Abstract

We report a mid-infrared, CW singly resonant optical parametric oscillator (OPO) with a thermally induced waveguide in its gain crystal. We measured a numerical aperture of 0.0062 for the waveguide at 80-W intracavity power at 3.2 μm. This thermal-guiding effect benefits to the stable operation of an OPO and improves the parametric conversion efficiency by more than a factor of two when compared with that without thermal guiding.

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References

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  1. S. Schiller and J. Mlynek, “Continuous-wave optical parametric oscillators, materials, devices, applications,” Appl. Phys. B 66, 664–764 (1998).
  2. M. van Herpen, S. Te Lintel Hekkert, S. E. Bisson, and F. J. M. Harren, “Wide single-mode tuning of a 3.0- 3.8-mum, 700-mW, continuous-wave Nd:YAG-pumped optical parametric oscillator based on periodically poled lithium niobate,” Opt. Lett. 27(8), 640–642 (2002).
    [CrossRef]
  3. A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
    [CrossRef]
  4. W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “Continuous-wave singly resonant optical parametric oscillator based on periodically poled LiNbO(3).,” Opt. Lett. 21(10), 713–715 (1996).
    [CrossRef] [PubMed]
  5. M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Singly resonant cw OPO with simple wavelength tuning,” Opt. Express 16(15), 11141–11146 (2008).
    [CrossRef] [PubMed]
  6. S. T. Lin, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and J. T. Shy, “Observation of thermal-induced optical guiding and bistability in a mid-IR continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 33(20), 2338–2340 (2008).
    [CrossRef] [PubMed]
  7. D. W. Chen and T. S. Rose, “Low noise 10-W generation near 3 μm with MgO doped PPLN,” Presented at CLEO 2005, CThQ2, Baltimore, USA, 22–27 May 2005.
  8. R. O. Moore, G. Biondini, and W. L. Kath, “Self-induced thermal effects and model competition in continuous-wave optical parametric oscillators,” J. Opt. Soc. Am. B 19(4), 802–811 (2002).
    [CrossRef]
  9. A. Henderson and R. Stafford, “Intra-cavity power effects in singly resonant cw OPOs,” Appl. Phys. B 85(2-3), 181–184 (2006).
    [CrossRef]
  10. A. Douillet, J. J. Zondy, A. Yelisseyev, S. Lobanov, and L. Isaenko, “Stability and frequency tuning of thermally loaded continuous-wave AgGaS2 optical parametric oscillators,” J. Opt. Soc. Am. B 16(9), 1481–1498 (1999).
    [CrossRef]
  11. M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
    [CrossRef]
  12. Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
    [CrossRef]
  13. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
    [CrossRef]
  14. B. E. A. Saleh, and M. C. Teich, Fundamental of Photonics (John Wiley & Sons, 1991), pp. 54–57.
  15. A. E. Siegman, Lasers (University Science Books, 1986), pp.786–787.
  16. O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
    [CrossRef]

2009

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

2008

2007

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

2006

A. Henderson and R. Stafford, “Intra-cavity power effects in singly resonant cw OPOs,” Appl. Phys. B 85(2-3), 181–184 (2006).
[CrossRef]

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

2003

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

2002

1999

1998

S. Schiller and J. Mlynek, “Continuous-wave optical parametric oscillators, materials, devices, applications,” Appl. Phys. B 66, 664–764 (1998).

1996

1990

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Alexander, J. I.

Anstett, G.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Bartschke, J.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Bauer, T.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Biondini, G.

Bisson, S. E.

Boller, K. J.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Bosenberg, W. R.

Byer, R. L.

Chiang, A. C.

Cristescu, S. M.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Douillet, A.

Drobshoff, A.

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Groß, P.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Halonen, L.

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Singly resonant cw OPO with simple wavelength tuning,” Opt. Express 16(15), 11141–11146 (2008).
[CrossRef] [PubMed]

Harren, F. J. M.

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Singly resonant cw OPO with simple wavelength tuning,” Opt. Express 16(15), 11141–11146 (2008).
[CrossRef] [PubMed]

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

M. van Herpen, S. Te Lintel Hekkert, S. E. Bisson, and F. J. M. Harren, “Wide single-mode tuning of a 3.0- 3.8-mum, 700-mW, continuous-wave Nd:YAG-pumped optical parametric oscillator based on periodically poled lithium niobate,” Opt. Lett. 27(8), 640–642 (2002).
[CrossRef]

Henderson, A.

A. Henderson and R. Stafford, “Intra-cavity power effects in singly resonant cw OPOs,” Appl. Phys. B 85(2-3), 181–184 (2006).
[CrossRef]

Huang, W. L.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

Huang, Y. C.

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Isaenko, L.

Kath, W. L.

Kosterev, A. A.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

L’ Huillier, J. A.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Lee, C. J.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Li, Z. G.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

Lim, G. C.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

Lin, S. T.

Lin, Y. Y.

Lindsay, I. D.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Lobanov, S.

Mlynek, J.

S. Schiller and J. Mlynek, “Continuous-wave optical parametric oscillators, materials, devices, applications,” Appl. Phys. B 66, 664–764 (1998).

Moore, N.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

Moore, R. O.

Myers, L. E.

Ngai, A. K. Y.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Nittmann, M.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Paul, O.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Peltola, J.

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Singly resonant cw OPO with simple wavelength tuning,” Opt. Express 16(15), 11141–11146 (2008).
[CrossRef] [PubMed]

Persijn, S.

