Abstract

Faraday polarization rotators are commonly used in laser experiments. Most Faraday materials have a nonnegligible absorption, which is a limiting factor for high power laser optical isolators or for intracavity optical diodes. By using a stronger magnetic field and a shorter length of Faraday material, one can obtain the same polarization rotation and a reduced absorption. In this paper, we describe two permanent magnet arrangements that are easy to build and produce magnetic fields up to 1.7T, substantially more than commonly used. The field homogeneity is largely sufficient for a 30dB isolation ratio. We finally discuss the prospects for producing even larger fields with permanent magnets.

© 2011 Optical Society of America

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    [CrossRef]
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  32. D. K. Wilson and A. Heiney, “Magnetic configuration for Faraday rotators,” U.S. patent 4,856,878 (15 August 1987).
  33. J. Vigué, G. Trénec, O. Cugat, and W. Volondat, “Magnetic field generator having permanent magnets,” Patent WO/2008/031935 A1 (30 March 2008).
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    [CrossRef]
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    [CrossRef]
  41. F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
    [CrossRef]
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2009 (1)

I. Mukhin, A. Voitovich, O. Palashov, and E. Khazanov, “2.1 Tesla permanent-magnet Faraday isolator for subkilowatt average power lasers,” Opt. Commun. 282, 1969–1972 (2009).
[CrossRef]

2008 (3)

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

2007 (4)

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

W. Zhang, F. Guo, and J. Chen, “Growth and characterization of Tb3Ga5−xAlxO12,” J. Cryst. Growth 306, 195–199 (2007).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

2002 (2)

2001 (2)

E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron. 31, 351–356 (2001).
[CrossRef]

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, and R. R. Reitze, Appl. Opt. 40, 366–374 (2001).
[CrossRef]

2000 (3)

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. H. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B 17, 99–102 (2000).
[CrossRef]

H. A. Leupold, A. Tilak, and E. Potenziani II, “Permanent magnet spheres: design, construction and application,” J. Appl. Phys. 87, 4730–4734 (2000).
[CrossRef]

1999 (2)

E. Khazanov, “Compensation of thermally induced polarisation distortions in Faraday isolators,” Quantum Electron. 29, 59–64 (1999).
[CrossRef]

F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
[CrossRef]

1998 (1)

F. Bloch, O. Cugat, G. Meunier, and J. C. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source,” IEEE Trans. Magn. 34, 2465–2468 (1998).
[CrossRef]

1997 (1)

1992 (2)

N. P. Barnes and L. B. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B 9, 1912–1915 (1992).
[CrossRef]

R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, “A compact tunable 60 dB Faraday optical isolator,” Rev. Sci. Instrum. 63, 5586–5590 (1992).
[CrossRef]

1989 (1)

1987 (1)

H. A. Leupold and E. Potenziani II, “Novel high-field permanent-magnet flux sources,” IEEE Trans. Magn. MAG-23, 3628–3629 (1987).
[CrossRef]

1986 (1)

1985 (1)

H. Zijlstra, “Permanent magnet systems for NMR tomography,” Philips J. Res. 40, 259–288 (1985).

1982 (1)

K. P. Birch, “A compact optical isolator,” Opt. Commun. 43, 79–84 (1982).
[CrossRef]

1981 (1)

K. Halbach, “Physical and optical properties of rare earth cobalt magnets,” Nucl. Instrum. Methods 187, 109–117 (1981).
[CrossRef]

1980 (2)

K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods 169, 1–10 (1980).
[CrossRef]

T. F. Johnston and W. Proffitt, “Design and performance of a broad-band optical diode to enforce one-direction traveling-wave operation of a ring laser,” IEEE J. Quantum Electron. QE-16, 483–488 (1980).
[CrossRef]

1979 (1)

F. Biraben, “Efficacité des systèmes unidirectionnels utilisables dans les lasers en anneau,” Opt. Commun. 29, 353–356(1979).
[CrossRef]

1978 (1)

A. Balbin Villaverde, D. A. Donatti, and D. G. Bozinis, “Terbium gallium garnet Verdet constant measurements with pulsed magnetic field,” J. Phys. C 11, L495–L498 (1978).
[CrossRef]

1977 (1)

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

1974 (1)

D. J. Dentz, R. C. Puttbach, and R. F. Belt, “Terbium gallium garnet for Faraday effect devices,” AIP Conf. Proc. 18, 954–958 (1974).

