Abstract

We present modeling studies of quasi-three-level laser oscillators, the validity of which was revealed by Ti:Al2O3-pumped Yb:YAG laser experiments, and these results are shown to be in excellent agreement with the theory. As much as 75% slope efficiency was obtained with a hemispherical laser cavity. Previous modeling calculations of laser performance have been valid only for certain special cases, restricting application to TEM00 Gaussian beam pumping and lasing profiles. This analysis may be applied to other longitudinally pumped quasi-three-level laser media in which the modes are not only TEM00 Gaussian beams but also other higher-order transverse modes, including the top-hat pumping profile that can be used to model transverse pumping schemes.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Y. Fan, R. L. Byer, “Modeling and CW operation of a quasi-three-level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron. QE-23, 605–612 (1987).
  2. P. J. M. Suni, S. W. Henderson, “1-mJ/pulse Tm:YAG laser pumped by a 3-W diode laser,” Opt. Lett. 16, 817–819 (1991).
    [CrossRef] [PubMed]
  3. T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
    [CrossRef]
  4. P. Lacovara, H. K. Choi, C. A. Wang, R. L. Aggarwal, T. Y. Fan, “Room-temperature diode-pumped Yb:YAG laser,” Opt. Lett. 16, 1089–1091 (1991).
    [CrossRef] [PubMed]
  5. T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level laser,” IEEE J. Quantum Electron. 28, 2692–2697 (1992).
    [CrossRef]
  6. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
    [CrossRef]
  7. T. Y. Fan, S. Klunk, G. Henein, “Diode-pumped Q-switched Yb:YAG laser,” Opt. Lett. 18, 423–425 (1993).
    [CrossRef] [PubMed]
  8. W. F. Krupke, L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1990).
    [CrossRef]
  9. L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
    [CrossRef]
  10. D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
    [CrossRef]
  11. A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
    [CrossRef]
  12. K. Kubodera, K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653–659 (1979).
    [CrossRef]
  13. P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21, 1582–1595 (1985).
    [CrossRef]
  14. W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5, 1412–1423 (1988).
    [CrossRef]
  15. D. S. Sumida, T. Y. Fan, “Effect of radiation trapping on fluorescence lifetime and emission cross section measurements in solid-state laser media,” Opt. Lett. 19, 1343–1345 (1994).
    [CrossRef] [PubMed]
  16. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
    [CrossRef] [PubMed]

1995 (1)

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

1994 (3)

1993 (3)

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

T. Y. Fan, S. Klunk, G. Henein, “Diode-pumped Q-switched Yb:YAG laser,” Opt. Lett. 18, 423–425 (1993).
[CrossRef] [PubMed]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

1992 (1)

T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level laser,” IEEE J. Quantum Electron. 28, 2692–2697 (1992).
[CrossRef]

1991 (2)

1990 (1)

W. F. Krupke, L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1990).
[CrossRef]

1988 (2)

T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
[CrossRef]

W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5, 1412–1423 (1988).
[CrossRef]

1987 (1)

T. Y. Fan, R. L. Byer, “Modeling and CW operation of a quasi-three-level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron. QE-23, 605–612 (1987).

1985 (1)

P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21, 1582–1595 (1985).
[CrossRef]

1979 (1)

K. Kubodera, K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653–659 (1979).
[CrossRef]

Aggarwal, R. L.

Brauch, U.

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

Byer, R. L.

T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
[CrossRef]

T. Y. Fan, R. L. Byer, “Modeling and CW operation of a quasi-three-level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron. QE-23, 605–612 (1987).

Chambaz, B.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Chartier, I.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Chase, L. L.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

W. F. Krupke, L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1990).
[CrossRef]

Choi, H. K.

DeLoach, L. D.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Fan, T. Y.

D. S. Sumida, T. Y. Fan, “Effect of radiation trapping on fluorescence lifetime and emission cross section measurements in solid-state laser media,” Opt. Lett. 19, 1343–1345 (1994).
[CrossRef] [PubMed]

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
[CrossRef] [PubMed]

T. Y. Fan, S. Klunk, G. Henein, “Diode-pumped Q-switched Yb:YAG laser,” Opt. Lett. 18, 423–425 (1993).
[CrossRef] [PubMed]

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level laser,” IEEE J. Quantum Electron. 28, 2692–2697 (1992).
[CrossRef]

P. Lacovara, H. K. Choi, C. A. Wang, R. L. Aggarwal, T. Y. Fan, “Room-temperature diode-pumped Yb:YAG laser,” Opt. Lett. 16, 1089–1091 (1991).
[CrossRef] [PubMed]

T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
[CrossRef]

T. Y. Fan, R. L. Byer, “Modeling and CW operation of a quasi-three-level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron. QE-23, 605–612 (1987).

