Abstract

We demonstrate the feasibility of passive compensation of the thermal lens effect in fused silica optics, placing suitable optical materials with negative dn/dT in the beam path of a high power near IR fiber laser. Following a brief overview of the involved mechanisms, photo-thermal absorption measurements with a Hartmann-Shack sensor are described, from which coefficients for surface/coating and bulk absorption in various materials are determined. Based on comprehensive knowledge of the 2D wavefront deformations resulting from absorption, passive compensation of thermally induced aberrations in complex optical systems is possible, as illustrated for an F-Theta objective. By means of caustic measurements during high-power operation we are able to demonstrate a 60% reduction of the focal shift in F-Theta lenses through passive compensation.

© 2014 Optical Society of America

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References

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  1. P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
    [Crossref]
  2. C. Wandera, “Performance of high power fibre laser cutting of thick-section steel and medium-section aluminium,“ Ph.D. thesis, Lappeenranta University of Technology (2010).
  3. O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
    [Crossref]
  4. S. Sinha, J. D. Mansell, and R. L. Byer, “Deformable mirrors for high-power lasers,” Proc. SPIE 4493, 55–63 (2002).
    [Crossref]
  5. G. Eberle, V. Chiron, and K. Wegener, “Simulation and realization of a focus shifting unit using a tunable lens for 3D laser material processing,” Phys. Procedia 41, 441–447 (2013).
    [Crossref]
  6. A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).
  7. D. Havrilla, M. Holzer, and P. Olschowsky, “Latest research results with high power solid state laser optics with minimized focus shift,” Proceedings of International Congress on Applications of Lasers & Electro-Optics (ICALEO), pp. 29–32 (2010).
  8. J. J. Kasinski and R. L. Burnham, “Near-diffraction-limited, high-energy, high-power, diode-pumped laser using thermal aberration correction with aspheric diamond-turned optics,” Appl. Opt. 35(30), 5949–5954 (1996).
    [Crossref] [PubMed]
  9. M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
    [Crossref]
  10. M. Scaggs and G. Haas, “Thermal lensing compensation objective for high power lasers,” Proc. SPIE 7913, 79130C (2011).
    [Crossref]
  11. S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
    [Crossref]
  12. B. Schäfer, J. Gloger, U. Leinhos, and K. Mann, “Photo-thermal measurement of absorptance losses, temperature induced wavefront deformation and compaction in DUV-optics,” Opt. Express 17(25), 23025–23036 (2009).
    [Crossref] [PubMed]
  13. B. Schäfer, M. Schöneck, A. Bayer, and K. Mann, “Absolute measurement of surface and bulk absorption in DUV optics from temperature induced wavefront deformation,” Opt. Express 18(21), 21534–21539 (2010).
    [Crossref] [PubMed]
  14. D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” J. Quantum Electron. 34(12), 2383–2392 (1998).
    [Crossref]
  15. A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
    [Crossref]
  16. P. G. Nelson, “A thermal analysis of a 1.5 meter f/5 fused silica primary lens for solar telescopes,” Coronal Solar Magnetism Observatory, Tech. Note 13, Rev. 2 (2007). http://www.cosmo.ucar.edu/publications/nelson_tech13r2_03-07.pdf
  17. D. E. Patent, 102010052471 B3, Messvorrichtung und Messverfahren zur Bestimmung einer optischen Qualität einer Prüfoptik“ (2012).
  18. M. Stubenvoll, B. Schäfer, K. Mann, A. Walter, and L. Zittel, “Photothermal absorption measurements for improved thermal stability of high-power laser optics,” Proc. SPIE 8885, 88851R (2013).
    [Crossref]
  19. ISO 1146, “Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios,” International Organization for Standardization (2005).