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Singly resonant cw OPO with simple wavelength tuning,” Opt. Express 16(15), 11141–11146 (2008).
[CrossRef] [PubMed]

Persijn, S. T.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Quosig, A.

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Schiller, S.

S. Schiller and J. Mlynek, “Continuous-wave optical parametric oscillators, materials, devices, applications,” Appl. Phys. B 66, 664–764 (1998).

Shy, J. T.

Stafford, R.

A. Henderson and R. Stafford, “Intra-cavity power effects in singly resonant cw OPOs,” Appl. Phys. B 85(2-3), 181–184 (2006).
[CrossRef]

Te Lintel Hekkert, S.

Tittel, F. K.

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

Vainio, M.

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Singly resonant cw OPO with simple wavelength tuning,” Opt. Express 16(15), 11141–11146 (2008).
[CrossRef] [PubMed]

van Herpen, M.

Xiong, Z.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

Yelisseyev, A.

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Zondy, J. J.

Appl. Phys. B

S. Schiller and J. Mlynek, “Continuous-wave optical parametric oscillators, materials, devices, applications,” Appl. Phys. B 66, 664–764 (1998).

A. K. Y. Ngai, S. T. Persijn, I. D. Lindsay, A. A. Kosterev, P. Groß, C. J. Lee, S. M. Cristescu, F. K. Tittel, K. J. Boller, and F. J. M. Harren, “Continuous wave optical parametric oscillator for quartz-enhanced photoacoustic trace gas sensing,” Appl. Phys. B 89(1), 123–128 (2007).
[CrossRef]

A. Henderson and R. Stafford, “Intra-cavity power effects in singly resonant cw OPOs,” Appl. Phys. B 85(2-3), 181–184 (2006).
[CrossRef]

M. Vainio, J. Peltola, S. Persijn, F. J. M. Harren, and L. Halonen, “Thermal effects in singly resonant continuous-wave optical parametric oscillators,” Appl. Phys. B 94(3), 411–427 (2009).
[CrossRef]

O. Paul, A. Quosig, T. Bauer, M. Nittmann, J. Bartschke, G. Anstett, and J. A. L’ Huillier, “Temperature-dependent Sellmeier equation in the MIR for the extraordinary refractive index of 5% MgO doped congruent LiNbO3,” Appl. Phys. B 86(1), 111–115 (2006).
[CrossRef]

Appl. Phys. Lett.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

IEEE J. Quantum Electron.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO/sub 4/ lasers,” IEEE J. Quantum Electron. 39(8), 979–986 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Other

D. W. Chen and T. S. Rose, “Low noise 10-W generation near 3 μm with MgO doped PPLN,” Presented at CLEO 2005, CThQ2, Baltimore, USA, 22–27 May 2005.

B. E. A. Saleh, and M. C. Teich, Fundamental of Photonics (John Wiley & Sons, 1991), pp. 54–57.

A. E. Siegman, Lasers (University Science Books, 1986), pp.786–787.

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Figures (5)

Fig. 1
Fig. 1

Experimental setup of the ring mid-IR SRO for characterizing the thermal-induced optical lensing and guiding in the nonlinear crystal, MgO:PPLN. The output pump mode is sensitive to the photothermal effect in the nonlinear crystal. A CCD behind a GaAs filter images the output pump beam as a function of the intracavity idler power to reveal the thermal effects in the SRO. B: a dichroic mirror to kick out the signal wave, HWP: half-wave plate, PBS: polarization beam splitter.

Fig. 2
Fig. 2

Intracavity idler power at 3.2 μm versus pump power at 1.064 μm. Thermal lensing in the low power state and thermal guiding in the high power state forms the bistability diagram.

Fig. 3
Fig. 3

Output pump-beam profiles at different intracavity idler powers. In (a), the SRO was not in operation. From (b) to (f), the SRO was in the low power state, and the intracavity idler power varied from 8 to 28 W. The images show asymmetric thermal lensing in the two transverse directions.

Fig. 4
Fig. 4

The focal length of the thermal lens versus intracavity idler power along the thickness (z) and width (y) directions of the crystal. The theoretical curve calculated from Eq. (4) fits better to the experimental data on the right half of the figure. At a low idler power, heating of the crystal by the signal and pump waves might not be ignored.

Fig. 5
Fig. 5

Pump beam profiles at the SRO output at 20-, 40-, and 60-W intracavity idler powers. At 60 W, the pump wave is tightly guided in the PPLN. During the formation of the thermal waveguide, an imperfectly aligned pump beam can be gradually pulled into the waveguide.

Tables (1)

Tables Icon

Table 1 Measured numerical aperture of the thermal waveguide versus pump power of the SRO. The corresponding intracavity idler power is also listed. As expected, the NA increases with the pump and intracavity idler powers. The NA provides a reference to the tolerance of the pump-laser and cavity alignment.

Equations (7)

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

Δ n p ( r ) = α i P i 4 π K 2 r 2 w i 2 d n p d T ,
Δ Φ ( r ) = k p Δ n p ( r ) Δ z ,
Δ Φ ( r ) = k p n p r 2 2 Δ ( 1 f ) ,
f = λ i K α i P i n p n i ( d n p / d T ) 1 2 tan 1 ( L / ( 2 z R , i ) ) .
Δ q Δ z = q 2 α i P i π n p K w i 2 d n p d T .
Δ q Δ z = q 2 α i P i π n p K w i 2 d n p d T + 1.
P c r = λ p K α i z R , p ( w 0 , i w 0 , p ) 2 1 d n p / d T .

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