Abstract

A complete analysis for a uniaxial core–uniaxial cladding step-index fiber is presented. Numerical results of a few lowest-order modes are presented for weakly guiding LiNbO3 single-crystal cladded fibers. We find that the fundamental mode in a uniaxial fiber is not linearly polarized but has significant orthogonal components, even though the fiber is under weak guidance and has no modal birefringence because the wave propagation direction is the crystalline Z axis. We conclude that large anisotropy can cause the relatively large minor field in a uniaxial fiber, but modal birefringence is not necessarily involved. The electrical lines of the fundamental mode for a uniaxial fiber are also considerably different from those of an isotropic one, but eigenvalues and fractions of power in the core are similar.

© 1991 Optical Society of America

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References

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  1. L. G. DeShazer, S. C. Rand, “Fabrication of fiber system for nonlinear optics,” in Infrared Optical Materials and Fibers IV, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.618, 95–102 (1986).
    [CrossRef]
  2. L. G. Deshazer, K. W. Kangas, “Tunable titanium doped sapphire fiber laser,” in Infrared Optical Materials and Fibers VP. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 118–121 (1987).
    [CrossRef]
  3. D. H. Jundt, M. M. Fejer, R. L. Byer, “Characterization of single-crystal sapphire fibers for optical power delivery applications,” Appl. Phys. Lett. 55, 2170–2172 (1989).
    [CrossRef]
  4. M. J. F. Digonnet, C. J. Gaeta, D. O’Meara, H. J. Shaw, “Clad Nd:YAG fibers for laser applications,” IEEE J. Lightwave Technol.LT-5, 642–646 (1987).
    [CrossRef]
  5. S. Sudo, A. Cordova-plaza, R. L. Byer, H. J. Shaw, “MgO:LiNbO3single-crystal fiber with magnesium-ion in-diffused cladding,” Opt. Lett. 12, 938–940 (1987).
    [CrossRef] [PubMed]
  6. F. J. Rosenbaum, “Hybrid modes on anisotropic dielectric rods,” IEEE J. Quantum Electron. QE-1, 367–374 (1965).
    [CrossRef]
  7. D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
    [CrossRef]
  8. N. Kapany, J. Burke, Optical Waveguides (Academic, New York, 1974).
  9. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  10. A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Technol. MTT-30, 790–794 (1982).
    [CrossRef]
  11. 1989 Catalogue of Sumitomo Cement Co. Ltd. (International Business Division, Sumitomo Cement, Chiyoda-ku, Tokyo, Japan).
  12. C. H. Bulmer, “Characteristics of Ti-indiffused waveguides in MgO-doped LiNbO3,” Electron. Lett. 20, 902–904 (1984).
    [CrossRef]
  13. M. P. Varnham, D. N. Payne, J. D. Love, “Fundamental limits to the transmission of linearly polarized light by birefringent optical fibers,” Electron. Lett. 20, 55–56 (1984).
    [CrossRef]
  14. C. L. Chen, “An analysis of high birefringence fibers,” IEEE J. Lightwave Technol. LT-5, 53–69 (1987).
    [CrossRef]

1989

D. H. Jundt, M. M. Fejer, R. L. Byer, “Characterization of single-crystal sapphire fibers for optical power delivery applications,” Appl. Phys. Lett. 55, 2170–2172 (1989).
[CrossRef]

1987

1984

C. H. Bulmer, “Characteristics of Ti-indiffused waveguides in MgO-doped LiNbO3,” Electron. Lett. 20, 902–904 (1984).
[CrossRef]

M. P. Varnham, D. N. Payne, J. D. Love, “Fundamental limits to the transmission of linearly polarized light by birefringent optical fibers,” Electron. Lett. 20, 55–56 (1984).
[CrossRef]

1982

A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Technol. MTT-30, 790–794 (1982).
[CrossRef]

1981

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

1965

F. J. Rosenbaum, “Hybrid modes on anisotropic dielectric rods,” IEEE J. Quantum Electron. QE-1, 367–374 (1965).
[CrossRef]

Bulmer, C. H.

C. H. Bulmer, “Characteristics of Ti-indiffused waveguides in MgO-doped LiNbO3,” Electron. Lett. 20, 902–904 (1984).
[CrossRef]

Burke, J.

N. Kapany, J. Burke, Optical Waveguides (Academic, New York, 1974).

Byer, R. L.

D. H. Jundt, M. M. Fejer, R. L. Byer, “Characterization of single-crystal sapphire fibers for optical power delivery applications,” Appl. Phys. Lett. 55, 2170–2172 (1989).
[CrossRef]

S. Sudo, A. Cordova-plaza, R. L. Byer, H. J. Shaw, “MgO:LiNbO3single-crystal fiber with magnesium-ion in-diffused cladding,” Opt. Lett. 12, 938–940 (1987).
[CrossRef] [PubMed]

Chen, C. L.