2014 (1)

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

2013 (3)

P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
[Crossref]

G. Eberle, V. Chiron, and K. Wegener, “Simulation and realization of a focus shifting unit using a tunable lens for 3D laser material processing,” Phys. Procedia 41, 441–447 (2013).
[Crossref]

M. Stubenvoll, B. Schäfer, K. Mann, A. Walter, and L. Zittel, “Photothermal absorption measurements for improved thermal stability of high-power laser optics,” Proc. SPIE 8885, 88851R (2013).
[Crossref]

2012 (1)

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

2011 (1)

M. Scaggs and G. Haas, “Thermal lensing compensation objective for high power lasers,” Proc. SPIE 7913, 79130C (2011).
[Crossref]

2010 (1)

2009 (1)

2008 (1)

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

2004 (1)

M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
[Crossref]

2003 (1)

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

2002 (1)

S. Sinha, J. D. Mansell, and R. L. Byer, “Deformable mirrors for high-power lasers,” Proc. SPIE 4493, 55–63 (2002).
[Crossref]

1998 (1)

D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” J. Quantum Electron. 34(12), 2383–2392 (1998).
[Crossref]

1996 (1)

Abdou Ahmed, M.

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

Abt, F.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Alexandrov, A.

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

Andrade, A. A.

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

Baesso, M. L.

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

Bayer, A.

Blomqvist, M.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Blomster, O.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Bodnar, I.

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

Brown, D. C.

D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” J. Quantum Electron. 34(12), 2383–2392 (1998).
[Crossref]

Burnham, R. L.

Byer, R. L.

S. Sinha, J. D. Mansell, and R. L. Byer, “Deformable mirrors for high-power lasers,” Proc. SPIE 4493, 55–63 (2002).
[Crossref]

Catunda, T.

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

Chiron, V.

G. Eberle, V. Chiron, and K. Wegener, “Simulation and realization of a focus shifting unit using a tunable lens for 3D laser material processing,” Phys. Procedia 41, 441–447 (2013).
[Crossref]

Dausinger, F.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Deininger, C.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Eberle, G.

G. Eberle, V. Chiron, and K. Wegener, “Simulation and realization of a focus shifting unit using a tunable lens for 3D laser material processing,” Phys. Procedia 41, 441–447 (2013).
[Crossref]

Gloger, J.

Graf, T.

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
[Crossref]

Haas, G.

M. Scaggs and G. Haas, “Thermal lensing compensation objective for high power lasers,” Proc. SPIE 7913, 79130C (2011).
[Crossref]

Huonker, M.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Kah, P.

P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
[Crossref]

Kasinski, J. J.

Kudryashov, A.

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

Leinhos, U.

Lu, J.

P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
[Crossref]

Mann, K.

Mansell, J. D.

S. Sinha, J. D. Mansell, and R. L. Byer, “Deformable mirrors for high-power lasers,” Proc. SPIE 4493, 55–63 (2002).
[Crossref]

Martikainen, J.

P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
[Crossref]

Mura, J.

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

Pålsson, M.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Piehler, S.

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

Romanov, P.

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

Roos, S.-O.

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

Roth, M.

M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
[Crossref]

Samarkin, V.

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

Scaggs, M.

M. Scaggs and G. Haas, “Thermal lensing compensation objective for high power lasers,” Proc. SPIE 7913, 79130C (2011).
[Crossref]

Schäfer, B.

Schöneck, M.

Sheldakova, J.

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

Sinha, S.

S. Sinha, J. D. Mansell, and R. L. Byer, “Deformable mirrors for high-power lasers,” Proc. SPIE 4493, 55–63 (2002).
[Crossref]

Stubenvoll, M.

M. Stubenvoll, B. Schäfer, K. Mann, A. Walter, and L. Zittel, “Photothermal absorption measurements for improved thermal stability of high-power laser optics,” Proc. SPIE 8885, 88851R (2013).
[Crossref]

Suoranta, R.

P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
[Crossref]

Thiel, C.

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

Voss, A.

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

Walter, A.

M. Stubenvoll, B. Schäfer, K. Mann, A. Walter, and L. Zittel, “Photothermal absorption measurements for improved thermal stability of high-power laser optics,” Proc. SPIE 8885, 88851R (2013).
[Crossref]

Weber, H.

