Abstract

We study thermally induced birefringence in crystalline Nd:YAG zigzag slab lasers and the associated depolarization losses. The optimum crystallographic orientation of the zigzag slab within the Nd:YAG boule and photoelastic effects in crystalline Nd:YAG slabs are briefly discussed. The depolarization is evaluated using the temperature and stress distributions, calculated using a finite element model, for realistically pumped and cooled slabs of finite dimensions. Jones matrices are then used to calculate the depolarization of the zigzag laser mode. We compare the predictions with measurements of depolarization, and suggest useful criteria for the design of the gain media for such lasers.

© 2006 Optical Society of America

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References

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  1. D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
    [CrossRef]
  2. K. Du, N. Wu, J. Xu, J. Giesekus, P. Loosen, and R. Poprawe, "Partially end-pumped Nd:YAG slab laser with a hybrid resonator," Opt. Lett. 23, 370-372 (1998).
    [CrossRef]
  3. H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
    [CrossRef]
  4. J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
    [CrossRef]
  5. T. J. Kane, R. C. Eckardt, and R. L. Byer, "Reduced thermal focusing and birefringence in zigzag slab geometry crystalline lasers," IEEE J. Quantum Electron. QE-19, 1351-1354 (1983).
    [CrossRef]
  6. Q. Lü, U. Wittrock, and S. Dong, "Photoelastic effect in Nd:YAG rod and slab lasers," Opt. Laser Technol. 27, 95-101 (1995).
    [CrossRef]
  7. A. Faulstich, H. J. Baker, and D. R. Hall, "Face pumping of thin, solid-state slab lasers with laser diodes," Opt. Lett. 21, 594-596 (1996).
    [CrossRef] [PubMed]
  8. D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
    [CrossRef]
  9. J. F. Nye, Physical Properties of Crystals (Oxford Science Publications, 2000).
  10. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).
  11. W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
    [CrossRef]
  12. R. W. Dixon, "Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners," J. Appl. Phys. 38, 5149-5153 (1967).
    [CrossRef]
  13. J. Richards and A. McInnes, "Versatile, efficient, diode-pumped miniature slab laser," Opt. Lett. 20, 371-373 (1995).
    [CrossRef] [PubMed]
  14. Teflon AF (1601) is an amorphous fluoropolymer solution developed by DuPont Fluoroproducts, P.O. Box 80711, Wilmington, Dela. 19880-0711.
  15. D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
    [CrossRef]
  16. D. C. Brown, "Nonlinear thermal and stress effects and scaling behavior of YAG slab amplifiers," IEEE J. Quantum Electron. 34, 2393-2402 (1998).
    [CrossRef]
  17. ANSYS finite-element software, Vers. 5.5.3, http://www.ansys.com/.

2002 (1)

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

2001 (1)

H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
[CrossRef]

2000 (1)

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

1998 (2)

D. C. Brown, "Nonlinear thermal and stress effects and scaling behavior of YAG slab amplifiers," IEEE J. Quantum Electron. 34, 2393-2402 (1998).
[CrossRef]

K. Du, N. Wu, J. Xu, J. Giesekus, P. Loosen, and R. Poprawe, "Partially end-pumped Nd:YAG slab laser with a hybrid resonator," Opt. Lett. 23, 370-372 (1998).
[CrossRef]

1997 (1)

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

1996 (1)

1995 (2)

Q. Lü, U. Wittrock, and S. Dong, "Photoelastic effect in Nd:YAG rod and slab lasers," Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

J. Richards and A. McInnes, "Versatile, efficient, diode-pumped miniature slab laser," Opt. Lett. 20, 371-373 (1995).
[CrossRef] [PubMed]

1984 (1)

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

1983 (1)

T. J. Kane, R. C. Eckardt, and R. L. Byer, "Reduced thermal focusing and birefringence in zigzag slab geometry crystalline lasers," IEEE J. Quantum Electron. QE-19, 1351-1354 (1983).
[CrossRef]

1970 (1)

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

1967 (1)

R. W. Dixon, "Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners," J. Appl. Phys. 38, 5149-5153 (1967).
[CrossRef]

Baker, H.

H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
[CrossRef]

Baker, H. J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).

Brown, D. C.

