Abstract

A model is developed that describes the power extraction in chemical oxygen-iodine lasers (COIL’s) and CO2 gasdynamic lasers with stable resonators when a large number of transverse Hermite–Gaussian eigenmodes oscillate. The extraction efficiency, mode intensities, and intensity distribution along the flow depend only on two parameters. The first is the ratio γ0 of the residence time of the gas in the resonator to the O2(1Δ) or N2(v) energy extraction time and the second is the ratio of the threshold to the small-signal gain. The efficiency is maximum for γ0 → ∞ and decreases rapidly as γ0 decreases. It is found that for a range of parameters corresponding to the highest efficiencies the intensity distribution along the flow is nonuniform and has two peaks near the upstream and downstream sections of the resonator. In this case only the highest-order modes that totally fill the resonator cross section oscillate (the so-called, experimentally observed sugar scooping bimodal intensity distribution). For the range of parameters corresponding to smaller efficiencies the intensity is uniform. In this case all the modes participate in lasing; however, the intensities of the high-order modes are larger than those of the low order. The current model is compared with the plane-mirror Fabry–Perot resonator model and with the constant intraresonator intensity and rooftop models of COIL’s with stable resonators. The extraction efficiency calculated with the last two models is close to that estimated from our model. However, the intensity distribution cannot be calculated correctly using the Fabry–Perot, the constant intraresonator intensity, or the rooftop model.

© 1998 Optical Society of America

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References

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  1. D. A. Copeland, A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29, 2525–2539 (1993).
    [CrossRef]
  2. B. D. Barmashenko, S. Rosenwaks, “Analysis of the optical extraction efficiency in gas-flow lasers with different types of resonator,” Appl. Opt. 35, 7091–7101 (1996).
    [CrossRef] [PubMed]
  3. M. V. Zagidullin, V. D. Nikolaev, “Gain saturation and the efficiency of energy conversion into radiation in a supersonic oxygen-iodine laser with a stable cavity,” Quantum Electron. 27, 411–416 (1997).
    [CrossRef]
  4. B. D. Barmashenko, S. Rosenwaks, “Analysis of lasing in COILs with wide aperture of the mirrors in the resonator,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, H. J. Baker, ed., Proc. SPIE3092, 682–685 (1997).
    [CrossRef]
  5. J. D. Anderson, Gasdynamic lasers: an Introduction (Academic, New York, 1976).
  6. J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).
  7. G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
    [CrossRef]
  8. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  9. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).
  10. V. K. Konyukhov, “Gasdynamic cw CO2 laser,” J. Sov. Laser Res. 3, 93–148 (1982).
    [CrossRef]
  11. Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
    [CrossRef]
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1992).
  13. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

1997 (1)

M. V. Zagidullin, V. D. Nikolaev, “Gain saturation and the efficiency of energy conversion into radiation in a supersonic oxygen-iodine laser with a stable cavity,” Quantum Electron. 27, 411–416 (1997).
[CrossRef]

1996 (2)

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

B. D. Barmashenko, S. Rosenwaks, “Analysis of the optical extraction efficiency in gas-flow lasers with different types of resonator,” Appl. Opt. 35, 7091–7101 (1996).
[CrossRef] [PubMed]

1993 (1)

D. A. Copeland, A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29, 2525–2539 (1993).
[CrossRef]

1982 (1)

V. K. Konyukhov, “Gasdynamic cw CO2 laser,” J. Sov. Laser Res. 3, 93–148 (1982).
[CrossRef]

1974 (1)

Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Anan’ev, Yu. A.

Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Anderson, J. D.

J. D. Anderson, Gasdynamic lasers: an Introduction (Academic, New York, 1976).

Barmashenko, B. D.

B. D. Barmashenko, S. Rosenwaks, “Analysis of the optical extraction efficiency in gas-flow lasers with different types of resonator,” Appl. Opt. 35, 7091–7101 (1996).
[CrossRef] [PubMed]

B. D. Barmashenko, S. Rosenwaks, “Analysis of lasing in COILs with wide aperture of the mirrors in the resonator,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, H. J. Baker, ed., Proc. SPIE3092, 682–685 (1997).
[CrossRef]

Bauer, A. H.

D. A. Copeland, A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29, 2525–2539 (1993).
[CrossRef]

Copeland, D. A.

D. A. Copeland, A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29, 2525–2539 (1993).
[CrossRef]

Crowell, P.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

Erkkila, J.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1992).

Hager, G. D.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).

Helms, C. A.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).

Konyukhov, V. K.

V. K. Konyukhov, “Gasdynamic cw CO2 laser,” J. Sov. Laser Res. 3, 93–148 (1982).
[CrossRef]

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Kovalchuk, L. V.

Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Nikolaev, V. D.

M. V. Zagidullin, V. D. Nikolaev, “Gain saturation and the efficiency of energy conversion into radiation in a supersonic oxygen-iodine laser with a stable cavity,” Quantum Electron. 27, 411–416 (1997).
[CrossRef]

Plummer, D.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1992).

