Abstract

Spectral beam combining (SBC) by volume Bragg gratings (VBGs) recorded in photo-thermo-refractive (PTR) glass is a powerful tool for laser applications that require higher radiance than a single laser unit can achieve. The beam-combining factor (BCF) is introduced as a tool to compare various beam-combining methods and experiments. It describes the change of radiance provided by a beam-combining system but is not affected by the initial beam quality of the combined lasers. A method of optimization of VBGs providing the maximum efficiency of SBC has been described for an arbitrary number of beams. An experiment confirming the proposed modeling for a two-beam SBC system by a single VBG has demonstrated a total combined power of 301 W with a channel separation of 0.25 nm, combining efficiency of 97%, close to diffraction limited divergence with M2=1.18, BCF of 0.77, and spectral radiance of 770TW/(sr·m2·nm), the highest to date for SBC.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. L. B. Glebov, “Photochromic and photo-thermo-refractive glasses,” in Encyclopedia of Smart Materials (Wiley, 2002).
  25. T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
    [CrossRef]
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2013 (2)

2012 (2)

A. Sridharan, P. H. Pax, J. E. Heebner, D. R. Drachenberg, P. J. Armstrong, and J. W. Dawson, “Mode-converters for rectangular-core fiber amplifiers to achieve diffraction-limited power scaling,” Opt. Express 20, 28792–28800 (2012).
[CrossRef]

I. V. Ciapurin, D. R. Drachenberg, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Modeling of phase volume diffractive gratings, part 2: reflecting sinusoidal uniform gratings, Bragg mirrors,” Opt. Eng. 51, 058001 (2012).
[CrossRef]

2011 (1)

2010 (1)

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

2009 (3)

L. B. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography Speckle 5, 77–84 (2009).
[CrossRef]

O. Andrusyak, V. Smirnov, G. Venus, and L. Glebov, “Beam combining of lasers with high spectral density using volume Bragg gratings,” Opt. Commun. 282, 2560–2563 (2009).
[CrossRef]

O. Schmidt, C. Wirth, D. Nodop, J. Limpert, T. Schreiber, T. Peschel, R. Eberhardt, and A. Tünnermann, “Spectral beam combination of fiber amplified ns-pulses by means of interference filters,” Opt. Express 17, 22974–22982 (2009).
[CrossRef]

2008 (3)

2007 (2)

2006 (2)

I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of phase volume diffractive gratings, part 1: transmitting sinusoidal uniform gratings,” Opt. Eng. 45, 015802 (2006).
[CrossRef]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31, 1797–1799 (2006).
[CrossRef]

2005 (2)

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Select. Top. Quantum Electron. 11, 567–577 (2005).

T. Y. Fan and A. Sanchez, “Coherent (phased array) and wavelength (spectral) beam combining compared (invited paper),”Proc. SPIE  5709, 157–164 (2005).

2003 (1)

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

2001 (1)

M. McCall, “Axial electromagnetic wave propagation in inhomogeneous dielectrics,” Mathemat. Comp. Model. 34, 1483–1497 (2001).
[CrossRef]

1999 (1)

1977 (1)

S. Sugimoto and K. Minemura, “High-speed digital-signal transmission experiments by optical wavelength-division multiplexing,” Electron. Lett. 13, 680–682 (1977).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Andrusyak, O.

O. Andrusyak, V. Smirnov, G. Venus, and L. Glebov, “Beam combining of lasers with high spectral density using volume Bragg gratings,” Opt. Commun. 282, 2560–2563 (2009).
[CrossRef]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
[CrossRef]

Armstrong, P. J.

Augst, S. J.

Cardinal, T.

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

Chu, X.

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

Ciapurin, I.

Ciapurin, I. V.

I. V. Ciapurin, D. R. Drachenberg, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Modeling of phase volume diffractive gratings, part 2: reflecting sinusoidal uniform gratings, Bragg mirrors,” Opt. Eng. 51, 058001 (2012).
[CrossRef]

I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of phase volume diffractive gratings, part 1: transmitting sinusoidal uniform gratings,” Opt. Eng. 45, 015802 (2006).
[CrossRef]

Dawson, J. W.

Dimarcello, F. V.

Drachenberg, D. R.

Eberhardt, R.

Efimov, O. M.

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

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

Fan, T. Y.

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers (invited),” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Select. Top. Quantum Electron. 11, 567–577 (2005).

T. Y. Fan and A. Sanchez, “Coherent (phased array) and wavelength (spectral) beam combining compared (invited paper),”Proc. SPIE  5709, 157–164 (2005).

Francois-Saint-Cyr, H. G.

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

Ghalmi, S.

Glebov, L.

O. Andrusyak, V. Smirnov, G. Venus, and L. Glebov, “Beam combining of lasers with high spectral density using volume Bragg gratings,” Opt. Commun. 282, 2560–2563 (2009).
[CrossRef]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
[CrossRef]

L. Glebov, “Volume diffractive elements in photosensitive inorganic glass for beam combining,” in Proceedings of Solid State and Diode Lasers Technical Review (2001), p. FA-5.

Glebov, L. B.

