Abstract

Guided-wave acousto-optic Bragg diffraction and surface acoustic-wave propagation in epitaxially matched SrTiO3/BaTiO3(001)/SrTiO3 thin-film heterostructures have been theoretically studied. The optimum electromechanical coupling and Bragg diffraction efficiency have been determined at several acoustic frequencies by means of varying the thickness of the SrTiO3 overlayer and the BaTiO3 waveguiding layer. A strain-controlling nonpiezoelectric SrTiO3 overlayer upon the BaTiO3/SrTiO3 structure is found to enhance the coupling coefficient (k 2) and the diffraction efficiency significantly. A comparison of asymmetric and symmetric structures shows an increase in the diffraction efficiency from 10.5% to 43.0% and a decrease in the untuned-transducer conversion efficiency from 36 to 23 dB at an operating frequency of 1 GHz with an interaction length of 1 mm and an acoustic power of 1 mW.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  24. G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics—Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. 9, Chap. 2, p. 35.
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1999

R. Nayak, V. Gupta, K. Sreenivas, “Studies on acousto-optical interaction in SrTiO3/BaTiO3/SrTiO3 epitaxial thin film heterostructures,” J. Phys. D 32, 380–387 (1999).
[CrossRef]

1997

C. Thompson, B. L. Weiss, “Acoustooptic interactions in AlGaAs–GaAs planar multilayer waveguide structures,” IEEE J. Quantum Electron. 33, 1601–1607 (1997).
[CrossRef]

M. Kimio, “Recent trend of surface acoustic wave (SAW) filters: high-frequency filter using zinc oxide-sapphire substrate,” Erekutoronikusu 42, 36–38 (1997) (in Japanese).

S. Shinichi, “Recent trend of surface acoustic wave (SAW) filters: high-frequency filter using diamond substrate,” Erekutoronikusu 42, 33–35 (1997) (in Japanese).

1996

A. M. Matteo, V. M. N. Passaro, M. N. Armenise, “High performance guided wave acoustooptic bragg cells in LiNbO3 and GaAs based structures,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 270–279 (1996).
[CrossRef]

D. M. Gill, B. A. Block, C. W. Conward, B. W. Wessels, S. T. Ho, “Thin film channel waveguides fabricated in metalorganic chemical vapor deposition grown BaTiO3 on MgO,” Appl. Phys. Lett. 69, 2968–2970 (1996).
[CrossRef]

1995

W. J. Lin, T. Y. Tseng, H. B. Lu, S. L. Tu, S. J. Yang, I. N. Lin, “Growth and ferroelectricity of epitaxial-like BaTiO3 films on single-crystal MgO, SrTiO3, and silicon substrates synthesized by pulse laser deposition,” J. Appl. Phys. 77, 6466–6471 (1995).
[CrossRef]

1994

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

B. K. Moon, H. Ishiwara, “Roles of buffer layers in epitaxial growth of SrTiO3 films on silicon substrates,” Jpn. J. Appl. Phys. 33, 1472–1477 (1994).
[CrossRef]

1993

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

1992

S. Jain, A. Mansingh, “Thin film layered structure for acousto-optic devices,” J. Phys. D 25, 1116–1121 (1992).
[CrossRef]

1989

Z. Surowiak, A. M. Margolin, I. N. Zakharchenko, S. V. Biryukov, “The influence of structure on the piezoelectric properties of BaTiO3 and (BaSr)TiO3 thin films with a diffuse phase transition,” Thin Solid Films 176, 227–246 (1989).
[CrossRef]

1985

K. Tsubouchi, N. Mikoshiba, “Zero-temperature coefficient SAW devices on A1N epitaxial films,” IEEE Trans. Sonics Ultrason. SU-32, 634–644 (1985).
[CrossRef]

1976

I. C. Chang, “Acousto-optic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2–21 (1976).
[CrossRef]

J. Kushibiki, N. Chubachi, K. Shibayama, “Improvement of diffraction efficiency in surface-acoustic-optic devices by means of multilayered structures,” Appl. Phys. Lett. 29, 333–335 (1976).
[CrossRef]

1975

A. H. Fahmy, E. L. Adler, “Multilayer acoustic surface wave program,” Proc. Inst. Electr. Eng. 122, 470–471 (1975).
[CrossRef]

1973

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

1971

K. Toda, K. Takahashi, K. Okazaki, “Effects of thermally diffused impurities on propagation of elastic waves on the ferroelectric ceramics,” Appl. Phys. Lett. 18, 396–398 (1971).
[CrossRef]

1970

Adler, E. L.

