Abstract

In this paper we present experiments and calculations of the property changes of a highly reflecting volume Bragg grating (VBG) when it is used as a laser cavity mirror. A small absorption of the reflected laser beam resulted in a laser output power roll-off, increased coupling through the VBG, and a change of the spectrum from a single to a double peak at high power. The simulations revealed that an inhomogeneous temperature distribution deformed the grating such that the diffraction efficiency was reduced and the light penetrated deeper into the VBG, which accelerated the deteriorating effects. We extrapolated the power limit found in our investigations for various beam radii and absorption coefficients.

© 2013 Optical Society of America

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  1. O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass,” Appl. Opt. 38, 619–627 (1999).
    [CrossRef]
  2. B. Volodin, S. Dolgy, E. Melnik, E. Downs, J. Shaw, and V. Ban, “Wavelength stabilization and spectral narrowing of high power multimode laser diodes and arrays by use of volume Bragg gratings,” Opt. Lett. 29, 1891–1893 (2004).
    [CrossRef]
  3. B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
    [CrossRef]
  4. T. Chung, A. Rapaport, V. Smirnov, L. Glebov, M. Richardson, and M. Bass, “Solid-state laser spectral narrowing using a volumetric photothermal refractive Bragg grating cavity mirror,” Opt. Lett. 31, 229–231 (2006).
    [CrossRef]
  5. P. Jelger and F. Laurell, “Efficient narrow-linewidth volume-Bragg grating-locked Nd:fiber laser,” Opt. Express 15, 11336–11340 (2007).
    [CrossRef]
  6. B. Jacobsson, M. Tiihonen, V. Pasiskevicius, and F. Laurell, “Narrowband bulk Bragg grating optical parametric oscillator,” Opt. Lett. 30, 2281–2283 (2005).
    [CrossRef]
  7. O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15, 344–353 (2009).
    [CrossRef]
  8. K.-H. Liao, M.-Y. Cheng, E. Flecher, V. Smirnov, L. Glebov, and A. Galvanauskas, “Large aperture chirped volume Bragg grating based CPA system,” Opt. Express 15, 4876–4882 (2007).
    [CrossRef]
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    [CrossRef]
  10. T. Waritanant, and T.-Y. Chung, “Influence of minute self-absorption of a volume Bragg grating used as a laser mirror,” IEEE J. Quantum Electron. 47, 390–397 (2011).
    [CrossRef]
  11. J. Lumeau, L. Glebova, and L. B. Glebov, “Influence of UV-exposure on the crystallization and optical properties of photo-thermo-refractive glass,” J. Non-Cryst. Solids 354, 425 (2008).
    [CrossRef]
  12. O. G. Andrusyak, “Dense spectral beam combining with volume Bragg gratings in photothermo-refractive glass,” Ph.D. thesis, University of Central Florida, 2009.
  13. J. Lumeau, L. Glebova, and L. Glebov, “Near-IR absorption in high-purity photothermorefractive glass and holographic optical elements: measurement and application for high-energy lasers,” Appl. Opt. 50, 5905–5911 (2011).
    [CrossRef]
  14. O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
    [CrossRef]
  15. I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
    [CrossRef]
  16. J. W. Kim, P. Jelger, J. K. Sahu, F. Laurell, and W. A. Clarkson, “High-power and wavelength-tunable operation of an Er, Yb fiber laser using a volume Bragg grating,” Opt. Lett. 33, 1204–1206 (2008).
    [CrossRef]
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    [CrossRef]
  18. P. Jelger, P. Wang, J. K. Sahu, F. Laurell, and W. A. Clarkson, “High-power linearly-polarized operation of a cladding-pumped Yb fibre laser using a volume Bragg grating for wavelength selection,” Opt. Express 16, 9507–9512 (2008).
    [CrossRef]
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    [CrossRef]
  20. H. Shu, S. Mokhov, B. Y. Zeldovich, and M. Bass, “More on analyzing the reflection of a laser beam by a deformed highly reflective volume Bragg grating using iteration of the beam propagation method,” Appl. Opt. 48, 22–27 (2009).
    [CrossRef]
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    [CrossRef]
  22. K. I. White and J. E. Midwinter, “An improved technique for the measurement of low optical absorption losses in bulk glass,” Opto-electronics 5, 323–334 (1973).
    [CrossRef]
  23. A. E. Siegman, Lasers (University Science Books, 1986).
  24. N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
    [CrossRef]
  25. K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
    [CrossRef]
  26. B. Jacobsson, “Experimental and theoretical investigation of a volume-Bragg-grating-locked Yb:KYW laser at selected wavelengths,” Opt. Express 16, 6443–6454 (2008).
    [CrossRef]
  27. J. Hong, W. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
    [CrossRef]
  28. COMSOL Inc., Stress-Optical Effects in a Silica-on-Silicon Waveguide (COMSOL, 2008).

