = p z
p s u r f a c e = T 2 H ' z 2 = 4 π 2 L 2 T H L sin ( 2 π L z )
β 2 π λ c l a d < L < β + 2 π λ c l a d
v z = 2 π L T H L μ × exp ( 2 π L x ) × cos ( 2 π L z )
v x = 2 π L T H L μ × exp ( 2 π L x ) × sin ( 2 π L z )
p = 4 π 2 L 2 T H L × exp ( 2 π L x ) × sin ( 2 π L z )
t = L μ 2 π T
h k ( t = 0 ) = 1 2 z 0 z 0 z 0 H ( z , t = 0 ) exp ( i π k z 0 z ) d z
H ( z , t ) = k = + h k ( t = 0 ) exp ( i π k z 0 z π T k z 0 μ t )
R ( z ' ) = σ 2 exp ( | z ' | L c )
α s = φ 2 ( d 2 ) ( n 1 2 n 2 2 ) k 0 3 8 π n 1 0 π R ¯ ( β n 2 k 0 cos θ ) d θ
φ ( x ) d x = 1
R ¯ ( ξ ) = R ( z ' ) exp ( i ξ z ' ) d z '
Q c r = λ 0 ( Δ λ ) 3 d B = π n g α λ 0
α R = 24 π 3 N p V 2 ( n 2 n 0 2 n 2 + 2 n 0 2 ) 2 ( n 0 λ ) 4 Γ
R ( z ' ) = H ˜ ( z ) H ˜ ( z + z ' ) z ' < < z 0 1 2 z 0 z 0 z 0 H ( z ) H ( z + z ' ) d z
k = | h k | 2 exp ( i π k z 0 z ' ) = 1 2 z 0 z 0 z 0 H ( z ) H ( z + z ' ) d z

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