1971 (1)

1964 (1)

1901 (1)

Lord Rayleigh, “On the magnetic rotation of light and the second law of thermo-dynamics,” Nature 64, 577–578 (1901).
[CrossRef]

1897 (1)

H. Becquerel, “Sur une interprétation applicable au phénomène de Faraday et au phénomène de Zeeman,” C. R. Acad. Sci. 125, 679–685 (1897).

1885 (1)

Lord Rayleigh, “On the constant of magnetic rotation of light in bisulphide of carbon,” Phil. Trans. R. Soc. London 176, 343–366 (1885).
[CrossRef]

Anastasiyev, A.

Andreev, N.

Andreev, N. F.

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

Babin, A.

Barnes, N. P.

Becquerel, H.

H. Becquerel, “Sur une interprétation applicable au phénomène de Faraday et au phénomène de Zeeman,” C. R. Acad. Sci. 125, 679–685 (1897).

Belt, R. F.

D. J. Dentz, R. C. Puttbach, and R. F. Belt, “Terbium gallium garnet for Faraday effect devices,” AIP Conf. Proc. 18, 954–958 (1974).

Biraben, F.

F. Biraben, “Efficacité des systèmes unidirectionnels utilisables dans les lasers en anneau,” Opt. Commun. 29, 353–356(1979).
[CrossRef]

Birch, K. P.

K. P. Birch, “A compact optical isolator,” Opt. Commun. 43, 79–84 (1982).
[CrossRef]

Bloch, F.

F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
[CrossRef]

F. Bloch, O. Cugat, G. Meunier, and J. C. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source,” IEEE Trans. Magn. 34, 2465–2468 (1998).
[CrossRef]

F. Bloch, “Source de champ intense 4 Tesla aimants permanents,” Ph.D. dissertation (Université de Grenoble, 1999).

Boyd, R. W.

Bozinis, D. G.

A. Balbin Villaverde, D. A. Donatti, and D. G. Bozinis, “Terbium gallium garnet Verdet constant measurements with pulsed magnetic field,” J. Phys. C 11, L495–L498 (1978).
[CrossRef]

Bretenaker, F.

Brillet, A.

A. Brillet and F. Cleva, ARTEMIS Observatoire Cote d’Azur, CNRS, Université de Nice Sophia Antipolis 06304 Nice (private communication, 2004).

Byer, R. L.

Canalias, C.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

Chen, J.

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

W. Zhang, F. Guo, and J. Chen, “Growth and characterization of Tb3Ga5−xAlxO12,” J. Cryst. Growth 306, 195–199 (2007).
[CrossRef]

Chen, Y.

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

Chevy, F.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

Cleva, F.

A. Brillet and F. Cleva, ARTEMIS Observatoire Cote d’Azur, CNRS, Université de Nice Sophia Antipolis 06304 Nice (private communication, 2004).

Clubley, D.

Cotteverte, J.-C.

Cugat, O.

F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
[CrossRef]

F. Bloch, O. Cugat, G. Meunier, and J. C. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source,” IEEE Trans. Magn. 34, 2465–2468 (1998).
[CrossRef]

J. Vigué, G. Trénec, O. Cugat, and W. Volondat, “Magnetic field generator having permanent magnets,” Patent WO/2008/031935 A1 (30 March 2008).

Dentz, D. J.

D. J. Dentz, R. C. Puttbach, and R. F. Belt, “Terbium gallium garnet for Faraday effect devices,” AIP Conf. Proc. 18, 954–958 (1974).

Diedrich, F.

R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, “A compact tunable 60 dB Faraday optical isolator,” Rev. Sci. Instrum. 63, 5586–5590 (1992).
[CrossRef]

Donatti, D. A.

A. Balbin Villaverde, D. A. Donatti, and D. G. Bozinis, “Terbium gallium garnet Verdet constant measurements with pulsed magnetic field,” J. Phys. C 11, L495–L498 (1978).
[CrossRef]

Douillet, A.

L. Hilico, A. Douillet, J.-P. Karr, and E. Tournié, “Faraday optical isolator in the 9.2 μm range for QCL applications” (personal communication, 2011).

Eismann, U.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

Fejer, M. M.

Frölich, D.

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

Fugger, B.

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

Fujimoto, Y.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Gao, Z.

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

Gauthier, D. J.

Gerbier, F.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

Guo, F.

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

W. Zhang, F. Guo, and J. Chen, “Growth and characterization of Tb3Ga5−xAlxO12,” J. Cryst. Growth 306, 195–199 (2007).
[CrossRef]

Gustafson, E. K.