Rosenwaks, S.

B. D. Barmashenko, S. Rosenwaks, “Analysis of the optical extraction efficiency in gas-flow lasers with different types of resonator,” Appl. Opt. 35, 7091–7101 (1996).
[CrossRef] [PubMed]

B. D. Barmashenko, S. Rosenwaks, “Analysis of lasing in COILs with wide aperture of the mirrors in the resonator,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, H. J. Baker, ed., Proc. SPIE3092, 682–685 (1997).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).

Scott, J. E.

J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).

Shaw, J.

J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).

Sherstobitov, V. E.

Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1992).

Truesdell, K. A.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).

Trusov, V. P.

Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1992).

Zagidullin, M. V.

M. V. Zagidullin, V. D. Nikolaev, “Gain saturation and the efficiency of energy conversion into radiation in a supersonic oxygen-iodine laser with a stable cavity,” Quantum Electron. 27, 411–416 (1997).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

D. A. Copeland, A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29, 2525–2539 (1993).
[CrossRef]

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32, 1525–1536 (1996).
[CrossRef]

J. Sov. Laser Res. (1)

V. K. Konyukhov, “Gasdynamic cw CO2 laser,” J. Sov. Laser Res. 3, 93–148 (1982).
[CrossRef]

Quantum Electron. (1)

M. V. Zagidullin, V. D. Nikolaev, “Gain saturation and the efficiency of energy conversion into radiation in a supersonic oxygen-iodine laser with a stable cavity,” Quantum Electron. 27, 411–416 (1997).
[CrossRef]

Sov. J. Quantum Electron. (1)

Yu. A. Anan’ev, L. V. Kovalchuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Other (7)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1992).

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

B. D. Barmashenko, S. Rosenwaks, “Analysis of lasing in COILs with wide aperture of the mirrors in the resonator,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, H. J. Baker, ed., Proc. SPIE3092, 682–685 (1997).
[CrossRef]

J. D. Anderson, Gasdynamic lasers: an Introduction (Academic, New York, 1976).

J. E. Scott, K. A. Truesdell, C. A. Helms, J. Shaw, G. D. Hager, “Design considerations for the chemical oxygen-iodine supersonic mixing nozzle,” paper AIAA94-2436, presented at the 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, 555 West 57th Street, New York, N.Y. 10019, 1994).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).

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Figures (12)

Fig. 1
Fig. 1

Dependencies of η m , η m,F–P*, and η m,Rig on g th/g 0 for COIL’s. γ0 → ∞, Y i = 0.5, and T = 170 K.

Fig. 2
Fig. 2

Dependencies of η m on g th/g 0 for COIL’s with stable and Fabry–Perot resonators for different values of γ0. Y i = 0.5 and T = 170 K.

Fig. 3
Fig. 3

Dependencies of η m on γ0 for COIL’s with stable and Fabry–Perot resonators for g th/g 0 = 0.5. Y i = 0.5, and T = 170 K.

Fig. 4
Fig. 4

Dependencies of I i on i/ N for COIL’s with different values of γ0 and g th/g 0 = 0.1. Y i = 0.5 and T = 170 K.

Fig. 5
Fig. 5

I(x) for COIL’s with different values of γ0 and g th/g 0= 0.1. Y i = 0.5 and T = 170 K.

Fig. 6
Fig. 6

Dependencies of I i on i/ N for COIL’s with different values of g th/g 0 and γ0 = 5. Y i = 0.5 and T = 170 K.

Fig. 7
Fig. 7

I(x) for COIL’s with different values of g th/g 0 and γ0 = 5. Y i = 0.5 and T = 170 K.

Fig. 8
Fig. 8

Dependencies of η m on g th/g 0 for COIL’s with different values of γ0. Y i = 0.5 and T = 170 K.

Fig. 9
Fig. 9

I(x) for COIL’s with g th/g 0 = 0.1 and γ0 = 20. Y i = 0.5 and Y = 170 K.

Fig. 10
Fig. 10

Dependencies of η m on g th/g 0 for GDL’s with different values of γ0.

Fig. 11
Fig. 11

Dependencies of I i on i/ N for GDL’s with different values of γ0 and g th/g 0 = 0.1.

Fig. 12
Fig. 12

I(x) for GDL’s with different values of γ0 and g th/g 0= 0.1.

