Abstract

The formulation of Schrödinger-like equations for nonlinear pulse propagation in a single-mode microstructured optical fiber with a strongly frequency-dependent guided-mode profile is investigated. A correct account of mode profile dispersion in general necessiates a generalization of the effective area concept commonly used in the generalized nonlinear Schrödinger equation (GNLSE). A numerical scheme to this end is developed, and applied to a solid-core photonic bandgap fiber as a test case. It is further shown, that a simple reformulation of the GNLSE, expressed only in terms of the traditional frequency-dependent effective area, yields a good agreement with the more complete theory.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
  2. A. Fuerbach, P. Steinvurzel, J. Bolger, A. Nulsen, and B. Eggleton, "Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers," Opt. Lett. 30, 830-832 (2005).
    [CrossRef]
  3. A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, "Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers," Opt. Express 13, 2977-2987 (2005).
    [CrossRef]
  4. D. Ouzounov, F. Ahmad, D. Muller, N. Venkataraman, M. Gallagher, M. Thomas, J. Silcox, K. Koch, and A. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
    [CrossRef]
  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]
  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]
  7. F. Gerome, K. Cook, A. George, W. Wadsworth, and J. Knight, "Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression," Opt. Express 15, 7126-7131 (2007).
    [CrossRef]
  8. 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]
  9. N. Karasawa, S. Nakamura, and N. Nakagawa, "Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband," IEEE J. Quantum Electron. 37, 398-404 (2001).
    [CrossRef]
  10. G. Chang, T. B. Norris, and H. G. Winful, "Optimization of supercontinuum generation in photonic crystal fibers for pulse compression," Opt. Lett. 28, 546-548 (2003).
    [CrossRef]
  11. B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
    [CrossRef]
  12. P. Mamyshev and S. Chernikov, "Ultrashort-pulse propagation in optical fibers," Opt. Lett. 15, 1076-1078 (1990).
    [CrossRef]
  13. M. Kolesik, E. Wright, and J. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers," Appl. Phys. B: Lasers Opt. 79, 293-300 (2004).
    [CrossRef]
  14. A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 16,601 (2005).
  15. Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, "Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect," Phys. Rev. A 72, 63,802 (2005).
  16. K. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
    [CrossRef]
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
  18. J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, "All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm," J. Opt. A: Pure and Applied Optics 6, 667-70 (2004).
    [CrossRef]
  19. A. Argyros, T. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005).
    [CrossRef] [PubMed]
  20. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005).
    [CrossRef] [PubMed]
  21. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320-1333 (2002).
    [PubMed]
  22. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003).
    [CrossRef] [PubMed]
  23. J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A: Pure and Applied Optics 6, 798-804 (2004).
    [CrossRef]
  24. J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 10, 662-664 (1986).
    [CrossRef]

2007 (2)

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).

2005 (7)

A. Fuerbach, P. Steinvurzel, J. Bolger, A. Nulsen, and B. Eggleton, "Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers," Opt. Lett. 30, 830-832 (2005).
[CrossRef]

A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, "Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers," Opt. Express 13, 2977-2987 (2005).
[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]

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, "Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect," Phys. Rev. A 72, 63,802 (2005).

A. Argyros, T. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005).
[CrossRef] [PubMed]

G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005).
[CrossRef] [PubMed]

2004 (3)

J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A: Pure and Applied Optics 6, 798-804 (2004).
[CrossRef]

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

J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, "All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm," J. Opt. A: Pure and Applied Optics 6, 667-70 (2004).
[CrossRef]

2003 (4)

2002 (1)

2001 (1)

N. Karasawa, S. Nakamura, and N. Nakagawa, "Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband," IEEE J. Quantum Electron. 37, 398-404 (2001).
[CrossRef]

1990 (1)

1989 (1)

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

1986 (1)

J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 10, 662-664 (1986).
[CrossRef]

Appl. Phys. B (1)

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Appl. Phys. B: Lasers Opt. (1)

M. Kolesik, E. Wright, and J. 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. (2)

N. Karasawa, S. Nakamura, and N. Nakagawa, "Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband," IEEE J. Quantum Electron. 37, 398-404 (2001).
[CrossRef]

K. 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. A: Pure and Applied Optics (2)

J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A: Pure and Applied Optics 6, 798-804 (2004).
[CrossRef]

J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, "All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm," J. Opt. A: Pure and Applied Optics 6, 667-70 (2004).
[CrossRef]

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

Opt. Express (8)

A. Argyros, T. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005).
[CrossRef] [PubMed]

A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, "Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers," Opt. Express 13, 2977-2987 (2005).
[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]

G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (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]

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

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003).
[CrossRef] [PubMed]

A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320-1333 (2002).
[PubMed]

Opt. Lett. (4)

Phys. Rev. A (1)

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, "Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect," Phys. Rev. A 72, 63,802 (2005).