M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
[Crossref]

Wegener, K.

G. Eberle, V. Chiron, and K. Wegener, “Simulation and realization of a focus shifting unit using a tunable lens for 3D laser material processing,” Phys. Procedia 41, 441–447 (2013).
[Crossref]

Wyss, E.

M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
[Crossref]

Zittel, L.

M. Stubenvoll, B. Schäfer, K. Mann, A. Walter, and L. Zittel, “Photothermal absorption measurements for improved thermal stability of high-power laser optics,” Proc. SPIE 8885, 88851R (2013).
[Crossref]

Appl. Opt. (1)

Eng. (1)

P. Kah, J. Lu, J. Martikainen, and R. Suoranta, “Remote laser welding with high hower fiber lasers,” Eng. 5(09), 700–706 (2013).
[Crossref]

J. Quantum Electron. (2)

M. Roth, E. Wyss, T. Graf, and H. Weber, “End-pumped Nd:YAG laser with self-adaptive compensation of the thermal lens,” J. Quantum Electron. 40(12), 1700–1703 (2004).
[Crossref]

D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” J. Quantum Electron. 34(12), 2383–2392 (1998).
[Crossref]

Opt. Express (2)

Phys. Procedia (1)

G. Eberle, V. Chiron, and K. Wegener, “Simulation and realization of a focus shifting unit using a tunable lens for 3D laser material processing,” Phys. Procedia 41, 441–447 (2013).
[Crossref]

Proc. SPIE (6)

A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE 8960, 89601G (2014).

M. Scaggs and G. Haas, “Thermal lensing compensation objective for high power lasers,” Proc. SPIE 7913, 79130C (2011).
[Crossref]

S. Piehler, C. Thiel, A. Voss, M. Abdou Ahmed, and T. Graf, “Self-compensation of thermal lensing in optics for high-brightness solid-state lasers,” Proc. SPIE 8239, 82390Z (2012).
[Crossref]

O. Blomster, M. Pålsson, S.-O. Roos, M. Blomqvist, F. Abt, F. Dausinger, C. Deininger, and M. Huonker, “Optics performance at high power levels,” Proc. SPIE 6871, 68712B (2008).
[Crossref]

S. Sinha, J. D. Mansell, and R. L. Byer, “Deformable mirrors for high-power lasers,” Proc. SPIE 4493, 55–63 (2002).
[Crossref]

M. Stubenvoll, B. Schäfer, K. Mann, A. Walter, and L. Zittel, “Photothermal absorption measurements for improved thermal stability of high-power laser optics,” Proc. SPIE 8885, 88851R (2013).
[Crossref]

Rev. Sci. Instrum. (1)

A. A. Andrade, T. Catunda, I. Bodnar, J. Mura, and M. L. Baesso, “Thermal lens determination of the temperature coefficient of optical path length in optical materials,” Rev. Sci. Instrum. 74(1), 877–880 (2003).
[Crossref]

Other (5)

P. G. Nelson, “A thermal analysis of a 1.5 meter f/5 fused silica primary lens for solar telescopes,” Coronal Solar Magnetism Observatory, Tech. Note 13, Rev. 2 (2007). http://www.cosmo.ucar.edu/publications/nelson_tech13r2_03-07.pdf

D. E. Patent, 102010052471 B3, Messvorrichtung und Messverfahren zur Bestimmung einer optischen Qualität einer Prüfoptik“ (2012).

D. Havrilla, M. Holzer, and P. Olschowsky, “Latest research results with high power solid state laser optics with minimized focus shift,” Proceedings of International Congress on Applications of Lasers & Electro-Optics (ICALEO), pp. 29–32 (2010).

C. Wandera, “Performance of high power fibre laser cutting of thick-section steel and medium-section aluminium,“ Ph.D. thesis, Lappeenranta University of Technology (2010).