2011 (2)

T. Waritanant, and T.-Y. Chung, “Influence of minute self-absorption of a volume Bragg grating used as a laser mirror,” IEEE J. Quantum Electron. 47, 390–397 (2011).
[CrossRef]

J. Lumeau, L. Glebova, and L. Glebov, “Near-IR absorption in high-purity photothermorefractive glass and holographic optical elements: measurement and application for high-energy lasers,” Appl. Opt. 50, 5905–5911 (2011).
[CrossRef]

2009 (3)

2008 (4)

2007 (4)

2006 (3)

2005 (2)

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

B. Jacobsson, M. Tiihonen, V. Pasiskevicius, and F. Laurell, “Narrowband bulk Bragg grating optical parametric oscillator,” Opt. Lett. 30, 2281–2283 (2005).
[CrossRef]

2004 (1)

2003 (1)

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

1999 (2)

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
[CrossRef]

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass,” Appl. Opt. 38, 619–627 (1999).
[CrossRef]

1997 (1)

1992 (1)

J. Hong, W. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

1973 (1)

K. I. White and J. E. Midwinter, “An improved technique for the measurement of low optical absorption losses in bulk glass,” Opto-electronics 5, 323–334 (1973).
[CrossRef]

Andrusyak, O.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15, 344–353 (2009).
[CrossRef]

Andrusyak, O. G.

O. G. Andrusyak, “Dense spectral beam combining with volume Bragg gratings in photothermo-refractive glass,” Ph.D. thesis, University of Central Florida, 2009.

Ban, V.

Bass, M.

Cheng, M.-Y.

Chung, T.

Chung, T.-Y.

T. Waritanant, and T.-Y. Chung, “Influence of minute self-absorption of a volume Bragg grating used as a laser mirror,” IEEE J. Quantum Electron. 47, 390–397 (2011).
[CrossRef]

Ciapurin, I. V.

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

Clarkson, W. A.

Dolgy, S.

Downs, E.

Efimov, O.

Efimov, O. M.

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
[CrossRef]

Fagundes-Peters, D.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Flecher, E.

Galvanauskas, A.

Giesen, A.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Glebov, L.

Glebov, L. B.

J. Lumeau, L. Glebova, and L. B. Glebov, “Influence of UV-exposure on the crystallization and optical properties of photo-thermo-refractive glass,” J. Non-Cryst. Solids 354, 425 (2008).
[CrossRef]

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
[CrossRef]

Glebova, L.

Glebova, L. N.

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

Hellström, J. E.

Hong, J.

J. Hong, W. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Huang, W.

J. Hong, W. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Huber, G.

Jacobsson, B.

Jelger, P.

Johannsen, J.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Kim, J. W.

Kuleshov, N. V.

Kutovoi, S.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Lagatsky, A. A.

Laurell, F.

Liao, K.-H.

Lumeau, J.

J. Lumeau, L. Glebova, and L. Glebov, “Near-IR absorption in high-purity photothermorefractive glass and holographic optical elements: measurement and application for high-energy lasers,” Appl. Opt. 50, 5905–5911 (2011).
[CrossRef]

J. Lumeau, L. Glebova, and L. B. Glebov, “Influence of UV-exposure on the crystallization and optical properties of photo-thermo-refractive glass,” J. Non-Cryst. Solids 354, 425 (2008).
[CrossRef]

Makino, T.

J. Hong, W. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Melnik, E.

Midwinter, J. E.

K. I. White and J. E. Midwinter, “An improved technique for the measurement of low optical absorption losses in bulk glass,” Opto-electronics 5, 323–334 (1973).
[CrossRef]

Mikhailov, V. P.

Mokhov, S.

Mond, M.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Papernov, S.

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
[CrossRef]

Pasiskevicius, V.

Petermann, K.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Peters, V.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Podlipensky, A. V.

Rapaport, A.

Richardson, K.

Richardson, M.

Romero, J. J.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Rotar, V.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15, 344–353 (2009).
[CrossRef]

Rotari, E. V.

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

Sahu, J. K.

Schmid, A. W.

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
[CrossRef]

Shaw, J.

Shu, H.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Smirnov, V.

Smirnov, V. I.

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

Speiser, J.

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

Tiihonen, M.

Venus, G.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15, 344–353 (2009).
[CrossRef]

Volodin, B.

Wang, P.

Waritanant, T.

T. Waritanant, and T.-Y. Chung, “Influence of minute self-absorption of a volume Bragg grating used as a laser mirror,” IEEE J. Quantum Electron. 47, 390–397 (2011).
[CrossRef]

White, K. I.

K. I. White and J. E. Midwinter, “An improved technique for the measurement of low optical absorption losses in bulk glass,” Opto-electronics 5, 323–334 (1973).
[CrossRef]

Zeldovich, B. Y.