I. V. Ciapurin, D. R. Drachenberg, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Modeling of phase volume diffractive gratings, part 2: reflecting sinusoidal uniform gratings, Bragg mirrors,” Opt. Eng. 51, 058001 (2012).
[CrossRef]

L. B. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography Speckle 5, 77–84 (2009).
[CrossRef]

L. B. Glebov, J. Lumeau, S. Mokhov, V. Smirnov, and B. Y. Zeldovich, “Reflection of light by composite volume holograms: Fresnel corrections and Fabry-Perot spectral filtering,” J. Opt. Soc. Am. A 25, 751–764 (2008).
[CrossRef]

I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of phase volume diffractive gratings, part 1: transmitting sinusoidal uniform gratings,” Opt. Eng. 45, 015802 (2006).
[CrossRef]

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

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

L. B. Glebov, “Photochromic and photo-thermo-refractive glasses,” in Encyclopedia of Smart Materials (Wiley, 2002).

Glebova, L. N.

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

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

Gowin, M.

Heebner, J. E.

Jung, M.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Lei, S.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

Leuchs, G.

Limpert, J.

Lindlein, N.

Ludewigt, K.

Lumeau, J.

McCall, M.

M. McCall, “Axial electromagnetic wave propagation in inhomogeneous dielectrics,” Mathemat. Comp. Model. 34, 1483–1497 (2001).
[CrossRef]

Messerly, M. J.

Minemura, K.

S. Sugimoto and K. Minemura, “High-speed digital-signal transmission experiments by optical wavelength-division multiplexing,” Electron. Lett. 13, 680–682 (1977).
[CrossRef]

Mokhov, S.

Monberg, E.

Nicholson, J. W.

Nodop, D.

Pax, P. H.

Peschel, T.

Ramachandran, S.

Ranka, J. K.

Richardson, K. C.

Sanchez, A.

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers (invited),” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

T. Y. Fan and A. Sanchez, “Coherent (phased array) and wavelength (spectral) beam combining compared (invited paper),”Proc. SPIE  5709, 157–164 (2005).

Schmidt, O.

Schreiber, T.

Sevian, A.

Shang-hong, Z.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

Sheng-bao, Z.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

Shi, L.

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), p. MQ1.

Smirnov, V.

Smirnov, V. I.

I. V. Ciapurin, D. R. Drachenberg, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Modeling of phase volume diffractive gratings, part 2: reflecting sinusoidal uniform gratings, Bragg mirrors,” Opt. Eng. 51, 058001 (2012).
[CrossRef]

I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of phase volume diffractive gratings, part 1: transmitting sinusoidal uniform gratings,” Opt. Eng. 45, 015802 (2006).
[CrossRef]

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

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

Sridharan, A.

Sridharan, A. K.

Sugimoto, S.

S. Sugimoto and K. Minemura, “High-speed digital-signal transmission experiments by optical wavelength-division multiplexing,” Electron. Lett. 13, 680–682 (1977).
[CrossRef]

Tassano, J.

ten Have, E.

Tsybin, I.

Tünnermann, A.

Venus, G.

O. Andrusyak, V. Smirnov, G. Venus, and L. Glebov, “Beam combining of lasers with high spectral density using volume Bragg gratings,” Opt. Commun. 282, 2560–2563 (2009).
[CrossRef]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
[CrossRef]

Venus, G. B.

I. V. Ciapurin, D. R. Drachenberg, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Modeling of phase volume diffractive gratings, part 2: reflecting sinusoidal uniform gratings, Bragg mirrors,” Opt. Eng. 51, 058001 (2012).
[CrossRef]

Wirth, C.

Wisk, P.

Wu, Z.

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

Xing-chun, C.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

Xu, J.

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

Yan, M. F.

Zeldovich, B. Y.

Zhan, S.

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

Zhao, S.

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

Zhuo-liang, W.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Electron. Lett. (1)

S. Sugimoto and K. Minemura, “High-speed digital-signal transmission experiments by optical wavelength-division multiplexing,” Electron. Lett. 13, 680–682 (1977).
[CrossRef]

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

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Select. Top. Quantum Electron. 11, 567–577 (2005).

J. Holography Speckle (1)

L. B. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography Speckle 5, 77–84 (2009).
[CrossRef]

J. Non-Cryst. Solids (1)

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of x-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325, 275–281 (2003).
[CrossRef]

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

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

Mathemat. Comp. Model. (1)

M. McCall, “Axial electromagnetic wave propagation in inhomogeneous dielectrics,” Mathemat. Comp. Model. 34, 1483–1497 (2001).
[CrossRef]

Opt. Commun. (2)

X. Chu, S. Zhao, L. Shi, S. Zhan, J. Xu, and Z. Wu, “Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings,” Opt. Commun. 281, 4099–4102 (2008).
[CrossRef]

O. Andrusyak, V. Smirnov, G. Venus, and L. Glebov, “Beam combining of lasers with high spectral density using volume Bragg gratings,” Opt. Commun. 282, 2560–2563 (2009).
[CrossRef]

Opt. Eng. (2)

I. V. Ciapurin, D. R. Drachenberg, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Modeling of phase volume diffractive gratings, part 2: reflecting sinusoidal uniform gratings, Bragg mirrors,” Opt. Eng. 51, 058001 (2012).
[CrossRef]

I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of phase volume diffractive gratings, part 1: transmitting sinusoidal uniform gratings,” Opt. Eng. 45, 015802 (2006).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (1)

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42, 308–312 (2010).
[CrossRef]

Opt. Lett. (3)

Proc. SPIE (1)

T. Y. Fan and A. Sanchez, “Coherent (phased array) and wavelength (spectral) beam combining compared (invited paper),”Proc. SPIE  5709, 157–164 (2005).