C. L. Chen, “An analysis of high birefringence fibers,” IEEE J. Lightwave Technol. LT-5, 53–69 (1987).
[CrossRef]

Cordova-plaza, A.

Deshazer, L. G.

L. G. Deshazer, K. W. Kangas, “Tunable titanium doped sapphire fiber laser,” in Infrared Optical Materials and Fibers VP. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 118–121 (1987).
[CrossRef]

L. G. DeShazer, S. C. Rand, “Fabrication of fiber system for nonlinear optics,” in Infrared Optical Materials and Fibers IV, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.618, 95–102 (1986).
[CrossRef]

Digonnet, M. J. F.

M. J. F. Digonnet, C. J. Gaeta, D. O’Meara, H. J. Shaw, “Clad Nd:YAG fibers for laser applications,” IEEE J. Lightwave Technol.LT-5, 642–646 (1987).
[CrossRef]

Fejer, M. M.

D. H. Jundt, M. M. Fejer, R. L. Byer, “Characterization of single-crystal sapphire fibers for optical power delivery applications,” Appl. Phys. Lett. 55, 2170–2172 (1989).
[CrossRef]

Gaeta, C. J.

M. J. F. Digonnet, C. J. Gaeta, D. O’Meara, H. J. Shaw, “Clad Nd:YAG fibers for laser applications,” IEEE J. Lightwave Technol.LT-5, 642–646 (1987).
[CrossRef]

Jundt, D. H.

D. H. Jundt, M. M. Fejer, R. L. Byer, “Characterization of single-crystal sapphire fibers for optical power delivery applications,” Appl. Phys. Lett. 55, 2170–2172 (1989).
[CrossRef]

Kangas, K. W.

L. G. Deshazer, K. W. Kangas, “Tunable titanium doped sapphire fiber laser,” in Infrared Optical Materials and Fibers VP. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 118–121 (1987).
[CrossRef]

Kapany, N.

N. Kapany, J. Burke, Optical Waveguides (Academic, New York, 1974).

Love, J. D.

M. P. Varnham, D. N. Payne, J. D. Love, “Fundamental limits to the transmission of linearly polarized light by birefringent optical fibers,” Electron. Lett. 20, 55–56 (1984).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

O’Meara, D.

M. J. F. Digonnet, C. J. Gaeta, D. O’Meara, H. J. Shaw, “Clad Nd:YAG fibers for laser applications,” IEEE J. Lightwave Technol.LT-5, 642–646 (1987).
[CrossRef]

Paul, D. K.

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

Payne, D. N.

M. P. Varnham, D. N. Payne, J. D. Love, “Fundamental limits to the transmission of linearly polarized light by birefringent optical fibers,” Electron. Lett. 20, 55–56 (1984).
[CrossRef]

Rand, S. C.

L. G. DeShazer, S. C. Rand, “Fabrication of fiber system for nonlinear optics,” in Infrared Optical Materials and Fibers IV, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.618, 95–102 (1986).
[CrossRef]

Rosenbaum, F. J.

F. J. Rosenbaum, “Hybrid modes on anisotropic dielectric rods,” IEEE J. Quantum Electron. QE-1, 367–374 (1965).
[CrossRef]

Shaw, H. J.

S. Sudo, A. Cordova-plaza, R. L. Byer, H. J. Shaw, “MgO:LiNbO3single-crystal fiber with magnesium-ion in-diffused cladding,” Opt. Lett. 12, 938–940 (1987).
[CrossRef] [PubMed]

M. J. F. Digonnet, C. J. Gaeta, D. O’Meara, H. J. Shaw, “Clad Nd:YAG fibers for laser applications,” IEEE J. Lightwave Technol.LT-5, 642–646 (1987).
[CrossRef]

Shevgaonkar, R. K.

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Sudo, S.

Tonning, A.

A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Technol. MTT-30, 790–794 (1982).
[CrossRef]

Varnham, M. P.

M. P. Varnham, D. N. Payne, J. D. Love, “Fundamental limits to the transmission of linearly polarized light by birefringent optical fibers,” Electron. Lett. 20, 55–56 (1984).
[CrossRef]

Appl. Phys. Lett.

D. H. Jundt, M. M. Fejer, R. L. Byer, “Characterization of single-crystal sapphire fibers for optical power delivery applications,” Appl. Phys. Lett. 55, 2170–2172 (1989).
[CrossRef]

Electron. Lett.