Ferrand, B.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Giesen, A.

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

Hanna, D. C.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Henderson, S. W.

Henein, G.

Huber, G.

T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
[CrossRef]

Hugel, H.

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

Klunk, S.

Krupke, W. F.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

W. F. Krupke, L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1990).
[CrossRef]

Kubodera, K.

K. Kubodera, K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653–659 (1979).
[CrossRef]

Kway, W. L.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Lacovara, P.

Large, A. C.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Mitzscherlich, P.

T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
[CrossRef]

Moulton, P. F.

P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21, 1582–1595 (1985).
[CrossRef]

Opower, H.

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

Otsuka, K.

K. Kubodera, K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653–659 (1979).
[CrossRef]

Payne, S. A.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Pelenc, D.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Risk, W. P.

Shepherd, D. P.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Smith, L. K.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Sumida, D. S.

Suni, P. J. M.

Tropper, A. C.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Voss, A.

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

Wang, C. A.

Witting, K.

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

Wyon, C.

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Appl. Phys. B (1)

A. Giesen, H. Hugel, A. Voss, K. Witting, U. Brauch, H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[CrossRef]

IEEE J. Quantum Electron. (6)

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, W. F. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

P. F. Moulton, “An investigation of the Co:MgF2 laser system,” IEEE J. Quantum Electron. 21, 1582–1595 (1985).
[CrossRef]

T. Y. Fan, R. L. Byer, “Modeling and CW operation of a quasi-three-level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron. QE-23, 605–612 (1987).

T. Y. Fan, G. Huber, R. L. Byer, P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of TmHo:YAG,” IEEE J. Quantum Electron. QE-24, 924–933 (1988).
[CrossRef]

T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level laser,” IEEE J. Quantum Electron. 28, 2692–2697 (1992).
[CrossRef]

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

J. Appl. Phys. (1)

K. Kubodera, K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653–659 (1979).
[CrossRef]

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

Opt. Commun. (1)

D. Pelenc, B. Chambaz, I. Chartier, B. Ferrand, C. Wyon, D. P. Shepherd, D. C. Hanna, A. C. Large, A. C. Tropper, “High slope efficiency and low-threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser,” Opt. Commun. 115, 491–497 (1995).
[CrossRef]

Opt. Lett. (5)

Opt. Quantum Electron. (1)

W. F. Krupke, L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1990).
[CrossRef]

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 (9)

Fig. 1
Fig. 1

Scheme of crystal splitting of the 2 F 7/2 and 2 F 5/2 levels of the Yb3+ ion in a YAG laser.8 The levels of the upper and lower manifolds and their energies are shown. The laser transition is noted by the heavy line. The populations of the upper and lower laser levels are given by N 2 and N 1, respectively.

Fig. 2
Fig. 2

Absorption spectra of 10-at. % Yb:YAG at room temperature with 0.1-nm resolution. The circles represent data points taken with a Ti:Al2O3 laser.

Fig. 3
Fig. 3

Schematic of a Ti:Al2O3 laser pumped with a Yb:YAG short cavity laser.

Fig. 4
Fig. 4

Yb:YAG threshold power (P th) and output power (P o) versus the cavity length that results in a change of the laser mode spot size. For the output power measurement the 243.8-mW incident pump power was held constant at 913 nm.

Fig. 5
Fig. 5

Yb:YAG output power versus absorbed power at 913 nm. The solid curves represent numerically generated plots. In the calculation we used the values of N 0 = 13.8 × 1020 ions/cm3, σ = 3.3 × 10-20 cm2, τ f = 0.951 ms,15 η p = 1.0, L i = 0.6%, and R p = 100%.

Fig. 6
Fig. 6

Calculated threshold ratio of the top-hat to the Gaussian beam pumping scheme. As a approaches 0 both values become equal. For a ≠ 0, the threshold of the top-hat pumping scheme is higher than that for the Gaussian beam. (The maximum value is 2.)

