Abstract

Photonic crystal bandgap fibers employing distributed mode filtering design provide near diffraction-limited light outputs, a critical property of fiber-based high-power lasers. Microstructure of the fibers is tailored to achieve single-mode operation at specific wavelength by resonant mode coupling of higher-order modes. We analyze the modal regimes of the fibers having a mode field diameter of 60 µm by the cross-correlated (C2) imaging method in different wavelength ranges and evaluate the sensitivity of the modal content to various input-coupling conditions. As a result, we experimentally identify regimes of resonant coupling between higher-order core modes and cladding band. We demonstrate a passive fiber design in which the higher-order modal content inside the single-mode guiding regime is suppressed by at least 20 dB even for significantly misaligned input-coupling configurations.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B27(11), B63–B92 (2010).
    [CrossRef]
  2. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express14(7), 2715–2720 (2006).
    [CrossRef] [PubMed]
  3. A. Galvanauskas, M. C. Swan, and C.-H. Liu, “Effectively single-mode large core passive and active fibers with chirally coupled-core structures,” paper CMB1, CLEO/QELS, San Jose (2008).
  4. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol.27(11), 1565–1570 (2009).
    [CrossRef]
  5. F. Jansen, F. Stutzki, H. J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express18(26), 26834–26842 (2010).
    [CrossRef] [PubMed]
  6. T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped Large-Mode-Area photonic bandgap rod fiber amplifier,” Opt. Express19(8), 7398–7409 (2011).
    [CrossRef] [PubMed]
  7. J. W. Nicholson, J. M. Fini, A. M. DeSantolo, E. Monberg, F. DiMarcello, J. Fleming, C. Headley, D. J. DiGiovanni, S. Ghalmi, and S. Ramachandran, “A higher-order-mode Erbium-doped-fiber amplifier,” Opt. Express18(17), 17651–17657 (2010).
    [CrossRef] [PubMed]
  8. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express20(5), 5742–5753 (2012).
    [CrossRef] [PubMed]
  9. M. Laurila, J. Saby, T. T. Alkeskjold, L. Scolari, B. Cocquelin, F. Salin, J. Broeng, and J. Lægsgaard, “Q-switching and efficient harmonic generation from a single-mode LMA photonic bandgap rod fiber laser,” Opt. Express19(11), 10824–10833 (2011).
    [CrossRef] [PubMed]
  10. F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express20(4), 3997–4008 (2012).
    [CrossRef] [PubMed]
  11. F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett.37(6), 1073–1075 (2012).
    [CrossRef] [PubMed]
  12. M. C. Swan, C. Liu, D. Guertin, N. Jacobsen, K. Tankala, and A. Galvanauskas, “33μm Core Effectively Single-Mode Chirally-Coupled-Core Fiber Laser at 1064-nm,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OWU2.
  13. T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express19(1), 255–260 (2011).
    [CrossRef] [PubMed]
  14. J. W. Nicholson, S. Ramachandran, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Propagation of femtosecond pulses in large-mode-area, higher-order-mode fiber,” Opt. Lett.31(21), 3191–3193 (2006).
    [CrossRef] [PubMed]
  15. S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
    [CrossRef]
  16. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  17. A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE1868, 2–12 (1993).
    [CrossRef]
  18. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express15(23), 15402–15409 (2007).
    [CrossRef] [PubMed]
  19. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
    [CrossRef] [PubMed]
  20. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
    [CrossRef]
  21. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17(11), 9347–9356 (2009).
    [CrossRef] [PubMed]
  22. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19(14), 13008–13019 (2011).
    [CrossRef] [PubMed]
  23. R. A. Barankov, “Cross-correlation imaging for waveguide characterization”, M.S. Thesis, http://arxiv.org/abs/1206.0666v1 [physics.optics]
  24. M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012).
    [CrossRef] [PubMed]
  25. S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” Photon. Technol. Lett.15(8), 1171–1173 (2003).
    [CrossRef]

2012 (4)

2011 (4)

2010 (3)

2009 (3)

2008 (2)

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
[CrossRef] [PubMed]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

2007 (1)

2006 (2)

2003 (1)

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” Photon. Technol. Lett.15(8), 1171–1173 (2003).
[CrossRef]

1993 (1)

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE1868, 2–12 (1993).
[CrossRef]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Alkeskjold, T. T.

Barankov, R. A.

Baumgartl, M.

Broeng, J.

Carstens, H.

Clarkson, W. A.

Cocquelin, B.

DeSantolo, A. M.

DiGiovanni, D. J.

DiMarcello, F.

