Abstract

Mode characteristics in the solid-state planar waveguide (PWG) laser amplifiers are investigated theoretically, in consideration of the temperature gradient generated by cooling across the thickness and by pumping inhomogeneity along the width direction. When variation of the refractive index along the width direction is dominated by the lower spatial frequencies, the vector wave equation is solved analytically by means of the perturbation method. It is similar to the zigzag slab amplifier in which the phase aberration depending on the width coordinate plays the most important role to cause degradation of the beam quality. The crossing mode distortions owing to two dimension nature of the index variations are illustrated, and that mode profile is varied by the index variation along both the thickness and the width directions. Modes in the single-mode or the few-mode PWGs are shown to suffer weaker thermal-induced distortion across the thickness than those in the multi-mode PWGs.

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References

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  1. N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.
  2. D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).
  3. C. Grivas, “Optically pumped planar waveguide lasers, Part I: Fundamentals and Fabrication techniques,” Prog. Quantum Electron.35, 159–239 (2011).
    [CrossRef]
  4. C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
    [CrossRef]
  5. J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron.13, 626–637 (2007)
    [CrossRef]
  6. H. J. Baker, J. R. Lee, and D. R Hall, “Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers,” Opt. Express10, 297–302 (2002).
    [CrossRef] [PubMed]
  7. G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.
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    [CrossRef] [PubMed]
  9. S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.
  10. L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008).
    [CrossRef]
  11. J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003).
    [CrossRef]
  12. J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
    [CrossRef]
  13. K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
    [CrossRef]
  14. C. F. Wang, T. H. Her, L. Zhao, X. Y. Ao, L. W. Casperson, C. H. Lai, and H. C. Chang, “Gain saturation and output characteristics of index-antiguided planar waveguide amplifiers with homogeneous broadening,” J. Lightwave Technol.29, 1958–1964 (2011).
    [CrossRef]

2011 (3)

2008 (2)

D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).

L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008).
[CrossRef]

2007 (1)

J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron.13, 626–637 (2007)
[CrossRef]

2005 (1)

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
[CrossRef]

2003 (1)

J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003).
[CrossRef]

2002 (1)

2000 (1)

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
[CrossRef]

1984 (1)

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

Alkeskjold, T. T.

Ao, X. Y.

Baker, H. J.

Bammert, S.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Bennett, G. T.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Bhutta, T.

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
[CrossRef]

Bonner, C. L.

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
[CrossRef]

Broeng, J.

Byer, R. L.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

Callicoatt, B. E.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Casperson, L. W.

Chang, H. C.

Cheng, X. J.

L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008).
[CrossRef]

Eggleston, J. M.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

Field, S.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Filgas, D.

D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).

Flegal, B.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Grivas, C.

C. Grivas, “Optically pumped planar waveguide lasers, Part I: Fundamentals and Fabrication techniques,” Prog. Quantum Electron.35, 159–239 (2011).
[CrossRef]

Hall, D. R

Hansen, K. R.

Her, T. H.

Hodgson, N.

N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

Hoffman, H. J.

N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

Injeyan, H.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Iwaki, L.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Jordan, W.

N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

Kane, T. J.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

Kawato, S.

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
[CrossRef]

Kobayashi, T.

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
[CrossRef]

Komine, H.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Kuhn, K.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

Lagsgaard, J.

Lai, C. H.

Lee, J. R.

Lee, T.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Li, C.

J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003).
[CrossRef]

Mackenzie, J. I.

J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron.13, 626–637 (2007)
[CrossRef]

J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003).
[CrossRef]

Malm, A. I. R.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

McNaught, S.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Redmond, S.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Rockwell, D.

D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).

Rubin, L.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Ryan, C.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Shepherd, D. P.

J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003).
[CrossRef]

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
[CrossRef]

Simpson, R.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Spariosu, K.

D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).

Sueda, K.

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
[CrossRef]

Szot, J.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Takahashi, H.

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
[CrossRef]

Tartaglia, M.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Ter-Mikirtychev, V. V.

N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

Tropper, A. C.

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
[CrossRef]

Unternahrer, J.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

Wagner, G. J.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

Wang, C. F.

Weiss, S. B.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Xiao, L.

L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008).
[CrossRef]

Xu, J. Q.

L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008).
[CrossRef]

Zamel, J.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

Zhao, L.

Appl. Phys. Lett. (1)

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000).
[CrossRef]

J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003).
[CrossRef]

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984).
[CrossRef]

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

J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron.13, 626–637 (2007)
[CrossRef]

J. Lightwave Technol. (1)

Opt. Commun. (1)

L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008).
[CrossRef]

Opt. Express (2)

Prog. Quantum Electron. (1)

C. Grivas, “Optically pumped planar waveguide lasers, Part I: Fundamentals and Fabrication techniques,” Prog. Quantum Electron.35, 159–239 (2011).
[CrossRef]

Raytheon Tech. Today (1)

D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).

