Abstract

Single-mode laser action has been demonstrated in a gain-guided, index-antiguided (GG IAG) Nd3+ doped phosphate glass fiber through the use of a diode side-pumping scheme. This pump technique can allow for length scaling of diode-pumped GG IAG fibers and therefore true power scaling of the GG IAG concept. Near-Gaussian beam quality was demonstrated with beam M2 of 1.4.

© 2010 Optical Society of America

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References

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  1. T. S. McComb, M. C. Richardson, and M. Bass, “High-power fiber lasers and amplifiers,” in Handbook of Optics, 3rd ed., M.Bass, ed. (McGraw-Hill, 2010) Vol. 5, Chap. 25.
  2. J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, P. Turner, and D. J. Richardson, “High-energy, high-power ytterbium-doped Q-switched fiber laser,” Opt. Lett. 25, 37–39 (2000).
    [CrossRef]
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    [CrossRef]
  4. J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  9. Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
    [CrossRef]
  10. Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
    [CrossRef] [PubMed]
  11. V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
    [CrossRef]
  12. V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).
  13. S. O. Kasap, Optoelectronics and Photonics: Principles and Practices (Prentice Hall, 2001).
  14. A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24, 1677–1682 (2007).
    [CrossRef]
  15. E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.
  16. W. Koechner, Solid-State Laser Engineering Fifth Revised and Updated Edition (Springer, 1999).

2010 (2)

T. S. McComb, M. C. Richardson, and M. Bass, “High-power fiber lasers and amplifiers,” in Handbook of Optics, 3rd ed., M.Bass, ed. (McGraw-Hill, 2010) Vol. 5, Chap. 25.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

2008 (1)

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

2007 (3)

2006 (1)

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

2005 (1)

2004 (1)

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

2003 (1)

2001 (1)

S. O. Kasap, Optoelectronics and Photonics: Principles and Practices (Prentice Hall, 2001).

2000 (2)

1999 (1)

W. Koechner, Solid-State Laser Engineering Fifth Revised and Updated Edition (Springer, 1999).

1997 (1)

1988 (1)

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

Alvarez-Chavez, J. A.

Ballato, J.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
[CrossRef]

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Bass, M.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

T. S. McComb, M. C. Richardson, and M. Bass, “High-power fiber lasers and amplifiers,” in Handbook of Optics, 3rd ed., M.Bass, ed. (McGraw-Hill, 2010) Vol. 5, Chap. 25.

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
[CrossRef] [PubMed]

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Beach, R. J.

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

Benoit, W.

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

Birks, T. A.

Chen, Y.

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
[CrossRef] [PubMed]

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Clarkson, W. A.

Dawson, J. W.

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

Dong, L.

Foy, P.

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Goldberg, L.

Hageman, W.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

Hakimi, F.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

Hawkins, W.

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Jovanovic, I.

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

Kasap, S. O.

S. O. Kasap, Optoelectronics and Photonics: Principles and Practices (Prentice Hall, 2001).

Kliner, D. A. V.

Knight, J. C.

Koechner, W.

W. Koechner, Solid-State Laser Engineering Fifth Revised and Updated Edition (Springer, 1999).

Koplow, J. P.

Liao, Z.

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

McCollum, B. C.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

McComb, T.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
[CrossRef] [PubMed]

Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
[CrossRef]

McComb, T. S.

T. S. McComb, M. C. Richardson, and M. Bass, “High-power fiber lasers and amplifiers,” in Handbook of Optics, 3rd ed., M.Bass, ed. (McGraw-Hill, 2010) Vol. 5, Chap. 25.

Mclaughlin, J. M.

Nilsson, J.

Offerhaus, H. L.

Payne, S.

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

Peng, W.

Po, H.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

Richardson, D. J.

Richardson, M.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
[CrossRef] [PubMed]

Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
[CrossRef]

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Richardson, M. C.

T. S. McComb, M. C. Richardson, and M. Bass, “High-power fiber lasers and amplifiers,” in Handbook of Optics, 3rd ed., M.Bass, ed. (McGraw-Hill, 2010) Vol. 5, Chap. 25.

Russell, P. S. J.

Siegman, A.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

Siegman, A. E.

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24, 1677–1682 (2007).
[CrossRef]

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20, 1617–1628 (2003).
[CrossRef]

Snitzer, E.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

Sudesh, V.

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Y. Chen, V. Sudesh, T. McComb, M. Richardson, M. Bass, and J. Ballato, “Lasing in a gain-guided index-antiguided fiber,” J. Opt. Soc. Am. B 24, 1683–1688 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
[CrossRef] [PubMed]

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Tumminelli, R.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

Turner, P.

Turner, P. W.

Wong, W. S.

Appl. Phys. B (1)

V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200 μm diameter core, gain-guided, index-antiguided single-mode fiber laser,” Appl. Phys. B 90, 369–372 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

A. E. Siegman, Y. Chen, V. Sudesh, M. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index-antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Opt. Lett. (5)

Proc. SPIE (1)

J. W. Dawson, R. J. Beach, I. Jovanovic, W. Benoit, Z. Liao, and S. Payne, “Large flattened mode optical fiber for reduction of nonlinear effects,” Proc. SPIE 5335, 132–139 (2004).
[CrossRef]

Other (5)

T. S. McComb, M. C. Richardson, and M. Bass, “High-power fiber lasers and amplifiers,” in Handbook of Optics, 3rd ed., M.Bass, ed. (McGraw-Hill, 2010) Vol. 5, Chap. 25.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Optical Fiber Sensors Conference Proceedings (1988), Vol. 1, PD5.