Dimarcello, F. V.

Dong, L.

Duparré, M.

Eidam, T.

Ermeneux, S.

Fermann, M. E.

Fini, J. M.

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, E. Monberg, F. DiMarcello, J. Fleming, C. Headley, D. J. DiGiovanni, S. Ghalmi, and S. Ramachandran, “A higher-order-mode Erbium-doped-fiber amplifier,” Opt. Express18(17), 17651–17657 (2010).
[CrossRef] [PubMed]

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Flamm, D.

Fleming, J.

Fu, L.

Ghalmi, S.

Hädrich, S.

Hansen, K. R.

Headley, C.

Jansen, F.

Jauregui, C.

Jørgensen, M. M.

Kaiser, T.

Lægsgaard, J.

Laurila, M.

Li, J.

Liem, A.

Limpert, J.

Marcinkevicius, A.

Mckay, H. A.

Mermelstein, M.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Mermelstein, M. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

Monberg, E.

Nicholson, J. W.

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, E. Monberg, F. DiMarcello, J. Fleming, C. Headley, D. J. DiGiovanni, S. Ghalmi, and S. Ramachandran, “A higher-order-mode Erbium-doped-fiber amplifier,” Opt. Express18(17), 17651–17657 (2010).
[CrossRef] [PubMed]

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
[CrossRef] [PubMed]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

J. W. Nicholson, S. Ramachandran, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Propagation of femtosecond pulses in large-mode-area, higher-order-mode fiber,” Opt. Lett.31(21), 3191–3193 (2006).
[CrossRef] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” Photon. Technol. Lett.15(8), 1171–1173 (2003).
[CrossRef]

Nilsson, J.

Otto, H. J.

Otto, H.-J.

Petersen, S. R.

Ramachandran, S.

Richardson, D. J.

Röser, F.

Rothhardt, J.

Saby, J.

Salin, F.

Schimpf, D. N.

Schmidt, O.

Schreiber, T.

Schröter, S.

Scolari, L.

Siegman, A. E.

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE1868, 2–12 (1993).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Stutzki, F.

Thomas, B. K.

Tünnermann, A.

Wielandy, S.

Wisk, P.

Yablon, A. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
[CrossRef] [PubMed]

Yan, M. E.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Yan, M. F.

J. W. Nicholson, S. Ramachandran, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Propagation of femtosecond pulses in large-mode-area, higher-order-mode fiber,” Opt. Lett.31(21), 3191–3193 (2006).
[CrossRef] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” Photon. Technol. Lett.15(8), 1171–1173 (2003).
[CrossRef]

Yvernault, P.

IEEE J. Sel. Top. Quantum Electron. (1)

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

J. Lightwave Technol. (1)

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

Laser Photonics Rev. (1)

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. E. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Opt. Express (13)

S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express15(23), 15402–15409 (2007).
[CrossRef] [PubMed]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
[CrossRef] [PubMed]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express14(7), 2715–2720 (2006).
[CrossRef] [PubMed]

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express19(1), 255–260 (2011).
[CrossRef] [PubMed]

F. Jansen, F. Stutzki, H. J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express18(26), 26834–26842 (2010).
[CrossRef] [PubMed]

T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped Large-Mode-Area photonic bandgap rod fiber amplifier,” Opt. Express19(8), 7398–7409 (2011).
[CrossRef] [PubMed]

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, E. Monberg, F. DiMarcello, J. Fleming, C. Headley, D. J. DiGiovanni, S. Ghalmi, and S. Ramachandran, “A higher-order-mode Erbium-doped-fiber amplifier,” Opt. Express18(17), 17651–17657 (2010).
[CrossRef] [PubMed]

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express20(5), 5742–5753 (2012).
[CrossRef] [PubMed]

M. Laurila, J. Saby, T. T. Alkeskjold, L. Scolari, B. Cocquelin, F. Salin, J. Broeng, and J. Lægsgaard, “Q-switching and efficient harmonic generation from a single-mode LMA photonic bandgap rod fiber laser,” Opt. Express19(11), 10824–10833 (2011).
[CrossRef] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express20(4), 3997–4008 (2012).
[CrossRef] [PubMed]

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17(11), 9347–9356 (2009).
[CrossRef] [PubMed]

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19(14), 13008–13019 (2011).
[CrossRef] [PubMed]

M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012).
[CrossRef] [PubMed]

Opt. Lett. (2)

Photon. Technol. Lett. (1)

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” Photon. Technol. Lett.15(8), 1171–1173 (2003).
[CrossRef]

Proc. SPIE (2)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE1868, 2–12 (1993).
[CrossRef]

Other (3)

A. Galvanauskas, M. C. Swan, and C.-H. Liu, “Effectively single-mode large core passive and active fibers with chirally coupled-core structures,” paper CMB1, CLEO/QELS, San Jose (2008).