Halbach, K.

K. Halbach, “Physical and optical properties of rare earth cobalt magnets,” Nucl. Instrum. Methods 187, 109–117 (1981).
[CrossRef]

K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods 169, 1–10 (1980).
[CrossRef]

Heiney, A.

D. K. Wilson and A. Heiney, “Magnetic configuration for Faraday rotators,” U.S. patent 4,856,878 (15 August 1987).

Hennawi, J.

Hilico, L.

L. Hilico, A. Douillet, J.-P. Karr, and E. Tournié, “Faraday optical isolator in the 9.2 μm range for QCL applications” (personal communication, 2011).

Johnston, T. F.

T. F. Johnston and W. Proffitt, “Design and performance of a broad-band optical diode to enforce one-direction traveling-wave operation of a ring laser,” IEEE J. Quantum Electron. QE-16, 483–488 (1980).
[CrossRef]

Kan, H.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Karr, J.-P.

L. Hilico, A. Douillet, J.-P. Karr, and E. Tournié, “Faraday optical isolator in the 9.2 μm range for QCL applications” (personal communication, 2011).

Katin, E. V.

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

Kawanaka, J.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Kawashima, T.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Khazanov, E.

Khazanov, E. A.

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron. 31, 351–356 (2001).
[CrossRef]

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

Kiselev, A.

Le Floch, A.

Leupold, H. A.

H. A. Leupold, A. Tilak, and E. Potenziani II, “Permanent magnet spheres: design, construction and application,” J. Appl. Phys. 87, 4730–4734 (2000).
[CrossRef]

H. A. Leupold and E. Potenziani II, “Novel high-field permanent-magnet flux sources,” IEEE Trans. Magn. MAG-23, 3628–3629 (1987).
[CrossRef]

H. A. Leupold and A. Tilak, “Field augmented permanent magnet structures,” U.S. patent 5,428,334 (27 June 1995).

Li, H.

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

Liu, J.

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

Mansell, J. D.

Mehl, O.

Meschede, D.

R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, “A compact tunable 60 dB Faraday optical isolator,” Rev. Sci. Instrum. 63, 5586–5590 (1992).
[CrossRef]

Meunier, G.

F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
[CrossRef]

F. Bloch, O. Cugat, G. Meunier, and J. C. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source,” IEEE Trans. Magn. 34, 2465–2468 (1998).
[CrossRef]

Mukhin, I.

I. Mukhin, A. Voitovich, O. Palashov, and E. Khazanov, “2.1 Tesla permanent-magnet Faraday isolator for subkilowatt average power lasers,” Opt. Commun. 282, 1969–1972 (2009).
[CrossRef]

Mukhin, I. B.

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

Nakatsuka, M.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Narum, P.

Nozawa, H.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Palashov, O.

Palashov, O. V.

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

Petway, L. B.

Poirson, J.

Potemkin, A. K.

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

Potenziani, E.

H. A. Leupold, A. Tilak, and E. Potenziani II, “Permanent magnet spheres: design, construction and application,” J. Appl. Phys. 87, 4730–4734 (2000).
[CrossRef]

H. A. Leupold and E. Potenziani II, “Novel high-field permanent-magnet flux sources,” IEEE Trans. Magn. MAG-23, 3628–3629 (1987).
[CrossRef]

Poteomkin, A.

Proffitt, W.

T. F. Johnston and W. Proffitt, “Design and performance of a broad-band optical diode to enforce one-direction traveling-wave operation of a ring laser,” IEEE J. Quantum Electron. QE-16, 483–488 (1980).
[CrossRef]

Puttbach, R. C.

D. J. Dentz, R. C. Puttbach, and R. F. Belt, “Terbium gallium garnet for Faraday effect devices,” AIP Conf. Proc. 18, 954–958 (1974).

Rayleigh, Lord

Lord Rayleigh, “On the magnetic rotation of light and the second law of thermo-dynamics,” Nature 64, 577–578 (1901).
[CrossRef]

Lord Rayleigh, “On the constant of magnetic rotation of light in bisulphide of carbon,” Phil. Trans. R. Soc. London 176, 343–366 (1885).
[CrossRef]

Reitze, D. H.

Reitze, R. R.

Robinson, C. C.

Ru, J.

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

Salomon, C.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

Sansalone, F. J.

Schröder, H. W.

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

Schulz, P. A.

Sergeev, A.

Sergeev, A. M.

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

Stein, L.