D. C. Brown, "Nonlinear thermal and stress effects and scaling behavior of YAG slab amplifiers," IEEE J. Quantum Electron. 34, 2393-2402 (1998).
[CrossRef]

Byer, R. L.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

T. J. Kane, R. C. Eckardt, and R. L. Byer, "Reduced thermal focusing and birefringence in zigzag slab geometry crystalline lasers," IEEE J. Quantum Electron. QE-19, 1351-1354 (1983).
[CrossRef]

Chesworth, A. A.

H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
[CrossRef]

Dixon, R. W.

R. W. Dixon, "Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners," J. Appl. Phys. 38, 5149-5153 (1967).
[CrossRef]

Dong, S.

Q. Lü, U. Wittrock, and S. Dong, "Photoelastic effect in Nd:YAG rod and slab lasers," Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Du, K.

Eckardt, R. C.

T. J. Kane, R. C. Eckardt, and R. L. Byer, "Reduced thermal focusing and birefringence in zigzag slab geometry crystalline lasers," IEEE J. Quantum Electron. QE-19, 1351-1354 (1983).
[CrossRef]

Eggleston, J. M.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

Faulstich, A.

Giesekus, J.

Hall, D. R.

H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
[CrossRef]

A. Faulstich, H. J. Baker, and D. R. Hall, "Face pumping of thin, solid-state slab lasers with laser diodes," Opt. Lett. 21, 594-596 (1996).
[CrossRef] [PubMed]

Hamilton, M. W.

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

Kane, T. J.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

T. J. Kane, R. C. Eckardt, and R. L. Byer, "Reduced thermal focusing and birefringence in zigzag slab geometry crystalline lasers," IEEE J. Quantum Electron. QE-19, 1351-1354 (1983).
[CrossRef]

Koechner, W.

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

Kuhn, K.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

Loosen, P.

Lü, Q.

Q. Lü, U. Wittrock, and S. Dong, "Photoelastic effect in Nd:YAG rod and slab lasers," Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

McInnes, A.

Middlemiss, B.

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

Millas, D. Pelaez

H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
[CrossRef]

Mudge, D.

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

Munch, J.

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford Science Publications, 2000).

Ostermeyer, M.

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

Ottaway, D.

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

Ottaway, D. J.

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

Poprawe, R.

Rice, D. K.

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

Richards, J.

Unternahrer, J.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

Veitch, P. J.

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

Wittrock, U.

Q. Lü, U. Wittrock, and S. Dong, "Photoelastic effect in Nd:YAG rod and slab lasers," Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).

Wu, N.

Xu, J.

Class. Quantum Grav. (1)

D. Mudge, M. Ostermeyer, D. J. Ottaway, P. J. Veitch, J. Munch, and M. W. Hamilton, "High-power Nd:YAG lasers using stable-unstable resonators," Class. Quantum Grav. 19, 1783-1792 (2002).
[CrossRef]

IEEE J. Quantum Electron. (4)

D. C. Brown, "Nonlinear thermal and stress effects and scaling behavior of YAG slab amplifiers," IEEE J. Quantum Electron. 34, 2393-2402 (1998).
[CrossRef]

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser--Part I: theory," IEEE J. Quantum Electron. QE-20, 289-301 (1984).
[CrossRef]

T. J. Kane, R. C. Eckardt, and R. L. Byer, "Reduced thermal focusing and birefringence in zigzag slab geometry crystalline lasers," IEEE J. Quantum Electron. QE-19, 1351-1354 (1983).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

D. Mudge, M. Ostermeyer, P. J. Veitch, J. Munch, B. Middlemiss, D. J. Ottaway, and M. W. Hamilton, "Power scalable TEMoo cw Nd:YAG laser with thermal lens compensation," IEEE J. Sel. Top. Quantum Electron. 6, 643-649 (2000).
[CrossRef]

D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W. Hamilton, "High-power diode-laser-pumped cw solid-state lasers using stable-unstable resonators," IEEE J. Sel. Top. Quantum Electron. 3, 19-26 (1997).
[CrossRef]

J. Appl. Phys. (1)

R. W. Dixon, "Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners," J. Appl. Phys. 38, 5149-5153 (1967).
[CrossRef]

Opt. Comm. (1)

H. Baker, A. A. Chesworth, D. Pelaez Millas, and D. R. Hall, "A planar waveguide Nd:YAG laser with a hybrid waveguide-unstable resonator," Opt. Comm. 191, 125-131 (2001).
[CrossRef]

Opt. Laser Technol. (1)

Q. Lü, U. Wittrock, and S. Dong, "Photoelastic effect in Nd:YAG rod and slab lasers," Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Opt. Lett. (3)

Other (4)

J. F. Nye, Physical Properties of Crystals (Oxford Science Publications, 2000).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).

