Abstract

Compression of linearly chirped picosecond pulses in hollow-core photonic bandgap fibers is investigated numerically. The modal properties of the fibers are modeled using the finite-element technique, whereas nonlinear propagation is described by a generalized nonlinear Schrödinger equation, which accounts both for the composite nature of the nonlinearity and the strong mode profile dispersion. Power limits for compression with more than 90% of the pulse energy in the main peak of the compressed pulse are investigated as a function of fiber design, and the temporal and spectral widths of the input pulse. The validity of approximate scaling rules is investigated, and figures of merit for fiber design are discussed.

© 2008 Optical Society of America

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  1. C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, "All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber," Opt. Express 11, 2832-2837 (2003).
    [CrossRef] [PubMed]
  2. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, "All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber," Opt. Express 11, 3332-3337 (2003).
    [CrossRef] [PubMed]
  3. C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006).
    [CrossRef]
  4. D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
    [CrossRef] [PubMed]
  5. D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, "Soliton pulse compression in photonic band-gap fibers," Opt. Express 13, 6153-6159 (2005).
    [CrossRef] [PubMed]
  6. C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007).
    [CrossRef] [PubMed]
  7. F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, "Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression," Opt. Express 15, 7126-7131 (2007).
    [CrossRef] [PubMed]
  8. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
    [CrossRef] [PubMed]
  9. M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
    [CrossRef]
  10. K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
    [CrossRef]
  11. J. Lægsgaard, "Mode profile dispersion in the generalised nonlinear Schr¨odinger equation," Opt. Express 15(24), 16,110-123 (2007).
    [CrossRef]
  12. J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, "Mode areas and field energy distribution in honeycomb photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2037-45 (2003).
    [CrossRef]
  13. J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, "Material effects in airguiding photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2046-51 (2003).
    [CrossRef]
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    [CrossRef]
  17. "JCMwave GmbH, www.jcmwave.com,".
  18. P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14, 7329-7341 (2006).
    [CrossRef] [PubMed]
  19. P. J. Roberts, "Birefringent hollow core fibers," Proc. SPIE 6782, 67821R (2007).
    [CrossRef]
  20. R. Amezcua-Correa, N. Broderick, M. Petrovich, F. Poletti, and D. Richardson, "Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation," Opt. Express 15, 17577-17586 (2007).
    [CrossRef] [PubMed]
  21. G. P. Agrawal, "Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers," Opt. Letters 15, 224-6 (1990).
    [CrossRef]
  22. F. Gérôme, J. Dupriez, J. C. Knight, and W. J. Wadsworth, "High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling," Opt. Express 16, 2381-2386 (2008).
    [CrossRef] [PubMed]

2008

2007

2006

C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006).
[CrossRef]

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14, 7329-7341 (2006).
[CrossRef] [PubMed]

2005

2004

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
[CrossRef]

2003

2000

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

1998

1997

1990

G. P. Agrawal, "Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers," Opt. Letters 15, 224-6 (1990).
[CrossRef]

1989

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, "Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers," Opt. Letters 15, 224-6 (1990).
[CrossRef]

Ahmad, F. R.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Amezcua-Correa, R.

Bird, D. M.

Birks, T. A.

Bjarklev, A.

Blow, K. J.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Broderick, N.

Broeng, J.

Cook, K.

De Matos, C. J. S.

Dudley, J. M.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

Dupriez, J.

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

Franco, M. A.

Gaeta, A. L.

Gallagher, M. T.

George, A. K.

Gerome, F.

Gérôme, F.

Grillon, G.

Hansen, K. P.

Hansen, T. P.

Harvey, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

Hensley, C. J.

Jespersen, K. G.

C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006).
[CrossRef]

Keiding, S. R.

C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006).
[CrossRef]

Knight, J. C.

Koch, K. W.

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
[CrossRef]

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

Lægsgaard, J.

Limpert, J.

Mangan, B. J.

Mlejnek, M.

Moloney, J. V.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, "Dynamic spatial replenishment of femtosecond pulses propagating in air," Opt. Lett. 23, 382-384 (1998).
[CrossRef]

Mortensen, N. A.

Muller, D.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Mysyrowicz, A.

Nibbering, E. T. J.

Nielsen, C. K.

C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006).
[CrossRef]

Nolte, S.

Ouzounov, D. G.

Petrovich, M.

Poletti, F.

Prade, B. S.

Richardson, D.

Riishede, J.

Roberts, P. J.

Russell, P. S. J.

Sabert, H.

Schreiber, T.

Silcox, J.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Taylor, J. R.

Thomas, M. G.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

Tunnermann, A.

Venkataraman, N.

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007).
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Venkateraman, N.

Wadsworth, W. J.

Williams, D. P.