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

Telle, H. R.

R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, “A compact tunable 60 dB Faraday optical isolator,” Rev. Sci. Instrum. 63, 5586–5590 (1992).
[CrossRef]

Tilak, A.

H. A. Leupold, A. Tilak, and E. Potenziani II, “Permanent magnet spheres: design, construction and application,” J. Appl. Phys. 87, 4730–4734 (2000).
[CrossRef]

H. A. Leupold and A. Tilak, “Field augmented permanent magnet structures,” U.S. patent 5,428,334 (27 June 1995).

Tokita, S.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Tournié, E.

L. Hilico, A. Douillet, J.-P. Karr, and E. Tournié, “Faraday optical isolator in the 9.2 μm range for QCL applications” (personal communication, 2011).

Toussaint, J. C.

F. Bloch, O. Cugat, G. Meunier, and J. C. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source,” IEEE Trans. Magn. 34, 2465–2468 (1998).
[CrossRef]

Toussaint, J.-C.

F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
[CrossRef]

Trénec, G.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

J. Vigué, G. Trénec, O. Cugat, and W. Volondat, “Magnetic field generator having permanent magnets,” Patent WO/2008/031935 A1 (30 March 2008).

Vigué, J.

J. Vigué, G. Trénec, O. Cugat, and W. Volondat, “Magnetic field generator having permanent magnets,” Patent WO/2008/031935 A1 (30 March 2008).

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

Villaverde, A. Balbin

A. Balbin Villaverde, D. A. Donatti, and D. G. Bozinis, “Terbium gallium garnet Verdet constant measurements with pulsed magnetic field,” J. Phys. C 11, L495–L498 (1978).
[CrossRef]

Voitovich, A.

I. Mukhin, A. Voitovich, O. Palashov, and E. Khazanov, “2.1 Tesla permanent-magnet Faraday isolator for subkilowatt average power lasers,” Opt. Commun. 282, 1969–1972 (2009).
[CrossRef]

Voitovich, A. V.

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

Volondat, W.

J. Vigué, G. Trénec, O. Cugat, and W. Volondat, “Magnetic field generator having permanent magnets,” Patent WO/2008/031935 A1 (30 March 2008).

Voytovich, A.

Welling, H.

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

Wilson, D. K.

D. K. Wilson and A. Heiney, “Magnetic configuration for Faraday rotators,” U.S. patent 4,856,878 (15 August 1987).

Wynands, R.

R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, “A compact tunable 60 dB Faraday optical isolator,” Rev. Sci. Instrum. 63, 5586–5590 (1992).
[CrossRef]

Yagi, H.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Yanagitani, T.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Yasuhara, R.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Yoshida, H.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007).
[CrossRef] [PubMed]

Yoshida, S.

Zhang, W.

W. Zhang, F. Guo, and J. Chen, “Growth and characterization of Tb3Ga5−xAlxO12,” J. Cryst. Growth 306, 195–199 (2007).
[CrossRef]

Zhao, B.

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

Zheleznov, D. S.

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

Zhuang, N.

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

Zijlstra, H.

H. Zijlstra, “Permanent magnet systems for NMR tomography,” Philips J. Res. 40, 259–288 (1985).

AIP Conf. Proc. (1)

D. J. Dentz, R. C. Puttbach, and R. F. Belt, “Terbium gallium garnet for Faraday effect devices,” AIP Conf. Proc. 18, 954–958 (1974).

Appl. Opt. (7)

Appl. Phys. (1)

H. W. Schröder, L. Stein, D. Frölich, B. Fugger, and H. Welling, “A high-power single-mode cw dye ring laser,” Appl. Phys. 14, 377–380 (1977).
[CrossRef]

C. R. Acad. Sci. (1)

H. Becquerel, “Sur une interprétation applicable au phénomène de Faraday et au phénomène de Zeeman,” C. R. Acad. Sci. 125, 679–685 (1897).