Fig. 7
Fig. 7

Numerically generated plots of the mode-matching efficiency η m = dS/dF for the top-hat pump condition as a function of a = w p /w o for different values of B, F/F th = 10. It is obvious that at a much higher pumping region (F/F th > 5) the mode-matching efficiency of the top-hat beam pumping will increase and perform well compared with the Gaussian beam pumping scheme.

Fig. 8
Fig. 8

Calculation of the mode-matching results as a function of pumping power ratio for top-hat beam pumping: a = 1.2.

Fig. 9
Fig. 9

Schematic of a longitudinally pumped solid-state laser.

Equations (37)

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

Pth=hvpVeff/lc*2ηpηaf1+f2στfLi+T+2N10σl,
Veff=1/crystalrpx, y, zϕox, y, zdV,
ηm  dSdF=1+Blc*/nlcrystalϕox, y, z1+Sϕox, y, zdVF crystalFrpx, y, z-Blc*/nlϕo2x, y, z1+Sϕox, y, z2dV,
rpr, z=2απwp2ηa exp-αzexp-2r2wp2=αlc*ηaa2C exp-αzexp-x,
ϕor, z=2πwo2lc* exp-2r2wo2=exp-a2xC,
C=πwo2lc*/2,
a=wp/wo,
x=2r2/wp2,
Pth=πhνpwo21+a24ηpηaf1+f2στfLi+T+2N10σl.
ηm=1+BS/C ln1+S/CF/C0F/Cexp-x-Ba21+S/Cexp-a2x2 exp-2a2xdx.
rpr, z=2απwp2ηa exp-2r2wp2×exp-αz+exp-αz-2l=αlc*ηaa2C exp-xexp-αz+exp-αz-2l,
ηa=1-exp-αl1+Rp exp-αl,
rpr, z=απwp2ηa exp-αz=αlc*2ηaa2C exp-αz, r wp,
=0   r > wp,
FC=21+BS/C ln1+S/C02exp-a2x1+S/Cexp-a2xdx.
FthC=2a21+B1-exp-2a2.
Pth=πhνpwo24ηpηaf1+f2στf2a21-exp2a2×Li+T+2N10σl.
Pth top-hatPth Gaussian=2a21+a21-exp2a2
ηm=21+BS/C ln1+S/CF/CF/C02exp-2a2x1+S/Cexp-a2x2dx-2Ba20exp-2a2x1+S/Cexp-a2x2dx.
ηmS0=1-exp-2a2a211+1+Bexp-2a2.
rpr, z=2πwp2ηa exp-2r2wp2=exp-xa2C  for Gaussian,
rpr, z=1πwp2l=12a2C,  r  wp  for top-hat.
dΔNx, y, zdt=0=f1+f2Rrpx, y, z-ΔNx, y, z-ΔN0τf-f1+f2cσΔNx, y, zn Φϕox, y, z,
dΦdt=0=cσncrystal ΔNx, y, zΦϕox, y, zdV-Φτc,
R=ηpηaPphνp,
Φ=2lc*PocThνL,
crystal rpx, y, zdV=x=0y=0z=0lrpx, y, zdxdydz=1,
cavityϕox, y, zdV=x=0y=00l nϕox, y, zdz+llc ϕox, y, zdzdxdy=1.
ΔNx, y, z=f1+f2τfRrpx, y, z-N101+f1+f2cστfn Φϕox, y, z.
2σlc*ncrystalf1+f2τfRrpx, y, z-N101+f1+f2cστfn Φϕox, y, z×ϕox, y, zdV=δ.
F=2f1+f2στflc*nδ R,
S=f1+f2cστfnΦ,
B=2N10σlδ.
F=1+Blc*/nlcrystalϕox, y, z1+Sϕox, y, zdVcrystalrpx, y, zϕox, y, z1+Sϕox, y, zdV,
Fth=1+Blc*/nlcrystal ϕox, y, zdVcrystalrpx, y, zϕox, y, zdV=1+Blc*/nlcrystalϕox, y, zdVVeff,
ηs=dPodPp=ηpηaTδνLνPdSdF,
dSdFS,B0=crystal rpx, y, zϕox, y, zdV2crystal rpx, y, zϕo2x, y, zdV

Metrics