A. H. Fahmy, E. L. Adler, “Multilayer acoustic surface wave program,” Proc. Inst. Electr. Eng. 122, 470–471 (1975).
[CrossRef]

G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics—Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. 9, Chap. 2, p. 35.

Armenise, M. N.

A. M. Matteo, V. M. N. Passaro, M. N. Armenise, “High performance guided wave acoustooptic bragg cells in LiNbO3 and GaAs based structures,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 270–279 (1996).
[CrossRef]

Auld, B. A.

B. A. Auld, in Acoustic Fields and Waves in Solids (Wiley, New York, 1973), Vol. 2, Chap. 12, p. 298.

Batson, P. E.

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

Biryukov, S. V.

Z. Surowiak, A. M. Margolin, I. N. Zakharchenko, S. V. Biryukov, “The influence of structure on the piezoelectric properties of BaTiO3 and (BaSr)TiO3 thin films with a diffuse phase transition,” Thin Solid Films 176, 227–246 (1989).
[CrossRef]

Block, B. A.

D. M. Gill, B. A. Block, C. W. Conward, B. W. Wessels, S. T. Ho, “Thin film channel waveguides fabricated in metalorganic chemical vapor deposition grown BaTiO3 on MgO,” Appl. Phys. Lett. 69, 2968–2970 (1996).
[CrossRef]

Burfoot, J. C.

J. C. Burfoot, G. W. Taylor, Polar Dielectrics and Their Applications (Macmillan, London, 1979).

Chang, I. C.

I. C. Chang, “Acousto-optic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2–21 (1976).
[CrossRef]

Chern, M. Y.

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

Chubachi, N.

J. Kushibiki, N. Chubachi, K. Shibayama, “Improvement of diffraction efficiency in surface-acoustic-optic devices by means of multilayered structures,” Appl. Phys. Lett. 29, 333–335 (1976).
[CrossRef]

Conward, C. W.

D. M. Gill, B. A. Block, C. W. Conward, B. W. Wessels, S. T. Ho, “Thin film channel waveguides fabricated in metalorganic chemical vapor deposition grown BaTiO3 on MgO,” Appl. Phys. Lett. 69, 2968–2970 (1996).
[CrossRef]

Eimura, T.

W. Yatsuda, T. Horishima, T. Eimura, T. Ooiwa, “Miniaturized SAW filters using a flip-chip technique,” in Proceedings of the 1994 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 4, pp. 159–162.

Fahmy, A. H.

A. H. Fahmy, E. L. Adler, “Multilayer acoustic surface wave program,” Proc. Inst. Electr. Eng. 122, 470–471 (1975).
[CrossRef]

Farnell, G. W.

G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics—Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. 9, Chap. 2, p. 35.

Fujito, K.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Ghatak, A.

A. Ghatak, K. Thyagarajan, in Optical Electronics (Cambridge U. Press, Cambridge, 1991).

Gill, D. M.

D. M. Gill, B. A. Block, C. W. Conward, B. W. Wessels, S. T. Ho, “Thin film channel waveguides fabricated in metalorganic chemical vapor deposition grown BaTiO3 on MgO,” Appl. Phys. Lett. 69, 2968–2970 (1996).
[CrossRef]

Gong, J.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Gupta, A.

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

Gupta, V.

R. Nayak, V. Gupta, K. Sreenivas, “Studies on acousto-optical interaction in SrTiO3/BaTiO3/SrTiO3 epitaxial thin film heterostructures,” J. Phys. D 32, 380–387 (1999).
[CrossRef]

Hirai, K.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Ho, S. T.