Other (3)

N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

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Figures (6)

Fig. 1
Fig. 1

Planar optical waveguide.

Fig. 2
Fig. 2

Parameter U and ψ0/ΔTco for various modes in first example.

Fig. 3
Fig. 3

Field profiles of several modes in the x-axis for the first PWG with ΔTco = 1.5K, 3K and 6K.

Fig. 4
Fig. 4

Mode field profiles in the x-axis for the second PWG with ΔTco = 0.6K, 1.2K and 2.4K.

Fig. 5
Fig. 5

Phase difference in the width direction vs propagation distance with (b) ΔTco=1.5K and (c) ΔTco=3K. Here (a) denotes a pumping distribution with 90% uniformity in central region of 90% of width.

Fig. 6
Fig. 6

Beam quality degradation of the fundamental mode caused by the y-axis phase aberration in the first PWG with ΔTco=1.5K. (a) Far field pattern without the phase aberration. (b)–(d) Far field patterns for the optical length L inside the PWG amplifiers, in consideration of the phase aberration induced by a pumping non-uniformity of 5% in y-axis. (e) BQ vs the pumping non-uniformity with different optical lengths.

Equations (36)

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

E ( x , t ) = e ( x ) e i ω t + i β z , H ( x , t ) = h ( x ) e i ω t + i β z ,
{ ( 2 x 2 + k 2 n ¯ 2 β 2 ) e = ( x ^ x + i β z ^ ) ( e x ln n 2 x ) , ( 2 x 2 + k 2 n ¯ 2 β 2 ) h = ln n 2 x [ ( x ^ x + i β z ^ ) × h ] × x ^ ,
ϕ ( x ) = { cos ( U x / ρ ) / cos U exp [ W ( | x | / ρ 1 ) ] or { sin ( U x / ρ ) / sin U | x | ρ x | x | exp [ W ( | x | / ρ 1 ) ] | x | > ρ
U = k ρ n co 2 n eff 2 , W = k ρ n eff 2 n cl 2 , n eff = β / k ,
E = i μ 0 ε 0 1 k n 2 × H , H = i ε 0 μ 0 1 k × E ,
tan U = { W / U ( Even TE ) or U / W ( Odd TE ) , n co 2 W / n cl 2 U ( Even TM ) or n co 2 W / n cl 2 U ( Odd TM ) .
V = k ρ n co 2 n cl 2 π / 2 .
κ 2 T x 2 + Q χ ( y ) = 0 , Q = { q , | x | ρ 0 , | x | > ρ ,
T ( x , y ) = T 0 + A ( x ) χ ( y ) .
n 2 ( x , y ) = ( n ¯ + d n d T Δ T ( x , y ) ) 2 n ¯ 2 + 2 d n d T ψ ( x ) χ ( y ) ,
e i = e ¯ i + 2 d n d T δ e i , h i = h ¯ i + 2 d n d T δ h i ,
χ ( y ) = m a m cos m π y w ,
{ δ e z ( x , y ) = m G z m ( x ) sin m π y w , δ h z ( x , y ) = m F z m ( x ) cos m π y w , for TE modes δ e z ( x , y ) = m G z m ( x ) cos m π y w , δ h z ( x , y ) = m F z m ( x ) sin m π y w , for TM modes
{ ( 2 x 2 + k 2 n ¯ 2 β 2 m 2 π 2 w 2 ) G z m = i a m β n ¯ 2 m π w ψ e ¯ y , ( 2 x 2 + k 2 n ¯ 2 β 2 m 2 π 2 w 2 ) F z m = i ε 0 μ 0 k a m ( ψ e ¯ y ) x ,