M. C. Swan, C. Liu, D. Guertin, N. Jacobsen, K. Tankala, and A. Galvanauskas, “33μm Core Effectively Single-Mode Chirally-Coupled-Core Fiber Laser at 1064-nm,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OWU2.

R. A. Barankov, “Cross-correlation imaging for waveguide characterization”, M.S. Thesis, http://arxiv.org/abs/1206.0666v1 [physics.optics]

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 (14)

Fig. 1
Fig. 1

(a) C2 imaging setup: SLD – superluminesent diode, BS – beam splitter, BC – beam combiner, LP – linear polarizer, HWP – half-wave plate [23]. (b) Optical micrograph of the DMF rod fiber.

Fig. 2
Fig. 2

Measured input spectra used for different C2 imaging measurements (pick-up fiber HI1060).

Fig. 3
Fig. 3

Simulated core overlap ratio (power) of different modes in the DMF rod fiber. The most relevant modes are shown in SM and MM guiding regimes. Insets illustrate the simulated electric field profiles (y-components) of different modes at specific wavelengths.

Fig. 4
Fig. 4

Simulated effective indices as a function of wavelength for the experimentally relevant modes of the DMF rod fiber.

Fig. 5
Fig. 5

Simulated C2 imaging trace of the DMF rod fiber at 1040 nm (98% in the FM, 1% in every of the two HOMs): solid line represents a trace accounting for the dispersion of the modes, dash-dotted line depicts the trace with digitally compensated dispersion. Inset: Gaussian spectrum centered at 1040 nm with FWHM = 10 nm, used in the simulation.

Fig. 6
Fig. 6

Simulated C2 imaging trace of the DMF rod fiber at 1060 nm (98% in the FM, 1% in every of the two HOMs): solid line represents a trace accounting for the dispersion of the modes, dash-dotted line depicts the trace with digitally compensated dispersion. Inset: Gaussian spectrum centered at 1060 nm with FWHM = 10 nm, used in the simulation.

Fig. 7
Fig. 7

Near-field images of three different coupling conditions: (a), (c), (e) – without filtering by the aperture, (b), (d), (f) – filtered by the aperture. Images are recorded in the SM regime of the DMF1040 fiber at the wavelength of 1040nm. The offset coupling conditions were x30 in (c), (d) and y30 in (e), (f), respectively. The small intensity peaks at the top right in (c) and bottom in (e) identify the resonator elements which guide the coupled HOMs from the core to the cladding band.

Fig. 8
Fig. 8

C2 trace of the DMF1040 fiber measured at the wavelength of 1040 nm, with and without the aperture, under the perfect coupling condition. The insets show the reconstructed mode images.

Fig. 9
Fig. 9

C2 trace of the DMF1040 fiber measured at the wavelength of 1040 nm, with and without the aperture, under X-offset coupling condition. The insets show the reconstructed mode images.

Fig. 10
Fig. 10

Input and output spectra of the DMF1040 fiber recorded at the central wavelength of 1040 nm. The output spectrum shows a beating pattern above 1050 nm, outside of the single-mode guiding region of the DMF1040 fiber. The modal content of this mode is measured by C2 imaging.

Fig. 11
Fig. 11

Summary of different C2 measurements carried out on the DMF1040 fiber with different coupling conditions (without cladding light) at the wavelength of 1040 nm.

Fig. 12
Fig. 12

C2 trace showing the total power of all the modes obtained from the envelope function of C2 trace measured at the wavelength of 1060 nm, at optimal coupling, without aperture. The LP11 mode has −14 dB less power than the fundamental mode (LP01). The inset shows the measured NF image of the fiber output. The white dotted circle identifies the core region in the reconstructed mode images.

Fig. 13
Fig. 13

Example of C2 trace of the DMF1040 fiber at the wavelength of 1060nm (offset coupling condition x20, no aperture). The red line represents the envelope of the trace (averaged over 8 points). The inset shows a beating pattern in the output spectrum.

Fig. 14
Fig. 14

Summary of different C2 measurements of the DMF1040 fiber in the multimode region (at 1060 nm) using different coupling conditions: perfect, x30 and y30 (all done without the aperture). The insets show the reconstructed mode images.

Equations (1)

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

P(x,y;τ)= m p m G mr 2 (τ τ mr ) I m (x,y),

Metrics