C. H. Bulmer, “Characteristics of Ti-indiffused waveguides in MgO-doped LiNbO3,” Electron. Lett. 20, 902–904 (1984).
[CrossRef]

M. P. Varnham, D. N. Payne, J. D. Love, “Fundamental limits to the transmission of linearly polarized light by birefringent optical fibers,” Electron. Lett. 20, 55–56 (1984).
[CrossRef]

IEEE J. Lightwave Technol.

C. L. Chen, “An analysis of high birefringence fibers,” IEEE J. Lightwave Technol. LT-5, 53–69 (1987).
[CrossRef]

IEEE J. Quantum Electron.

F. J. Rosenbaum, “Hybrid modes on anisotropic dielectric rods,” IEEE J. Quantum Electron. QE-1, 367–374 (1965).
[CrossRef]

IEEE Trans. Microwave Theory Technol.

A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Technol. MTT-30, 790–794 (1982).
[CrossRef]

Opt. Lett.

Radio Sci.

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

Other

N. Kapany, J. Burke, Optical Waveguides (Academic, New York, 1974).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

L. G. DeShazer, S. C. Rand, “Fabrication of fiber system for nonlinear optics,” in Infrared Optical Materials and Fibers IV, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.618, 95–102 (1986).
[CrossRef]

L. G. Deshazer, K. W. Kangas, “Tunable titanium doped sapphire fiber laser,” in Infrared Optical Materials and Fibers VP. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 118–121 (1987).
[CrossRef]

1989 Catalogue of Sumitomo Cement Co. Ltd. (International Business Division, Sumitomo Cement, Chiyoda-ku, Tokyo, Japan).

M. J. F. Digonnet, C. J. Gaeta, D. O’Meara, H. J. Shaw, “Clad Nd:YAG fibers for laser applications,” IEEE J. Lightwave Technol.LT-5, 642–646 (1987).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Eigenvalues of a few lowest modes of a uniaxial step-index fiber with parameters given in Table 6.

Fig. 2
Fig. 2

Electrical fields of HE11 mode at V = 2 for a uniaxial fiber with parameters given in Table 6: (a) Ex, (b) − Ey. [20 log α = −41.4, where α = (Eminor)max/(Emain)max.]

Fig. 3
Fig. 3

Electrical fields of HE11 mode at V = 2 for an isotropic fiber with nco = 2.2866, ncl = 2.2815: (a) Ex, (b) −Ey. (20 log α = −60.7.)

Fig. 4
Fig. 4

Electrical lines of HE11 in one quarter of a uniaxial fiber with parameters given in Table 6. (V = 2; Ey is enlarged 10 times.)

Fig. 5
Fig. 5

Electrical lines of HE11 mode in one quarter of an isotropic fiber with nco = 2.2866 and ncl = 2.2815. (V = 2; Ey is enlarged 100 times.)

Fig. 6
Fig. 6

Fraction of modal power in the core: solid curves, uniaxial fibers with parameters given in Table 6; dotted curve, isotropic fiber with nco = 2.2866 and ncl = 2.2815; dashed curve, isotropic fiber with nco = 2.5 and ncl = 1.5.

Tables (6)

Tables Icon

Table 1 Longitudinal Field Equations

Tables Icon

Table 2 Field Components of HEνm and EHνm Modes

Tables Icon

Table 3 Field Components of TE0m Modesa

Tables Icon

Table 4 Field Components of TM0m Modesa

Tables Icon

Table 5 Modal Power of TE0m and TM0m Modes

Tables Icon

Table 6 Refractive Indices of Pure and 5% MgO-Doped LiNbO3 Bulk Crystals

Equations (35)