{ ( 2 x 2 + k 2 n ¯ 2 β 2 m 2 π 2 w 2 ) G z m = i μ 0 ε 0 i a m k n ¯ 2 ( ψ h ¯ y ) x , ( 2 x 2 + k 2 n ¯ 2 β 2 m 2 π 2 w 2 ) F z m = i a m β n ¯ 2 m π w ψ h ¯ y .
e ¯ x , z TM = 1 n ¯ 2 μ 0 ε 0 h ¯ x , z TE , G i TM = 1 n ¯ 2 μ 0 ε 0 F i TE , F i TM = G i TE .
k 2 n ¯ 2 β 2 ~ { U ¯ 2 ρ 2 or W ¯ 2 ρ 2 } m 2 π 2 w 2 , and m π w x ~ 1 ρ ,
G y m k β μ 0 ε 0 F x m i k k 2 n ¯ 2 β 2 μ 0 ε 0 F z m x k 2 a m k 2 n ¯ 2 β 2 ψ e ¯ y .
F z m i k a m μ 0 ε 0 = { M co ( x ) + d 1 sin ( U ¯ x / ρ ) / sin U ¯ , | x | ρ , M cl ( x ) + d 2 x | x | exp [ W ¯ ( | x | / ρ 1 ) ] , | x | > ρ
M co ( x ) = Re { e i U ¯ x / ρ d x [ e 2 i U ¯ x / ρ d x e i U ¯ x / ρ ( ψ e ¯ y , co ) x ] } , M cl ( x ) = e W ¯ x / ρ d x [ e 2 W ¯ x / ρ d x e W ¯ x / ρ ( ψ e ¯ y , cl ) x ] ,
{ d 1 + M co ( ρ ) = d 2 + M cl ( ρ ) , d 1 + ρ W ¯ U 2 ( M co x ψ ) | | x | = ρ = d 2 ρ W ( M cl x ψ ) | | x | = ρ
Δ ψ ( ρ ) = ψ ( ρ ) ψ 0 = ( W ¯ 2 V 2 M co x + U ¯ 2 V 2 M cl x ) | | x | = ρ U ¯ 2 W ¯ ρ V 2 [ M co ( ρ ) M cl ( ρ ) ] .
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 n ¯ 2 + 2 k 2 d n d T ψ 0 χ ( y ) ) ( E H ) = 0 .
β β ( y ) = β ¯ ( 1 + d n d T ψ 0 n ¯ 2 χ ( y ) ) .
1 k 2 n ¯ 2 β 2 2 e i β ( y ) z y 2 ~ ρ 2 w 2 k L γ [ f 1 ( y ) + k L γ f 2 ( y ) ] ,
k ρ γ L w 1 .
β β ( y , z ) = β ¯ [ 1 + d n d T ψ 0 n ¯ 2 χ ( y ) z 2 6 ( d n d T ψ 0 n ¯ 2 χ y ) 2 ] .
( 2 y 2 + 2 z 2 + k 2 n ¯ 2 + 2 k 2 d n d T ψ 0 χ ( y ) ) e i β ( y , z ) z ~ k 2 γ 3 L 4 w 4 g 1 ( y ) + k 2 γ 2 g 2 ( y ) .
d x | e ¯ y | 2 = d x | e ¯ y + 2 d n d T δ e y | 2 d x e ¯ y G y m = 0 .
E y ( x , t ) = exp [ i ω t + i z β ( y , z ) ] × { e ¯ y + 2 d n d T G y ( x ) χ ( y ) } ,
ψ ( x ) = { n ¯ Δ T cl + n ¯ Δ T co ( 1 x 2 / ρ 2 ) , | x | ρ , n ¯ Δ T cl + 2 n ¯ Δ T co ( 1 | x | / ρ ) , | x | > ρ .
Δ T cl = q ρ ( 1 h + D PWG D in 2 κ out + D in 2 ρ 2 κ YAG ) , Δ T co = q ρ 2 2 κ YAG ,
M co ( x ) = ( a 2 b 4 U 2 b x 2 6 ρ 2 ) x cos ( U ¯ x / ρ ) cos U ( 3 a 32 b 16 U 2 + b x 2 4 ρ 2 ) ρ sin ( U ¯ x / ρ ) U cos U , M cl ( x ) = ( a W ¯ + b ( W ¯ + 1 ) 4 W 2 ( ρ + 2 W ¯ | x | b x 2 2 ρ ) x | x | exp [ W ¯ ( | x | / ρ 1 ) ] ,
M co ( x ) = ( a 2 b 4 U 2 b x 2 6 ρ 2 ) x sin ( U ¯ x / ρ ) sin U + ( 11 a 32 b 16 U 2 + b x 2 4 ρ 2 ) ρ cos ( U ¯ x / ρ ) U sin U , M cl ( x ) = ( a W ¯ + b ( W ¯ + 1 ) 4 W 2 ( ρ + 2 W ¯ | x | b x 2 2 ρ ) exp [ W ¯ ( | x | / ρ 1 ) ] .
ψ 0 = n ¯ Δ T cl + n ¯ Δ T co W ¯ 2 ( 4 U ¯ 2 + 3 + 3 W ¯ / V ¯ 2 ) 6 U ¯ 2 6 U 2 W ( 1 + W ) .
B Q = ( PIB DL / PIB R ) 1 / 2 ,

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