D. M. Gill, B. A. Block, C. W. Conward, B. W. Wessels, S. T. Ho, “Thin film channel waveguides fabricated in metalorganic chemical vapor deposition grown BaTiO3 on MgO,” Appl. Phys. Lett. 69, 2968–2970 (1996).
[CrossRef]

Horiguchi, K.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Horishima, T.

W. Yatsuda, T. Horishima, T. Eimura, T. Ooiwa, “Miniaturized SAW filters using a flip-chip technique,” in Proceedings of the 1994 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 4, pp. 159–162.

Ishiwara, H.

B. K. Moon, H. Ishiwara, “Roles of buffer layers in epitaxial growth of SrTiO3 films on silicon substrates,” Jpn. J. Appl. Phys. 33, 1472–1477 (1994).
[CrossRef]

Ishizawa, N.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Jain, S.

S. Jain, A. Mansingh, “Thin film layered structure for acousto-optic devices,” J. Phys. D 25, 1116–1121 (1992).
[CrossRef]

Kawasaki, M.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Kimio, M.

M. Kimio, “Recent trend of surface acoustic wave (SAW) filters: high-frequency filter using zinc oxide-sapphire substrate,” Erekutoronikusu 42, 36–38 (1997) (in Japanese).

Kushibiki, J.

J. Kushibiki, N. Chubachi, K. Shibayama, “Improvement of diffraction efficiency in surface-acoustic-optic devices by means of multilayered structures,” Appl. Phys. Lett. 29, 333–335 (1976).
[CrossRef]

Laibowitz, R. B.

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

Lin, I. N.

W. J. Lin, T. Y. Tseng, H. B. Lu, S. L. Tu, S. J. Yang, I. N. Lin, “Growth and ferroelectricity of epitaxial-like BaTiO3 films on single-crystal MgO, SrTiO3, and silicon substrates synthesized by pulse laser deposition,” J. Appl. Phys. 77, 6466–6471 (1995).
[CrossRef]

Lin, W. J.

W. J. Lin, T. Y. Tseng, H. B. Lu, S. L. Tu, S. J. Yang, I. N. Lin, “Growth and ferroelectricity of epitaxial-like BaTiO3 films on single-crystal MgO, SrTiO3, and silicon substrates synthesized by pulse laser deposition,” J. Appl. Phys. 77, 6466–6471 (1995).
[CrossRef]

Lu, H. B.

W. J. Lin, T. Y. Tseng, H. B. Lu, S. L. Tu, S. J. Yang, I. N. Lin, “Growth and ferroelectricity of epitaxial-like BaTiO3 films on single-crystal MgO, SrTiO3, and silicon substrates synthesized by pulse laser deposition,” J. Appl. Phys. 77, 6466–6471 (1995).
[CrossRef]

Mansingh, A.

S. Jain, A. Mansingh, “Thin film layered structure for acousto-optic devices,” J. Phys. D 25, 1116–1121 (1992).
[CrossRef]

Margolin, A. M.

Z. Surowiak, A. M. Margolin, I. N. Zakharchenko, S. V. Biryukov, “The influence of structure on the piezoelectric properties of BaTiO3 and (BaSr)TiO3 thin films with a diffuse phase transition,” Thin Solid Films 176, 227–246 (1989).
[CrossRef]

Matteo, A. M.

A. M. Matteo, V. M. N. Passaro, M. N. Armenise, “High performance guided wave acoustooptic bragg cells in LiNbO3 and GaAs based structures,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 270–279 (1996).
[CrossRef]

Meguro, T.

H. Odagawa, T. Meguro, K. Yamanouchi, “GHz-range low-loss wide band SAW filters using narrow-gap floating electrode type unidirectional transducers,” in Proceedings of the 1996 IEEE International Frequency Control Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 50, pp. 188–193.
[CrossRef]

Mikoshiba, N.