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).

Science (1)

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

Other (2)

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 16,601 (2005).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Guided-mode profiles in the all-silica PBG fiber described in the text. Black circles denote high-index rods. Profiles for V=3.95 (left), V=4.5 (middle) and V=5 (right) are shown. The computational domain shown in the figure is discretized in 250×250 gridpoints.

Fig. 2.
Fig. 2.

Basis functions, as in Eq. (28), derived by fitting the fields calculated by the finitedifference method. The plots show F 0 (left), F 1 (middle) and F 2 (right) for an M=2 expansion.

Fig. 3.
Fig. 3.

Effective area for a PBG fiber as described in the text with Λ=12 µm. The black curve denotes the results of the semivectorial finite-difference calculation, whereas the red and green curves denote results with M=2 and M=3 polynomial expansions for the fields.

Fig. 4.
Fig. 4.

Dispersion curve for the fiber structure described in the text with Λ=12 µm, and material dispersion included in the calculation.

Fig. 5.
Fig. 5.

Average wavelength of a soliton with a peak power of 100 kW launched at 1100 nm as a function of propagation distance. Inset shows the shifts after 10 meters of propagation as a function of peak power, P 0, calculated with the GNLSE and M-GNLSE.

Fig. 6.
Fig. 6.

Spectral weight of a soliton launched at 1100 nm with a peak power of 100 kW after 10 meters of propagation. |G(z,ω)|2 has been multiplied by c/λ 2 to convert into units of energy per wavelength.

Fig. 7.
Fig. 7.

Spectrum of a Gaussian pulse of peak power 50 kW, initial width of 0.93 ps and a center wavelength of 1025 nm after 60 cm of propagation.

Fig. 8.
Fig. 8.

Spectrum of a Gaussian pulse of peak power 40 kW, initial width of 11 ps and a center wavelength of 1120 nm after 1.2 m of propagation. Inset show the cumulative integral of the spectral functions, starting from the long wavelength edge.

Equations (49)