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[ J ν ( U ) U J ν ( U ) + K ν ( W ) W K ν ( W ) ] [ κ co J ν ( κ co U ) U J ν ( κ co U ) + κ cl W ( n cl t n co t ) 2 K ν ( κ cl W ) K ν ( κ cl W ) ] = ( ν β k n co t ) 2 ( V U W ) 4 .
b 1 = 1 2 U [ J ν 1 ( U ) J ν ( U ) J ν + 1 ( U ) J ν ( U ) ] , b 2 = 1 2 W [ K ν 1 ( W ) K ν ( W ) + K ν + 1 ( W ) K ν ( W ) ] , F 2 = ( V U W ) 2 ν b 1 + b 2 , Δ = ( n co t ) 2 ( n cl t ) 2 2 ( n co t ) 2 , R = r a ,
η = P co P co + P cl = 0 1 ( e r h ϕ e ϕ h r ) R d R 0 1 ( e r h ϕ e ϕ h r ) R d R + 0 ( e r h ϕ e ϕ h r ) R d R .
f ν ( ϕ ) { κ co [ J ν 1 ( κ co U R ) J ν + 1 ( κ co U R ) ] 2 J ν ( κ co U ) ν F 2 U R J ν ( U R ) J ν ( U ) }
f ν ( ϕ ) ( U W ) { κ cl [ K ν 1 ( κ cl W R ) + K ν + 1 ( κ cl W R ) ] 2 K ν ( κ cl W ) + ν F 2 W R K ν ( W R ) K ν ( W ) }
g ν ( ϕ ) { F 2 [ J ν 1 ( U R ) J ν + 1 ( U R ) ] 2 J ν ( U ) + ν J ν ( κ co U R ) U R J ν ( κ co U ) }
g ν ( ϕ ) ( U W ) { F 2 [ K ν 1 ( W R ) + K ν + 1 ( W R ) ] 2 K ν ( W ) + ν K ν ( κ cl W R ) W R K ν ( κ cl E ) }
i U α β J ν ( κ co U R ) J ν ( κ co U ) f ν ( ϕ )
i U a β K ν ( κ cl W R ) K ν ( κ cl W ) f ν ( ϕ )
( 0 μ 0 ) 1 / 2 g ν ( ϕ ) k ( n co t ) 2 β { β 2 F 2 ( k n co t ) 2 [ J ν 1 ( U R ) J ν + 1 ( U R ) ] 2 J ν ( U ) ν J ν ( κ co U R ) U R J ν ( κ co U ) }
( 0 μ 0 ) 1 / 2 ( U W ) g ν ( ϕ ) k ( n co t ) 2 β { β 2 F 2 ( k n co t ) 2 [ K ν 1 ( W R ) + K ν + 1 ( W R ) ] 2 K ν ( W ) + ( n cl t n co t ) 2 ν K ν ( κ cl W R ) W R K ν ( κ cl W ) }
( 0 μ 0 ) 1 / 2 f ν ( ϕ ) k ( n co t ) 2 β { κ co [ J ν 1 ( κ co U R ) J ν + 1 ( κ co U R ) ] 2 J ν ( κ co U ) ( β 2 F 2 ( k n co t ) 2 ) ν J ν ( U R ) U R J ν ( U ) }
( 0 μ 0 ) 1 / 2 ( U W ) f ν ( ϕ ) k ( n co t ) 2 β { κ cl ( n cl t n co t ) 2 [ K ν 1 ( κ cl W R ) + K ν + 1 ( κ cl W R ) ] 2 K ν ( κ cl W ) + [ β 2 F 2 ( k n co t ) 2 ] ν K ν ( W R ) W R K ν ( W ) }
i ( 0 μ 0 ) 1 / 2 U F 2 a k J ν ( U R ) J ν ( U ) g ν ( ϕ )
i ( 0 μ 0 ) 1 / 2 U F 2 a k k ν ( W R ) K ν ( W ) g ν ( ϕ )
J 1 ( U R ) J 1 ( U )
K 1 ( W R ) K 1 ( W )
β k ( 0 μ 0 ) 1 / 2 J 1 ( U R ) J 1 ( U )
β k ( 0 μ 0 ) 1 / 2 K 1 ( W R ) K 1 ( W )
i ( 0 μ 0 ) 1 / 2 U J 0 ( U R ) k a J 1 ( U )
i ( 0 μ 0 ) 1 / 2 W K 0 ( W R ) k a K 1 ( W )
J 1 ( κ co U R ) J 1 ( κ co U )
( n co t n cl t ) 2 K 1 ( κ cl W R ) K 1 ( κ cl W )
i U a β κ co J 0 ( κ co U R ) J 1 ( κ co U )
i W a β κ cl ( n co t n cl t ) 2 K 0 ( κ cl W R ) K 1 ( κ cl W )
( 0 μ 0 ) 1 / 2 k ( n co t ) 2 β J 1 ( κ co U R ) J 1 ( κ co U )
( 0 μ 0 ) 1 / 2 k ( n co t ) 2 β K 1 ( κ cl W R ) K 1 ( κ cl W )
π 2 a 2 ( 0 μ 0 ) 1 / 2 β k [ 1 J 0 ( U ) J 2 ( U ) J 1 2 ( U ) ]
π 2 a 2 ( 0 μ 0 ) 1 / 2 k ( n co t ) 2 β [ 1 J 0 ( κ co U ) J 2 ( κ co U ) J 1 2 ( κ co U ) ]
π 2 a 2 ( 0 μ 0 ) 1 / 2 β k [ 1 K 0 ( W ) K 2 ( W ) K 1 2 ( W ) ]
π 2 a 2 ( 0 μ 0 ) 1 / 2 k ( n co t ) 2 β ( 1 2 Δ ) [ 1 K 0 ( κ cl W ) K 2 ( κ cl W ) K 1 2 ( κ cl W ) ]
P co P co + P cl
P co P co + P cl
δ = ( n 0 n e ) n 0
D max δ max = 0.119

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