Wood, D.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, "Dynamic spatial replenishment of femtosecond pulses propagating in air," Opt. Lett. 23, 382-384 (1998).
[CrossRef]

Zellmer, H.

Appl. Phys. B: Lasers Opt.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
[CrossRef]

IEEE J. Quantum Electron.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14, 7329-7341 (2006).
[CrossRef] [PubMed]

J. Lægsgaard, "Mode profile dispersion in the generalised nonlinear Schr¨odinger equation," Opt. Express 15(24), 16,110-123 (2007).
[CrossRef]

C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, "All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber," Opt. Express 11, 2832-2837 (2003).
[CrossRef] [PubMed]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, "All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber," Opt. Express 11, 3332-3337 (2003).
[CrossRef] [PubMed]

C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006).
[CrossRef]

D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, "Soliton pulse compression in photonic band-gap fibers," Opt. Express 13, 6153-6159 (2005).
[CrossRef] [PubMed]

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007).
[CrossRef] [PubMed]

F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, "Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression," Opt. Express 15, 7126-7131 (2007).
[CrossRef] [PubMed]

R. Amezcua-Correa, N. Broderick, M. Petrovich, F. Poletti, and D. Richardson, "Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation," Opt. Express 15, 17577-17586 (2007).
[CrossRef] [PubMed]

F. Gérôme, J. Dupriez, J. C. Knight, and W. J. Wadsworth, "High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling," Opt. Express 16, 2381-2386 (2008).
[CrossRef] [PubMed]

Opt. Lett.

Opt. Letters

G. P. Agrawal, "Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers," Opt. Letters 15, 224-6 (1990).
[CrossRef]

Phys. Rev. Lett.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000).
[CrossRef] [PubMed]

Proc.SPIE

P. J. Roberts, "Birefringent hollow core fibers," Proc. SPIE 6782, 67821R (2007).
[CrossRef]

Science

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Other

"JCMwave GmbH, www.jcmwave.com,".

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

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Figures (10)

Fig. 1.
Fig. 1.

Hollow-core PBG fiber structures investigated in the present work. More cladding layers than shown were included in the simulations to ensure confinement loss within the probed band gap region does not affect dispersion.

Fig. 2.
Fig. 2.

Dispersion curves for the HC-PBG fibers investigated in the present work. The scaling factor S=Λ/Λ0, where Λ0=2.53 µm for HC, 2.6 µm for HC2, and 2.5 µm for HC3. Red dots denote the position of the input center wavelength (1064 nm) on the dispersion curves of the fibers with scaled pitches investigated here.

Fig. 3.
Fig. 3.

Left panel: Total effective area at atmospheric pressure of the three HC-PBG designs studied. Right panel: Silica effective area of the HC-PBG fibers, i.e. effective area at zero air pressure. Red dots indicate position of the input wavelength in the scaled fibers. The areas of the fiber with a 19-cell core have been reduced by factors of 3 (left panel) and 10 (right panel) to facilitate comparison.

Fig. 4.
Fig. 4.

Values of the figure of merit F=|β 2|Aeff for the different fiber designs. The results for the evacuated HC3 fiber have been scaled down by a factor of 3 to facilitate comparison. Note that F scales with S 3, so for S≠1 the values in the figure should be scaled to compare with figures in the following subsections.

Fig. 5.
Fig. 5.

Normalized power of compressed pulses for different values of P 0. Input pulse parameters are t 0=6 ps, W=5 nm (left) and 10 nm (right).

Fig. 6.
Fig. 6.

Pulse energies (top row), peak powers (middle row) and temporal FWHM (bottom row), at maximal compression for Q=0.9 in the HC1 and HC2 fiber structures as a function of the figure of merit |β 2|Aeff . Left and right columns give data for air-filled and evacuated fibers, respectively. All input pulses have t 0=3 ps.

Fig. 7.
Fig. 7.

Same as Fig. 6 for the HC3 fiber.

Fig. 8.
Fig. 8.

Compression factors (upper row) and maximal peak power (lower row) versus Q for compression of t 0=3 ps pulses in the HC2 fiber. Left column shows results for air-filled fibers, right column for evacuated fibers. Data points from all fiber scalings investigated have been included in the plots.

Fig. 9.
Fig. 9.

Spectral densities of pulses compressed in the HC2 fiber, calculated with or without Raman scattering. The P 0 value in the air-filled and evacuated examples were chosen so that P 0/F was the same in the two cases.

Fig. 10.
Fig. 10.

Maximal peak powers for Q=0.9 compression in the HC1 (upper row) and HC2 (lower row) fibers for different input pulse durations scaled so that W t 0 and P 0 t 2 0 are constants. Left column shows results for air-filled fibers, right column for evacuated fibers. Legends indicate W and t 0. Note the logarithmic scale in the lower right panel.