W. Koechner, Solid-State Laser Engineering Fifth Revised and Updated Edition (Springer, 1999).

V. Sudesh, T. McComb, M. Richardson, W. Hageman, M. Bass, J. Ballato, and A. Siegman, “Waveguide-pumping gain-guided index-antiguided fiber laser,” U.S. patent 7,668,211 (February 23, 2010).

S. O. Kasap, Optoelectronics and Photonics: Principles and Practices (Prentice Hall, 2001).

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Figures (11)

Fig. 1
Fig. 1

Calculated optimal GG IAG fiber lengths that would result in single-mode operation with a 90% R output coupler and HR mirror for various fiber core sizes. Calculations assume Δ n of 4.5 × 10 3 and α = 0 .

Fig. 2
Fig. 2

Diagram of gain thresholds pertinent to the maintenance of single LP 01 mode lasing in a GG IAG fiber. The crosshatched region denotes the allowed range of average gain in the fiber core resulting in laser threshold. The broader gray region denotes the allowed range of gain that can be allowed throughout the fiber to assure lossless LP 01 mode propagation.

Fig. 3
Fig. 3

Calculated normalized absorbed pump power along the rod axis for 1% and 10% Nd 3 + doping for pumping at 802 nm. To maintain only a 60% variation in gain (shown by the dotted vertical lines), the pump length cannot be more than 1   mm even for 1% doping.

Fig. 4
Fig. 4

Diagram of side-pumped fiber concept. (A) The fiber is pressed into molten indium filled copper V-groove. (B) The fiber sides are completely coated with molten indium which is then allowed to solidify. (C) Entire fixture is polished until fiber and indium are level with copper surface. The fiber now has a polished window for pump light to enter.

Fig. 5
Fig. 5

Images of simulated pump energy distribution in the core of the GG IAG fiber. Three different models are shown for the type of reflection at the fiber–indium interface: pure specular reflection, a Gaussian roughness model that simulates the roughness of ground glass, and pure random scattering at the interface. All images in row (a) were obtained using a cylinder rod with diameter 1.2 cm as a focusing lens. Images in row (b) were obtained with a cylinder rod with diameter 0.6 cm. In all images, light is initially incident from below the image.

Fig. 6
Fig. 6

Pump efficiency versus the reflectivity of the cladding–metal interface. The simulation was conducted for a side-pump system with 6 mm diameter rod lens focused at a 120 μ m pump window.

Fig. 7
Fig. 7

Photograph of GG IAG side-pump setup with four-bar linear array and 4 cm fiber.

Fig. 8
Fig. 8

Beam profile of the GG IAG laser beam pumped with a Dilas linear array. The profile was captured with Spiricon LBA-PC software and a Pulnix 745 camera.

Fig. 9
Fig. 9

Measured output versus input energy for the 4 cm long GG IAG laser pumped with a Dilas linear array with repetition rate of 1 Hz and pulse duration of 3 ms.

Fig. 10
Fig. 10

Modeled and experimental laser output curves. The modeled curve utilized a pump efficiency of 60% and a waveguide loss of 0.1388 cm 1 .

Fig. 11
Fig. 11

Calculated optical pump efficiency versus the fiber core diameter. These results are for 100% reflectivity at the fiber–metal interface, 10% Nd doped Kigre Q100 glass core, and pump light at the peak absorption wavelength of 802 nm.

Equations (14)

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

V ̃ 2 Δ N + j G ,
Δ N ( 2 π a λ ) 2 2 n 0 Δ n ,
G ( 2 π a λ ) 2 ( n 0 λ 2 π ) g .
G 01 4 j 01 4 Δ N 133.8 Δ N ,
G 11 4 j 11 4 Δ N 862.2 Δ N .
g t h osc > g g g 01 .
g t h osc = 1 2 l g ln ( R 1 R 2 ) + α ,
g g g 01 = 133.8 2 n 0 3 ( Δ n ) λ 2 ( 2 π ) 2 a 3 .
g t h osc + g g g 01 < g g g 11 .
2 g g g 01 < g tot osc < g g g 11 ,
g g g 01 < g t h osc < ( g g g 11 g g g 10 ) .
ln ( R 1 R 2 ) 133.8 2 n 0 3 ( Δ n ) λ 2 2 π 2 a 3 2 α > l g > ln ( R 1 R 2 ) 316.7 2 n 0 3 ( Δ n ) λ 2 2 π 2 a 3 2 α .
Δ abs_average = G max G min G max = G 11 2 G 01 G 11 = 862.2 Δ N 2 133.8 Δ N 862.2 Δ N = 21 % .
Δ abs_total = G max G min G max = G 11 G 01 G 11 = 862.2 Δ N 133.8 Δ N 862.2 Δ N = 61 % .

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