Teflon AF (1601) is an amorphous fluoropolymer solution developed by DuPont Fluoroproducts, P.O. Box 80711, Wilmington, Dela. 19880-0711.

ANSYS finite-element software, Vers. 5.5.3, http://www.ansys.com/.

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Figures (13)

Fig. 1
Fig. 1

Ideal slab geometry showing the axes and notation used in this paper. The axes are the same as those used by Eggleston et al. (Ref. 4) but different from those used by Lü et al. (Ref. 6).

Fig. 2
Fig. 2

Suitable and unsuitable orientations for zigzag slabs in a Nd:YAG boule, elevation (left) and top view (right). The slab (b) in the left-hand diagram is located at the perimeter of the boule. The double-arrowed line in slabs (a) and (b) in the right-hand diagram, represent the zigzag direction. The 〈110〉 and 〈211〉 crystal axes and the equivalent directions are marked. The section of the boule from where cylindrical laser rods are typically harvested is also shown for completeness. The cut angle, φ, is measured relative to the [10 1 ¯ ] direction.

Fig. 3
Fig. 3

Imaged transmission at the exit face of a zigzag slab with suitable (left) and unsuitable (right) orientations. The growth striations in the right image are visible as some parts of the zigzag path is almost parallel to the growth striations.

Fig. 4
Fig. 4

(a) Pump architecture and beam path through a side-pumped, side-cooled CPFS laser. The waveguides smooth the intensity profile of the pump distribution in the plane of the page but preserve the numerical aperture of the pump light in the perpendicular (x) direction. (b) Geometrical parameters used in the FEM of the side-pumped, side cooled CPFS. The slab is pumped and cooled on both sides. w pump is the full height of the pump profile in the slab. L p is the length of the pumped region. The cooling water flows along a channel that is w cool high and L p long. All other slab surfaces are assumed to be insulated.

Fig. 5
Fig. 5

(Color online) (a) temperature, (b) σ xx , and (c) σ xy for the h = 4.3 mm high slab.

Fig. 6
Fig. 6

Predicted transmission through parallel polarizers for an infinitely long, uniformly pumped, and cooled slab, h = 43 mm and w = 3.0 mm. Cut angles φ = 0° and 30°.

Fig. 7
Fig. 7

Predicted transmission through parallel polarizers for an infinitely long, uniformly pumped, and cooled slab, h = 4.3 mm and w = 3 mm. Cut angles φ = 0°, 10°, and 20°.

Fig. 8
Fig. 8

Integrated transmission through parallel polarizers as a function of the cut angle, for the h = 43 mm and h = 4.3 mm slabs.

Fig. 9
Fig. 9

Predicted transmission through parallel polarizers for different pump and cooling heights and a Gaussian pump profile. The parameters for gw11, gw22, and gw33 are given in Table 2. Slab h = 4.3 mm and w = 3 mm.

Fig. 10
Fig. 10

Predicted transmission through parallel polarizers resulting from a top-hat pump profile (uw22) and a Gaussian pump profile (gw22), for slab h = 4.3 mm. The parameters for uw22 and gw22 are given in Table 3.

Fig. 11
Fig. 11

Predicted transmission through parallel polarizers resulting from a top-hat pump profile (uw66) and a Gaussian pump profile (gw66), for slab h = 8.6 mm. The parameters for uw66 and gw66 are given in Table 3.

Fig. 12
Fig. 12

Experiment used to measure the depolarization of a HeNe probe beam. P1 and P2, input and output crossed polarizers; IF, interference filter; PM, power meter; IL, imaging lens; HR, high reflectivity laser mirror at 1064 nm; CCD, charge coupled-device camera; PR, partially reflective laser mirror at 1064 nm; HRHT, dichroic beam splitter having high reflectivity at 1064 nm and high transmission at 632.8 nm.