Appl. Opt. (5)

IEEE J. Quantum Electron. (1)

T. Waritanant, and T.-Y. Chung, “Influence of minute self-absorption of a volume Bragg grating used as a laser mirror,” IEEE J. Quantum Electron. 47, 390–397 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15, 344–353 (2009).
[CrossRef]

J. Cryst. Growth (1)

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Hong, W. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

J. Non-Cryst. Solids (1)

J. Lumeau, L. Glebova, and L. B. Glebov, “Influence of UV-exposure on the crystallization and optical properties of photo-thermo-refractive glass,” J. Non-Cryst. Solids 354, 425 (2008).
[CrossRef]

Opt. Express (6)

Opt. Lett. (6)

Opto-electronics (1)

K. I. White and J. E. Midwinter, “An improved technique for the measurement of low optical absorption losses in bulk glass,” Opto-electronics 5, 323–334 (1973).
[CrossRef]

Proc. SPIE (2)

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical holographic element writing,” Proc. SPIE 3578, 564–575 (1999).
[CrossRef]

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100 W Yb-fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[CrossRef]

Other (3)

O. G. Andrusyak, “Dense spectral beam combining with volume Bragg gratings in photothermo-refractive glass,” Ph.D. thesis, University of Central Florida, 2009.

A. E. Siegman, Lasers (University Science Books, 1986).

COMSOL Inc., Stress-Optical Effects in a Silica-on-Silicon Waveguide (COMSOL, 2008).

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Figures (12)

Fig. 1.
Fig. 1.

Schematic picture of the w-type Yb:KYW laser cavity. Either a VBG or a highly reflecting mirror (M) was used as the cavity end mirror, as seen in the dashed box to the right. The flat input coupler (IC) was highly transmitting (R<2%) for 980 nm and highly reflecting (R>99.5%) for 1030 nm and was placed at a 22.5° incident angle. HR1 and HR2 were both highly reflecting (R>99.9%) at 1030 nm, where HR1 was flat and HR2 had a radius of curvature of 200 mm and was placed at 10° incident angle. The OC had a radius of curvature of 100 mm and transmitted 15% at 1030 nm.

Fig. 2.
Fig. 2.

Output power from the reference laser (triangles) and the powers coupled out through the output coupling mirror (crosses) and the grating (bullets) for the VBG laser. The dashed line is the output of the VBG laser calculated using Eq. (1).

Fig. 3.
Fig. 3.

(a) Output power versus launched pump power. The power coupled out through the OC and the VBG are marked by crosses and bullets, respectively. (b) Output power versus intracavity power. The power coupled out through the OC and the VBG are marked by crosses and bullets, respectively. In both figures the measured temporal fluctuations are visualized by the lines (maximum and minimum values).

Fig. 4.
Fig. 4.

Reflectivity of the VBG extracted from the experiment plotted against the product of the absorption coefficient and the incident intensity (αabsI0,inc).

Fig. 5.
Fig. 5.

Position of spectral peaks versus thermal load on the VBG, αabsI0,inc.

Fig. 6.
Fig. 6.

Spectra for the VBG laser for three different pump levels corresponding to αabsI0,inc values of 10, 150, and 180W/cm3.

Fig. 7.
Fig. 7.

(a) Spectrum of the signal in each channel (Ch. A. and Ch. B.) at αabsI0,inc150W/cm3. Each spectrum was normalized by dividing it by its maximum value. (b) Traces of the oscillations in the two channels at αabsI0,inc150W/cm3.

Fig. 8.
Fig. 8.

(a) Normalized spectrum of the signal in each channel (Ch. A. and Ch. B.) at αabsI0,inc180W/cm3. (b) Traces of the oscillations in the two channels at αabsI0,inc180W/cm3.

Fig. 9.
Fig. 9.

Schematic of the simulation process. The simulation finds steady-state solutions of R(λ), I(z), and T(z) for incremental values of αabs·I0,inc. For each αabsI0,inc the simulation model iterates between the two parts until the solutions converged.

Fig. 10.
Fig. 10.

(a) Calculations of the maximum reflectivity and the wavelength shift of the reflectivity peak compared to an unchirped grating for various levels of absorbed energy (αabsI0,inc). The simulation points are marked by “x.” (b) shows the normalized intensity distribution along the beam path inside the VBG for an unchirped grating and for chirped gratings where the value of αabsI0,inc is equal to 125 and 137W/cm3, respectively. Position 0 mm is the surface of the grating closest to the laser cavity. The normalized intensity is the sum of the intensity of the forward- and the backward-propagating wave inside the VBG divided by the incoming intensity, that is, the value at the VBG surface closest to the laser cavity is 1+R.

Fig. 11.
Fig. 11.

(a) Reflectivity as a function of wavelength for an unchirped grating and for a chirped grating with αabsI0,inc=137W/cm3. In (b) the temperature distribution in the center of the laser beam is shown versus z coordinate in the VBG with z=0mm as the entrance/exit facet for the beam.

Fig. 12.
Fig. 12.

Stability point in power incident on the VBG, Plimit, versus I(1/e2) beam radius, ω, for various values of the absorption coefficient, αabs, spanning from 0.2% to 1% cm1 in steps of 0.2% cm1.

Equations (6)

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

Iout=δ1[2αm0pmδ1+δ2+δ01]Isat2.
Isat=ωστ.
2αm0pm=(2Ioutδ1Isat+1)(δ1+δ2+δ0).
(k·T)=Q.
Cp(T)=9.36×109T31.54×105T2+8.22×103T0.637.
Q(x,y,z)=αabsI0,incexp{2x2+y2ω2}I(z).

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