Other (3)

L. Glebov, “Volume diffractive elements in photosensitive inorganic glass for beam combining,” in Proceedings of Solid State and Diode Lasers Technical Review (2001), p. FA-5.

L. B. Glebov, “Photochromic and photo-thermo-refractive glasses,” in Encyclopedia of Smart Materials (Wiley, 2002).

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), p. MQ1.

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Figures (8)

Fig. 1.
Fig. 1.

Diffraction efficiency spectrum of a reflecting VBG with following parameters: t=4.7mm, and δn=320ppm, naver=1.486, d=358.9nm, Bragg angle in air 4° for 1064 nm. Probe beam: monochromatic plane wave (blue-dots), monochromatic divergent wave (θ=0.5mrad, half-angle at 1/e2) (red solid), polychromatic plane wave with 50 pm full spectral width (green dash).

Fig. 2.
Fig. 2.

Illustration of a single-stage SBC of two beams using a single reflecting VBG. Beam 1 (λ1) transmits through a VBG diffraction minimum with some diffraction loss. Beam 2 (λ2) diffracts at the Bragg condition with some transmission loss.

Fig. 3.
Fig. 3.

Combining efficiency of monochromatic plane waves combined by a single VBG (material losses are not considered) versus thickness and refractive index modulation.

Fig. 4.
Fig. 4.

Combining efficiency of two beams by a single VBG with 0.25 nm spectral separation and 0.5 mrad divergence including the effects of material losses.

Fig. 5.
Fig. 5.

Illustration of four-stage SBC of five beams. Numbers correspond to those in Fig. 7.

Fig. 6.
Fig. 6.

Efficiency of five-channel SBC by four VBGs as a function of VBG thickness and refractive index modulation for 0.25 nm spectral separation, and 0.5 mrad of divergence including the effects of material losses.

Fig. 7.
Fig. 7.

Diffraction efficiency spectra of VBGs for an optimized five-channel system. Vertical solid lines represent the wavelengths of five combined beams, where beam 1 transmits through all four VBGs, and beam five diffracts off of the final VBG. The diffraction efficiency spectrum of each VBG is only extended far enough to overlap with the associated interacting beams. Numbers correspond to beams in Fig. 5.

Fig. 8.
Fig. 8.

Two-beam SBC experiment with 300 W combined output power and M2 of 1.16. (Left) Image of combined beam at the waist. (Right) M2 measurement derived from the beam-diameter measurements at five points through the waist of the combined beam after passing through a focusing lens.

Equations (19)

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B=Pπω2πθBeam2=Pπω2πθideal-Gaussian2(M2)2=Pπω2π(λπω)2(M2)2=Pλ2(M2)2,
M2=θλ/(πω)=θbeamθideal-Gaussian.
BCF=BcombinedBunit=[P/(M2)2]combined[P/(M2)2]unit.
BCF=ηBC(M2)unit2(M2)combined2,
ηBC=PcombPunit.
η(ξ)=sin2(ξ2S2)ξ2/S2cos2(ξ2S2),
S=πtδnλ0cos(θm*)
f=2navcos(θm)λ0.
ξ(dθ,Δλ)=(2πnavtcos(θm)λ0)×(dθsin(θm)cos(θm)+dθ22+Δλλ0).
ηBC(t,δn)=1PB+PT1×{PB[η(λB,t,δn)]+PT1[1η(λT1,t,δn)]},
η0=tanh2(πtδnλ0|cos(θm)|).
G1(Δλ,w)=e2(Δλλ0w)2,
G2(Δθm,θBeam)=e2(ΔθmθmθBeam)2,
α10(λ)cm1=α10(λ0)(λ0λ)4cm1,
α10(λ0=750nm)=200δn.
Ltot(λ,t,δn)=110α10(λ)t=110200δnλ04tλ4.
ηBC(t,δn)=(1Ltot(λ,t,δn)PB+PT1)×{PB[ηdθ(λB,t,δn,θbeam)]+PT1[1ηdθ(λT1,t,δn,θbeam)]}.
ηBC(t,δn)=(1Ltot(λ,t,δn)Pin)×{PB[ηdθ(λB,t,δn,θbeam)]+PT1[1ηdθ(λT1,t,δn,θbeam)]++PTN[1ηdθ(λTN,t,δn,θbeam)]}.
ηBC(t,δn)=(1LtotPin)×n=1N{PB[ηdθ(λB,t,δn,θbeam)]+PTn[1ηdθ(λTn,t,δn,θbeam)]},

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