Fig. 13
Fig. 13

Predicted (upper) and measured (lower) transmission profiles through crossed polarizers for the Nd:YAG slab shown in Fig. 4. The measured depolarization used the setup shown in Fig. 12. High transmission indicates strong depolarization.

Tables (4)

Tables Icon

Table 1 Maximum Temperature and Stresses for Two Infinitely Long, Optimum Cut Angle, Uniformly Pumped and Face-Cooled Slabs a

Tables Icon

Table 2 Maximum Stress Levels for Slabs with Different Pump and Cooling Heights, for slab h = 4.3 mm and w = 3.0 mm a

Tables Icon

Table 3 Maximum Stresses for Slabs ( h = 4.3 mm and h = 8.6 mm) with Different Pump and Cooling Geometries Using Top-Hat (uw) and Gaussian (gw) Pump Profiles a

Tables Icon

Table 4 Dimensions of the Side-Pumped, Side-Cooled Zigzag CPFS Slab Used in the Experiment

Equations (20)

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

x 2 ε x + y 2 ε y + z 2 ε z = 1.
B x x 2 + B y y 2 + B z z 2 = 1 ,
B i j x i x j = 1.
B i j = B o , i j + π i j k l σ k l ,
B o , i j = δ i j / { n o + d n d T [ T ( x , y , z ) T o ] } 2 ,
[ B 11 B 22 B 33 B 23 B 13 B 12 ] = [ B 0 , 11 B 0 , 22 B 0 , 33 0 0 0 ] + [ π 11     π 12 π 13     π 14     π 15     0 π 12     π 11 π 13 π 14 π 15     0 π 13     π 13 π 33     0     0     0 π 14 π 14 0     π 44     0 π 15 π 15 π 15 0     0     π 44       π 14 0     0 0 π 15     π 14       π 66 ] [ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] ,
π 11 = 0.30285 × 10 12 m 2 N 1 , π 12 = + 0.11158 × 10 12 m 2 N 1 , π 13 = + 0.17187 × 10 12 m 2 N 1 , π 33 = 0.36313 × 10 12 m 2 N 1 , π 44 = 0.14693 × 10 12 m 2 N 1 , π 66 = 0.20722 × 10 12 m 2 N 1 , π 14 = 0.08525 × 10 12 cos ( 3 φ ) m 2 N 1 , π 15 = 0.08525 × 10 12 sin ( 3 φ ) m 2 N 1 ,
B 11 = B 11 ,
B 22 = B 22 cos 2 γ + 2 B 23 cos γ sin γ + B 33 sin 2 γ ,
B 12 = B 21 = B 12 cos γ + B 13 sin γ .
B 12 = cos γ [ σ 23 π 15 + σ 13 π 14 + σ 12 π 66 ] + sin γ [ ( σ 11 σ 22 ) π 15 + σ 13 π 44 + σ 12 π 14 ] = cos γ [ σ y z π 15 + σ x z π 14 + σ x y π 66 ] + sin γ [ ( σ x x σ y y ) π 15 + σ x z π 44 + σ x y π 14 ] .
[ B 11 B 12 B 12 B 22 ] ,
λ ± = 1 2 [ ( B 11 + B 22 ) ± ( B 11 B 22 ) 2 + 4 B 11 2 ] , and  
u ± = [ 1 ( λ ± B 11 B 12 ) ] .
J = [ exp ( i k n Δ ) 0 0 exp ( i k n + Δ ) ] ,
J = R ( β ) J R ( β ) ,
R ( β ) = [ cos β sin β sin β cos β ]
cos β = u [ 1 0 ] | u | = 1 / [ 1 + ( λ B 11 B 12 ) 2 ] 1 / 2
J TIR = [ sin ( γ ) i cos ( γ ) 2 n r 2 sin ( γ ) + i cos ( γ ) 2 n r 2 0 0 n r 2 sin ( γ ) i cos ( γ ) 2 n r 2 n r 2 sin ( γ ) + i cos ( γ ) 2 n r 2 ] ,
J total = [ Π m = p 1 ( J 20 m J 19 m , . . . , J 1 m J TIR ) ] × ( J 20 m , . . . , J 11 m J TIR J 10 m , . . .    , J 1 m ) × [ Π m = 1 p ( J TIR J 20 m J 19 m , . . . , J 1 m ) ] ,

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