ISO 1146, “Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios,” International Organization for Standardization (2005).

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Figures (9)

Fig. 1
Fig. 1 Thermal lensing mechanisms and important material parameters.
Fig. 2
Fig. 2 Numerically calculated first order optical power of a plane N-PK51 substrate (diameter 35 mm) as a function of sample length for three diameters of a Gaussian laser beam. Beam power is 100 W and the used absorption coefficients for bulk and surface were µ = 0.09 1/m and β = 15 ppm, respectively. Stress-free mechanical boundary conditions and a constant heat transfer coefficient of κ = 10 W·m−2K−1 were applied for the calculations.
Fig. 3
Fig. 3 Comparison between measurements (left) and simulations (right) of cuboid samples of different optical materials investigated in the crossed-beam setup. The x and y axes represent the spatial arrangement of the cuboid sample. The wavefront is displayed on the z axis for a sample height of 25 mm and reaching 20 mm (measurement) or 23 mm (simulation) into the bulk from the entry surface.
Fig. 4
Fig. 4 Schematic illustration of the procedure to determine absolute surface/coating and bulk absorption coefficients (cf. text).
Fig. 5
Fig. 5 Experimental setup for the determination of focal shifts in F-Theta lenses and other optical systems using pivoted mirrors for a coaxial arrangement of probe beam and heating beam (left). The graph on the right demonstrates the linear dependence of the wavefront deformation wRMS on the laser power in a planar sample.
Fig. 6
Fig. 6 Wavefront deformation in plane AR coated fused silica and N-PK51 samples (length 12 mm, diameter 25 mm, laser power 168 W, beam diameter 1.2 mm). The measured area covers approx. 10x10 mm2. The effect of the negative thermal dispersion of N-PK51 is clearly visible leading to an inversion of the wavefront.
Fig. 7
Fig. 7 Thermal wavefront distortions for a combination of an F-Theta lens (AR2) and N-PK51 elements of different lengths at P = 100 W and d = 3.8 mm. The measured area covers approx. 10x10 mm2.
Fig. 8
Fig. 8 Caustic measurement setup for the determination of thermal beam waist displacements of high power laser optics. Displayed are also some beam profiles around the waist, taken at high power operation and with a 70 mm compensating element implemented.
Fig. 9
Fig. 9 Thermal focal shifts of an F-Theta lens (AR1) with and without compensating element (left). It is assumed that focal shifts at P = 25 W are very small and are therefore set to zero. The graph on the right shows average beam diameters at a fixed working distance of 500 mm.

Tables (2)

Tables Icon

Table 1 dn/dT values and thermal parameter K/λ @1060nm for fused silica and some dn/dT < 0 materials.

Tables Icon

Table 2 Coating/surface (β) and bulk (µ) absorption coefficients obtained through curve-fitting of simulated wavefront data to measurements for two different AR coatings and an uncoated N-PK51 sample.

Equations (8)

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

δ w r , φ ( r ) = 0 L [ ( n 0 1 ) Δ L L + Δ n T + Δ n σ ] d z = 0 L [ ( n 0 1 ) ε z z ( r , z ) + d n d T δ T ( r , z ) n 0 3 2 ( B σ ) r , φ ] d z
1 f r , φ = d 2 δ w r , φ ( r ) d r 2 | r = 0
1 f = K L d 2 d r 2 δ T ( r ) | r = 0
K p l , s t r e s s = d n d T + ( n 0 1 ) α ( 1 + ν ) + n 0 3 4 ( B + B ) α E K p l . s t r a i n = d n d T + n 0 3 4 ( B + ( 1 + 2 ν ) B )   α E
δ T ( r ) = δ T 0 c S μ P 4 π r b 2 λ r 2 +
1 f = K L c S μ P 2 λ π r b 2
L I I = L I μ I λ I I μ I I λ I K I K I I
| w e ( x , z , t ) P μ V ( x , z , t ) β S ' ( x , z , t ) | 2 d x d z = ! m i n

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