K. Tsubouchi, N. Mikoshiba, “Zero-temperature coefficient SAW devices on A1N epitaxial films,” IEEE Trans. Sonics Ultrason. SU-32, 634–644 (1985).
[CrossRef]

Moon, B. K.

B. K. Moon, H. Ishiwara, “Roles of buffer layers in epitaxial growth of SrTiO3 films on silicon substrates,” Jpn. J. Appl. Phys. 33, 1472–1477 (1994).
[CrossRef]

Nayak, R.

R. Nayak, V. Gupta, K. Sreenivas, “Studies on acousto-optical interaction in SrTiO3/BaTiO3/SrTiO3 epitaxial thin film heterostructures,” J. Phys. D 32, 380–387 (1999).
[CrossRef]

Odagawa, H.

H. Odagawa, T. Meguro, K. Yamanouchi, “GHz-range low-loss wide band SAW filters using narrow-gap floating electrode type unidirectional transducers,” in Proceedings of the 1996 IEEE International Frequency Control Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 50, pp. 188–193.
[CrossRef]

Okazaki, K.

K. Toda, K. Takahashi, K. Okazaki, “Effects of thermally diffused impurities on propagation of elastic waves on the ferroelectric ceramics,” Appl. Phys. Lett. 18, 396–398 (1971).
[CrossRef]

Ooiwa, T.

W. Yatsuda, T. Horishima, T. Eimura, T. Ooiwa, “Miniaturized SAW filters using a flip-chip technique,” in Proceedings of the 1994 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 4, pp. 159–162.

Passaro, V. M. N.

A. M. Matteo, V. M. N. Passaro, M. N. Armenise, “High performance guided wave acoustooptic bragg cells in LiNbO3 and GaAs based structures,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 270–279 (1996).
[CrossRef]

Scott, B. A.

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

Shaw, T. M.

T. M. Shaw, A. Gupta, M. Y. Chern, P. E. Batson, R. B. Laibowitz, B. A. Scott, “Atomic scale oxide superlattices grown by RHEED controlled pulsed laser deposition,” J. Mater. Res. 9, 2566–2573 (1994).
[CrossRef]

Shibayama, K.

J. Kushibiki, N. Chubachi, K. Shibayama, “Improvement of diffraction efficiency in surface-acoustic-optic devices by means of multilayered structures,” Appl. Phys. Lett. 29, 333–335 (1976).
[CrossRef]

Shinichi, S.

S. Shinichi, “Recent trend of surface acoustic wave (SAW) filters: high-frequency filter using diamond substrate,” Erekutoronikusu 42, 33–35 (1997) (in Japanese).

Sreenivas, K.

R. Nayak, V. Gupta, K. Sreenivas, “Studies on acousto-optical interaction in SrTiO3/BaTiO3/SrTiO3 epitaxial thin film heterostructures,” J. Phys. D 32, 380–387 (1999).
[CrossRef]

Surowiak, Z.

Z. Surowiak, A. M. Margolin, I. N. Zakharchenko, S. V. Biryukov, “The influence of structure on the piezoelectric properties of BaTiO3 and (BaSr)TiO3 thin films with a diffuse phase transition,” Thin Solid Films 176, 227–246 (1989).
[CrossRef]

Takahashi, K.

K. Toda, K. Takahashi, K. Okazaki, “Effects of thermally diffused impurities on propagation of elastic waves on the ferroelectric ceramics,” Appl. Phys. Lett. 18, 396–398 (1971).
[CrossRef]

Tanaka, U.

J. Gong, M. Kawasaki, K. Fujito, U. Tanaka, N. Ishizawa, K. Hirai, K. Horiguchi, “Heteroepitaxial growth of c-axis-oriented BaTiO3 thin films with an atomically smooth surface,” Jpn. J. Appl. Phys. 32, L687–L689 (1993).
[CrossRef]

Taylor, G. W.

J. C. Burfoot, G. W. Taylor, Polar Dielectrics and Their Applications (Macmillan, London, 1979).

Thompson, C.