Equations (30)

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

× E = μ 0 H t
× H = ε 0 ε ( r ) E t + P NL t
H ( r , t ) = 1 2 π m d ω G m ( z , ω ) h m ( r , ω ) exp ( i ( β m ( ω ) z ω t ) )
E ( r , t ) = 1 2 π m d ω G m ( z , ω ) e m ( r , ω ) exp ( i ( β m ( ω ) z ω t ) )
d r [ e m × h n * + e n * × h m ] · z ̂ = N m δ mn
N m G m ( z , ω ) z = i ω d r d t e m * ( r , t ; ω ) · P NL ( r , t ) ,
e m ( r , t ; ω ) = e m ( r , ω ) exp ( i ( β m ( ω ) z ω t ) ) .
P NL ( r , t ) = ε 0 d t R ( r , t t ) χ ( 3 ) ( r ) E ( r , t ) E ( r , t ) E ( r , t )
P NL ( r , t ) = ε 0 χ ( 3 ) ( r ) E ( r , t ) d t R ( r , t t ) E ( r , t ) 2
P NL ( r , t ) = ε 0 ( 2 π ) 3 npq d ω 1 3 d t G n ( z , ω 1 ) G p ( z , ω 2 ) G q * ( z , ω 3 ) ×
e n ( r , t ; ω 1 ) [ e p ( r , t ; ω 2 ) · e q * ( r , t ; ω 3 ) ] χ ( 3 ) ( r ) R ( r , t t )
N m G m ( z , ω ) z = i ω ε 0 ( 2 π ) 2 exp ( i β m ( ω ) z ) npq v = 1 N χ ν ( 3 )
d ω 1 2 G ˜ n ( z , ω 1 ) G ˜ p ( z , ω 2 ) G ˜ q * ( z , ω 1 + ω 2 ω ) R ν ( ω ω 1 ) ×
v d r [ e m * ( r , ω ) · e n ( r , ω 1 ) ] [ e p ( r , ω 2 ) · e q * ( r , ω 1 + ω 2 ω ) ] ,
G ˜ m ( z , ω ) = G m ( z , ω ) exp ( i β m ( ω ) z )
G m ( z , ω ) z = i ω ε 0 exp ( i β m ( ω ) z ) npq v = 1 N 3 χ v ( 3 )
1 ( 2 π ) 2 d ω 1 2 G ˜ n ( z , ω 1 ) G ˜ p ( z , ω 2 ) G ˜ q * ( z , ω 1 + ω 2 ω ) R v ( ω ω 1 ) ×
v d r [ e m * ( r , ω ) · e n ( r , ω 1 ) ] [ e p ( r , ω 2 ) · e q * ( r , ω 1 + ω 2 ω ) ]
d r [ e * ( r , ω ) · e ( r , ω 1 ) ] [ e ( r , ω 2 ) · e * ( r , ω 1 + ω 2 ω ) ]
[ C ( ω ) C ( ω 1 ) C ( ω 2 ) C ( ω 1 + ω 2 ω ) ] 1 4
C ( ω ) = [ d r e * ( r , ω ) 4 ] 1
n 2 ( v ) = 3 χ v ( 3 ) 4 n v 2 ε 0 c
G ( z , ω ) z = i ω c exp ( i β ( ω ) z ) v = 1 N n 2 ( v ) [ A eff ( v ) ( ω ) ] 1 4 ×
1 ( 2 π ) 2 d ω 1 2 G ̂ ( v ) ( z , ω 1 ) G ̂ ( v ) ( z , ω 2 ) G ̂ ( v ) * ( z , ω 1 + ω 2 ω ) R v ( ω ω 1 )
G ̂ ( v ) ( z , ω ) = G ˜ ( z , ω ) [ A eff ( v ) ( ω ) ] 1 4 , A eff ( v ) = C ( v ) 4 c 2 n v 2 ε 0 2 = μ 0 [ R e d r e × h * · z ̂ ] 2 ε 0 n v 2 v d r e ( r ) 4
G ˜ ( z c , τ ) z c = i 2 2 G ˜ ( z c , τ ) τ 2 + i n 2 S i O 2 ω 0 P 0 t 0 2 c β 2 A eff G ˜ ( z c , τ ) G ˜ ( z c , τ ) 2 , τ = t t 0 , z c = z β 2 t 0 2
1 A eff = n 2 air n 2 S i O 2 A eff air + 1 A eff S i O 2
β 3 6 t 0 β 2 3 G ˜ ( z c , τ ) τ 3 , β 3 = d 3 β d ω 3 ω 0
G ( t ) = P 0 [ 1 ( t t 0 ) 2 ] exp ( i ( C t 2 + ω 0 t ) ) , t < t 0 ; G ( t ) = 0 , t > t 0
C = ω 0 2 W 8 π c t 0

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