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

γ = 3 χ xxxx ( 3 ) ω 2 8 c 2 β ( ω ) A eff
× H = ε ( r ) E t + P NL t , × E = μ 0 H t
E ( r , t ) = m 1 2 π d ω G ˜ m ( z , ω ) e m ( r , ω ) exp [ i ( β m ( ω ) z ω t ) ] ,
H ( r , t ) = m 1 2 π d ω G ˜ m ( z , ω ) h m ( r , ω ) exp [ i ( β m ( ω ) z ω t ) ] ,
G ˜ m ( z , ω ) z = 2 π i ω 2 N m ( ω ) d r exp ( i β m ( ω ) z ) e m * ( r , ω ) · P NL ( r , ω )
P NL ( r , ω ) = d t exp ( i ω t ) P NL ( r , t )
d r [ e m × h n * + e n * × h m ] = 2 N m ( ω ) δ mn
P NL ( r , t ) = ε 0 d t ' R ( t t ' ) χ ( 3 ) E ( r , t ) E ( r , t ' ) E ( r , t ' )
P NL ( r , ω ) = ε 0 ( 2 π ) 2 nqp d ω 1 d ω 2 G n ( z , ω 1 ) G q * ( z , ω 1 + ω 2 ω ) G p ( z , ω 2 ) ×
R ( ω ω 1 ) χ ( 3 ) e n ( r , ω 1 ) e q * ( r , ω 1 + ω 2 ω ) · e p ( r , ω 2 )
G m ( z , ω ) = G ˜ m ( z , ω ) exp ( i β m ( ω ) z )
G ˜ m ( z , ω ) z = i ω ε 0 4 π N m ( ω ) exp ( i β m ( ω ) z ) nqp d ω 1 d ω 2 G n ( z , ω 1 ) ×
G q * ( z , ω 1 + ω 2 ω ) G p ( z , ω 2 ) R ( ω ω 1 ) K mnqp ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 )
K mnqp ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 ) = d r e m * ( r , ω ) · χ ( 3 ) e n ( r , ω 1 ) e q * ( r , ω 1 + ω 2 ω ) e p ( r , ω 2 )
e ( r , ω ) = F ( r , ω ) x ̂ , h ( r , ω ) = n eff ( ω ) ε 0 c F ( r , ω ) y ̂
N ( ω ) = n eff ( ω ) ε 0 c , G ˜ ( z , ω ) 2 n 0 ε 0 c G ˜ ( z , ω )
d r F ( r , ω ) 2 = 1 , E p ( z ) = 1 2 π 0 d ω G ˜ ( z , ω ) 2 n eff ( ω ) n 0
G ˜ ( z , ω ) z = i n 2 ω 2 exp ( i β ˜ ( ω ) z ) 2 π c 2 β ( ω ) d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) ×
G ( z , ω 2 ) R ( ω ω 1 ) K ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 )
K ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 ) = d r F ( ω ) F ( ω 1 ) F ( ω 1 + ω 2 ω ) F ( ω 2 )
β ˜ ( ω ) = β ( ω ) β ( ω 0 ) β 1 ( ω 0 ) × ( ω ω 0 )
n 2 = 3 χ xxxx ( 3 ) 4 n 0 ε 0 c
G ˜ ( z , ω ) z = i n 2 ω 2 exp ( i β ˜ ( ω ) z ) 2 π c 2 β ( ω ) A eff d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) G ( z , ω 2 ) R ( ω ω 1 )
A eff 1 = d r F 4 ( r ) [ d r F 2 ( r ) ] 2
G ˜ ( z , ω ) z = i n 2 ω 2 exp ( i β ˜ ( ω ) z ) 2 π c 2 β ( ω ) A eff ( ω ) d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) G ( z , ω 2 ) R ( ω ω 1 )
K ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 ) = μ ν γ δ K μ ν γ δ k μ ( ω ) k v ( ω 1 ) k γ ( ω 1 + ω 2 ω ) k δ ( ω 2 )
d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) G ( z , ω 2 ) R ( ω ω 1 ) K ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 ) =
μ ν γ δ K μ ν γ δ k μ ( ω ) d ω 1 d ω 2 G v ( z , ω 1 ) G γ * ( z , ω 1 + ω 2 ω ) G δ ( z , ω 2 ) R ( ω ω 1 )
G μ ( z , ω ) = G ( z , ω ) k μ ( ω )
F ( r , ω ) μ a μ ( ω ) F μ ( r )
k μ ( ω ) = a μ ( ω ) , K μ ν γ δ = d r F μ ( r ) F ν ( r ) F γ ( r ) F δ ( r )
F ( r , ω ) F 0 ( r ) + μ = 1 M F μ ( r ) ( ω ω c ) μ
V = π d λ n rod 2 n S i O 2 2
d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) G ( z , ω 2 ) R ( ω ω 1 ) K ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 ) =
K 0000 a 0 ( ω ) d ω 1 d ω 2 G ¯ ( z , ω 1 ) G ¯ * ( z , ω 1 + ω 2 ω ) G ¯ ( z , ω 2 ) R ( ω ω 1 )
G ¯ ( z , ω ) = G ( z , ω ) a 0 ( ω )
A eff 1 ( ω ) = K 0000 a 0 4 ( ω )
G ˜ ( z , ω ) z = i n 2 ω 2 exp ( i β ˜ ( ω ) z ) 2 π c 2 β ( ω ) A eff 1 4 ( ω ) d ω 1 d ω 2 G ¯ ( z , ω 1 ) G ¯ * ( z , ω 1 + ω 2 ω ) G ¯ ( z , ω 2 ) R ( ω ω 1 )
G ¯ ( z , ω ) = G ( z , ω ) A eff 1 4 ( ω )
N phot d ω n eff ( ω ) G ( z , ω ) 2 ω
d ω n eff ( ω ) A eff ( ω ) G ( z , ω ) 2 ω
F ( r , ω ) F ( r , ω c ) + F ' ( r , ω c ) ( ω ω c )
K ( ω , ω 1 , ω 1 + ω 2 ω , ω 2 ) K 0 + 2 K 1 ( ω 1 + ω 2 2 ω c )
K 0 = d r F 4 ( r , ω c ) , K 1 = d r F ' ( r , ω c ) F 3 ( r , ω c )
A eff 1 ( ω ) K 0 + 4 K 1 ( ω ω c )
1 A eff 1 4 ( ω ) d ω 1 d ω 2 G ¯ ( z , ω 1 ) G ¯ * ( z , ω 1 + ω 2 ω ) G ¯ ( z , ω 2 ) R ( ω ω 1 ) A eff 3 4 ( ω c ) 4 A eff 3 4 ( ω c ) ×
d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) G ( z , ω 2 ) R ( ω ω 1 ) [ 4 K 0 + 4 K 1 2 ( ω 1 + ω 2 2 ω c ) ] =
d ω 1 d ω 2 G ( z , ω 1 ) G * ( z , ω 1 + ω 2 ω ) G ( z , ω 2 ) R ( ω ω 1 ) [ K 0 + 2 K 1 ( ω 1 + ω 2 2 ω c ) ]
P 0 = β 2 γ T 0 2

Metrics