C. Thompson, B. L. Weiss, “Acoustooptic interactions in AlGaAs–GaAs planar multilayer waveguide structures,” IEEE J. Quantum Electron. 33, 1601–1607 (1997).
[CrossRef]

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, in Optical Electronics (Cambridge U. Press, Cambridge, 1991).

Tien, P. K.

Toda, K.

K. Toda, K. Takahashi, K. Okazaki, “Effects of thermally diffused impurities on propagation of elastic waves on the ferroelectric ceramics,” Appl. Phys. Lett. 18, 396–398 (1971).
[CrossRef]

Tsai, C. S.

C. S. Tsai, Guided Wave Acousto-Optics: Interactions, Devices and Applications (Springer-Verlag, Berlin, 1990).
[CrossRef]

Tseng, T. Y.

W. J. Lin, T. Y. Tseng, H. B. Lu, S. L. Tu, S. J. Yang, I. N. Lin, “Growth and ferroelectricity of epitaxial-like BaTiO3 films on single-crystal MgO, SrTiO3, and silicon substrates synthesized by pulse laser deposition,” J. Appl. Phys. 77, 6466–6471 (1995).
[CrossRef]

Tsubouchi, K.

K. Tsubouchi, N. Mikoshiba, “Zero-temperature coefficient SAW devices on A1N epitaxial films,” IEEE Trans. Sonics Ultrason. SU-32, 634–644 (1985).
[CrossRef]

Tu, S. L.

W. J. Lin, T. Y. Tseng, H. B. Lu, S. L. Tu, S. J. Yang, I. N. Lin, “Growth and ferroelectricity of epitaxial-like BaTiO3 films on single-crystal MgO, SrTiO3, and silicon substrates synthesized by pulse laser deposition,” J. Appl. Phys. 77, 6466–6471 (1995).
[CrossRef]

Ulrich, R.

Weiss, B. L.

C. Thompson, B. L. Weiss, “Acoustooptic interactions in AlGaAs–GaAs planar multilayer waveguide structures,” IEEE J. Quantum Electron. 33, 1601–1607 (1997).
[CrossRef]

Wessels, B. W.

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Figures (6)

Fig. 1
Fig. 1

Basic AO device with a symmetric SrTiO3/BaTiO3/SrTiO3 heterostructure.

Fig. 2
Fig. 2

Electromechanical coupling coefficient as a function of SAW frequency for the asymmetric structure with (a) 0.5-, (b) 1.0-, (c) 1.5-, and (d) 2.0-µm BT waveguiding layers and the symmetric structure with (e) 0.5-µm ST overlayer and 1.0-µm BT layer.

Fig. 3
Fig. 3

Diffraction efficiency as a function of SAW frequency for the asymmetric and symmetric waveguide structures.

Fig. 4
Fig. 4

Untuned-transducer conversion efficiency as a function of SAW frequency for the asymmetric and symmetric waveguide structures.

Fig. 5
Fig. 5

Diffraction efficiency as a function of ST overlayer thickness in a symmetric ST/BT(1.0-µm)/ST structure and as a function of BT waveguide layer thickness in an asymmetric BT/ST structure.

Fig. 6
Fig. 6

Relative acoustic attenuation as a function of SAW frequency.

Tables (1)

Tables Icon

Table 1 Comparison of Diffraction Efficiency and Loss for Symmetric and Asymmetric Structures

Equations (8)

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

αr=8.686 ω2ηeff2 ρνr3dB/m,
ρ2Uj/t2=Cijkl2Uk/xixl+ekij2φ/xkxi,
eikl2Uk/xixl-εik2φ/xixk=0, i, j, k, l=1, 2, 3  ,
k2=2×ν-ν/ν,
M2=n6p2/ρν3,
ηf=η02fsinη02f+KΔθL/22η02f+KΔθL/222,
η02f=π2λ02 nm2nn2|Γmnf|2Lcos θm cos θn,
|Γmnf|2=0 UmzUnzp:Saz+r:EpUpzdz2-+ Um2zdz-+ Un2zdz,

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