Eur. Phys. J. Appl. Phys. (1)

F. Bloch, O. Cugat, J.-C. Toussaint, and G. Meunier, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents,” Eur. Phys. J. Appl. Phys. 5, 85–89 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. S. Zheleznov, I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and A. V. Voitovich, “Faraday rotators with short magneto-optical elements for 50 kW laser power,” IEEE J. Quantum Electron. 43, 451–457 (2007).
[CrossRef]

T. F. Johnston and W. Proffitt, “Design and performance of a broad-band optical diode to enforce one-direction traveling-wave operation of a ring laser,” IEEE J. Quantum Electron. QE-16, 483–488 (1980).
[CrossRef]

IEEE Trans. Magn. (2)

H. A. Leupold and E. Potenziani II, “Novel high-field permanent-magnet flux sources,” IEEE Trans. Magn. MAG-23, 3628–3629 (1987).
[CrossRef]

F. Bloch, O. Cugat, G. Meunier, and J. C. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source,” IEEE Trans. Magn. 34, 2465–2468 (1998).
[CrossRef]

J. Appl. Phys. (1)

H. A. Leupold, A. Tilak, and E. Potenziani II, “Permanent magnet spheres: design, construction and application,” J. Appl. Phys. 87, 4730–4734 (2000).
[CrossRef]

J. Cryst. Growth (3)

W. Zhang, F. Guo, and J. Chen, “Growth and characterization of Tb3Ga5−xAlxO12,” J. Cryst. Growth 306, 195–199 (2007).
[CrossRef]

J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310, 2613–2616 (2008).
[CrossRef]

F. Guo, J. Ru, H. Li, N. Zhuang, J. Liu, B. Zhao, and J. Chen, “Growth and magneto-optical properties of NaTb(MoO4)2 crystals,” J. Cryst. Growth 310, 4390–4393 (2008).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. C (1)

A. Balbin Villaverde, D. A. Donatti, and D. G. Bozinis, “Terbium gallium garnet Verdet constant measurements with pulsed magnetic field,” J. Phys. C 11, L495–L498 (1978).
[CrossRef]

J. Phys. Conf. Ser. (1)

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Development of cryogenic TGG ceramic based Faraday rotator for inertial fusion driver,” J. Phys. Conf. Ser. 112, 032059 (2008).
[CrossRef]

Nature (1)

Lord Rayleigh, “On the magnetic rotation of light and the second law of thermo-dynamics,” Nature 64, 577–578 (1901).
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K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods 169, 1–10 (1980).
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I. Mukhin, A. Voitovich, O. Palashov, and E. Khazanov, “2.1 Tesla permanent-magnet Faraday isolator for subkilowatt average power lasers,” Opt. Commun. 282, 1969–1972 (2009).
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Opt. Express (1)

Opt. Lett. (1)

Phil. Trans. R. Soc. London (1)

Lord Rayleigh, “On the constant of magnetic rotation of light in bisulphide of carbon,” Phil. Trans. R. Soc. London 176, 343–366 (1885).
[CrossRef]

Philips J. Res. (1)

H. Zijlstra, “Permanent magnet systems for NMR tomography,” Philips J. Res. 40, 259–288 (1985).

Quantum Electron. (4)

N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “A 45 dB Faraday isolator for 100 W average radiation power,” Quantum Electron. 30, 1107–1108 (2000).
[CrossRef]

E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron. 31, 351–356 (2001).
[CrossRef]

E. Khazanov, “Compensation of thermally induced polarisation distortions in Faraday isolators,” Quantum Electron. 29, 59–64 (1999).
[CrossRef]

A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37, 471–474 (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

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[CrossRef]

Other (10)

Terbium gallium garnet on Northrop Grumman website, http://www.st.northropgrumman.com.

L. Hilico, A. Douillet, J.-P. Karr, and E. Tournié, “Faraday optical isolator in the 9.2 μm range for QCL applications” (personal communication, 2011).

D. K. Wilson and A. Heiney, “Magnetic configuration for Faraday rotators,” U.S. patent 4,856,878 (15 August 1987).

J. Vigué, G. Trénec, O. Cugat, and W. Volondat, “Magnetic field generator having permanent magnets,” Patent WO/2008/031935 A1 (30 March 2008).

F. Bloch, “Source de champ intense 4 Tesla aimants permanents,” Ph.D. dissertation (Université de Grenoble, 1999).

H. A. Leupold and A. Tilak, “Field augmented permanent magnet structures,” U.S. patent 5,428,334 (27 June 1995).

ChenYang Technologies GmbH & Co, http://www.chenyang-ism.com/.

U. Eismann, F. Gerbier, C. Canalias, G. Trénec, J. Vigué, F. Chevy, and C. Salomon, “An all-solid-state laser source at 671 nm for cold atom experiments with lithium,” Appl. Phys. B (to be published).

A. Brillet and F. Cleva, ARTEMIS Observatoire Cote d’Azur, CNRS, Université de Nice Sophia Antipolis 06304 Nice (private communication, 2004).