Equations (59)

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O 2 1 Δ + I     O 2 3 Σ + I * ,
N 2 v = 1 + CO 2     N 2 + CO 2 00 0 1 ,
U   d Y d x = - g O 2 I h ν ,
g = σ   I 0 2 2 K e + 1 Y - 1 K e - 1 Y + 1 1 1 + I / I SD ,
I SD h ν 2 k f O 2 3 σ K e K e - 1 Y + 1
I x = i = 1 N   I i f i x ,
f i x = 2 / π 1 / 2 2 i i ! 1 w exp 2 x 2 w 2 H i 2 x 2 w ,
w = λ L m π g 1 - g 1 / 2 ,   g = 1 - L m / R 2 .
f i x = 2 π w cos 2 0 x 2 w 2 i - y 2 d y - π i / 2 i - x 2 / w 2 ,   | x | < x i , f i x = 1 2 π w exp - 2   2 i | x | 2 w y 2 - 2 i d y x 2 / w 2 - i ,   | x | > x i ,
x i = w i = l res 2 i N 1 / 2
N = 0.5 l res / w 2 .
f i x = 1 π 1 x i 2 - x 2 , | x | < x i 0 , | x | > x i
-   gf i x d x = g th , I i > 0 g th , I i = 0 ,
δ dif i = exp - 2 N 2 1 - i / N - i / N ln   1 + 1 - i / N i / N N 2 1 - i / N .
P out = P av η ext ,
P av = h ν O 2 ULH Y 0 - 1 2 K e + 1 1 - exp - γ 0 ,
γ 0 l res Ut e ,
t e = 3 K e k f I 0 2 K e + 1
η r = t t + a + i = 1 N   δ dif i I i i = 1 N   I i ,
η m = Y 0 - Y f Y 0 - 1 / 2 K e + 1 1 1 - exp - γ 0 ,
U   d CO 2 * d x = - k f N 2 CO 2 * + k f CO 2 N 2 * - σ I / h ν CO 2 * ,
U   d N 2 * d x = k f N 2 CO 2 * - k f CO 2 N 2 * ,
g = σ N 2 * CO 2 / N 2 1 + I / I S ,
I S = h ν k f N 2 σ .
U   d N 2 * d x = - g I h ν .
I S , g = h ν 1 σ l res / U ,
d ν d ξ = - φ   ν 1 + φ / γ 0 ,     ν ξ = - 1 / 2 = 1 ,
- 1 / 2 1 / 2 ν 1 + φ / γ 0   f i ξ d ξ   = g th / g 0 ,   φ i > 0 , g th / g 0 ,   φ i = 0 ,
t e = 1 k f CO 2 0 .
P av = h ν N 2 * 0 ULH 1 - exp - γ 0 ,
η m = 1 - ν ξ = 1 / 2
η m + 1 - η m , F P * ln 1 - η m = 0 ,
η m , F P * = 1 - g th g 0 1 - g th g 0 Y 0 - 1 2 K e + 1 Y 0 + 1 K e - 1
η m , Rig = 1 - g th / g 0 .
η m , F P = η m , F P *   1 - exp - γ 0 1 - g th / g 0 / η m , F P * 1 - exp - γ 0 .
I x P = 1 π l res 2 2 - x 2 1 / 2 .
I i / P / N = 1 2 1 - i / N .
- l res / 2 l res / 2 g d x l res = g th
g - x + g x 2 = g th .
η m + 1 - η m , F P * ln 1 - η m = - η m , F P * γ 0 1 - g th g 0 × η m ln 1 - η m ,
η m , F P * = η m , F P = η m , Rig = 1 - g th g 0 .
I S , g h ν σ l res / U I 0 / O 2 2 K e - 1 Y i + 1 2 K e + 1
d Y d ξ = - ϕ   Y - 1 2 K e + 1 y + ϕ / γ 0 ,     Y ξ = - 1 / 2 = Y 0 ,
- 1 / 2 1 / 2 F Y F Y i 1 + ϕ / y γ 0   f i ξ d ξ   = g th / g 0 , φ i > 0 g th / g 0 , φ i = 0 ,
y = K e - 1 Y + 1 K e - 1 Y 0 + 1 ,
F Y = Y - 1 / 2 K e + 1 Y + 1 / K e + 1 ,
f i ξ = 1 π 1 ξ i 2 - ξ 2 , 0 ,   | ξ | < ξ i   | ξ | > ξ i ,
ξ i = 1 2 i N 1 / 2 ,
φ ξ = i = 1 N   φ i f i ξ .
d Y d ξ = - ψ   Y - 1 2 K e + 1 y + ψ / γ 0 ,   Y ξ = - 1 / 2 = Y 0 ,
- 1 / 2 1 / 2 F Y F Y i 1 + ψ / y γ 0   f i ξ exp - ψ i 1 - θ ψ i d ξ = g th / g 0 ,   i = 1 , ,   N ,
ψ ξ = i = 1 N   ψ i f i ξ θ ψ i ,
θ x 1 , x 0 0 , x < 0 .
φ i = ψ i , ψ i 0 0 , ψ i < 0 ,
φ ξ ,   η = i , j = 1 i = N , j = N y   φ ij f i ξ f j η ,
| ξ | , | η | < 1 / 2 F Y F Y i 1 + φ / y γ 0   f i ξ f j η × d ξ d η = g th / g 0 , φ ij > 0 g th / g 0 , φ ij = 0 .
j = 1 N y   φ ij f j η = φ i ,
4 η 2 1 N y φ ij d j / N y j / N y / 4 - η 2 1 / 2 = φ i .
φ ij = φ i 2 N y 1 - j / N y .

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