Flux2D/3D, finite elements software package available from CEDRAT Company, www.cedrat.com.

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Figures (7)

Fig. 1
Fig. 1

Plot of the functions f i ( θ ) as a function of θ expressed in degrees. The solid black curve is the optimum function f BCTM ( θ ) = 1 + 3 cos 2 θ , while f 1 ( θ ) = 3 cos 2 θ 1 is represented by the dashed red curve, f 2 ( θ ) = f 1 ( θ ) is represented by the dotted–dashed blue curve, and f 3 ( θ ) = 3 sin ( 2 θ ) / 2 is represented by the dotted purple curve. We have plotted these functions only when they are positive and this explains why f 4 ( θ ) does not appear in this figure.

Fig. 2
Fig. 2

Meridian cut of an axial-only cylindrical magnet. The magnetized material extends between an internal cylinder of radius r i and an external cylinder of radius r e . The arrows indicate the magnetization vector M.

Fig. 3
Fig. 3

Meridian cut of an axial & radial cylindrical magnet. The magnetized material extends between an internal cylinder of radius r i and an external cylinder of radius r e . The arrows indicate the magnetization M.

Fig. 4
Fig. 4

Drawing of our axial-only prototypes: magnetization is indicated by arrows. The ring dimensions are given in Table 1 for the two prototypes. Aluminum alloy rings are placed on the border line between opposite magnetization.

Fig. 5
Fig. 5

Magnetic field component B z in Tesla measured as a function of z for our two prototypes following the two-orientation design: squares for prototype 1 and dots for prototype 2. In both cases, the data points are fitted by equation B z ( z ) = B z ( O ) + B z , 2 z 2 / 2 + B z , 4 z 4 / 24 (blue curve for prototype 1, red curve for prototype 2). The values of B z ( O ) and B z , 2 are discussed in the text; the contribution of the B z , 4 term is almost negligible.

Fig. 6
Fig. 6

Drawing of our axial & radial prototype 3 (dimensions in millimeters). This magnet is made of three axial rings and of four radial rings (two with inward magnetization and two with outward magnetization). Magnetization is indicated by arrows.

Fig. 7
Fig. 7

Magnetic field component B z in Tesla measured as a function of z for our axial & radial prototype 3. The data points are fitted by an equation B z ( z ) = B z ( O ) + B z , 2 ( z z c ) 2 / 2 + B z , 4 ( z z c ) 4 / 24 . The values of B z ( O ) and B z , 2 are discussed in the text; B z , 4 is almost negligible.

Tables (2)

Tables Icon

Table 1 Dimensions (in Millimeters) of the Rings Used for the Axial-Only Prototypes, Following the Design of Fig. 4

Tables Icon

Table 2 Values of B z ( O ) , B z , 2 , r i , r e , and h, and the Chosen Value of ρ max for Each Prototype

Equations (15)

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θ F = V B z ( z ) d z V B 0 L .
M = V α .
M * = V α × K ( d n / d T ) ,
d B z = μ 0 M ( r ) d 3 r 4 π r 3 × f ( θ , ψ ) , w ith f ( θ , ψ ) = 2 cos θ cos ψ + sin θ sin ψ ,
B z ( O ) = 4 3 μ 0 M ln ( r e r i ) .
B z , ( O ) = 4 3 3 B r ln ( r e r i ) = 0.77 B r ln ( r e r i ) .
δ B z ( O ) B r [ 1 1 1 + ( r e / h ) 2 ] ,
B z ( x , y , z ) = B z ( O ) + B z , 2 2 z 2 ( x 2 + y 2 ) 4 ,
B z , 2 16 B r 9 3 r i 2 ,
B z , ( O ) = 1.23 B r ln ( r e r i ) .
T = d S d P d S sin 2 ( θ F θ 0 ) / d S d P d S .
d P d S exp [ 2 ρ 2 w 2 ] .
I ( ρ ) B 0 L + B z , 2 ( L 3 6 ρ 2 L ) 24 I 0 [ 1 B z , 2 4 B 0 ρ 2 ] .
T = [ θ 0 B z , 2 w 0 2 8 B 0 ] 2 × X 0 2 K 0 2 X 0 K 1 + K 2 K 0 with K n = 0 X max X n exp ( X ) d X .
T min = [ θ 0 B z , 2 w 2 8 B 0 ] 2 ( K 2 K 0 K 1 2 K 0 2 ) [ θ 0 B z , 2 w 2 8 B 0 ] 2 ,

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