Abstract

Whereas core-mode reflection and core-mode-to-radiation-mode coupling in tilted fiber Bragg gratings is well understood, as is coupling between a core mode and higher-order core and cladding modes in untilted gratings, here we analyze in detail the coupling among core modes and cladding modes in reflective and transmissive tilted fiber gratings. We show that strong coupling between an LP01 core mode and the exact (1m) cladding modes occurs in a transmissive tilted grating for nearly any tilt angle except angles close to 90°, whereas the LP01-to-(lm) cladding mode coupling (l ≠ 1) is appreciable only for tilt angles just below 90° (∼88°). In a reflective grating, strong coupling between the LP01 core mode and the exact (1m) cladding modes occurs only for angles less than ∼5°, whereas coupling to (1m) cladding modes for m > 1 occurs only for angles greater than ∼5°. Coupling among bound core modes exhibits a similar behavior, except that in general the coupling is stronger. Experimentally we show coupling to both higher-order bound core modes and cladding modes in a transmissive tilted grating at visible and near-infrared wavelengths.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  2. K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  3. H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
    [CrossRef]
  4. K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photon. Technol. Lett. 11, 1286–1288 (1999).
    [CrossRef]
  5. F. Ouellette, “Photorefractive intermodal exchangers in optical fibers,” IEEE J. Quantum Electron. 27, 796–803 (1991).
    [CrossRef]
  6. T. Strasser, J. R. Pedrazzani, M. Andrejco, “Reflective-mode conversion with UV-induced phase gratings in two-mode fiber,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper FB3, pp. 348–349.
  7. A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
    [CrossRef]
  8. H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.
  9. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]
  10. T. Erdogan, J. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  11. L. Dong, B. Ortega, L. Reekie, “Coupling characteristics of cladding modes in tilted optical fiber Bragg gratings,” Appl. Opt. 37, 5099–5105 (1998).
    [CrossRef]
  12. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), p. 85.
  13. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  14. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991).
  15. C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, Oxford, 1992).
  16. T. Erdogan, D. Stegall, “Impact of dispersion on the bandwidth of long-period fiber-grating filters,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 280–282.

1999 (1)

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photon. Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (2)

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

T. Erdogan, J. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

1991 (1)

F. Ouellette, “Photorefractive intermodal exchangers in optical fibers,” IEEE J. Quantum Electron. 27, 796–803 (1991).
[CrossRef]

1990 (1)

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1989 (2)

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

Andrejco, M.

T. Strasser, J. R. Pedrazzani, M. Andrejco, “Reflective-mode conversion with UV-induced phase gratings in two-mode fiber,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper FB3, pp. 348–349.

Bhatia, V.

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

Bilodo, F.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Bucholtz, F.

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

Dong, L.

Erdogan, T.

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photon. Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

T. Erdogan, J. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

T. Erdogan, D. Stegall, “Impact of dispersion on the bandwidth of long-period fiber-grating filters,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 280–282.

Ewing, K.

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

Glenn, W. H.

Hill, K.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Johnson, D.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Judkins, J.

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

Kersey, A.

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

Kim, B.

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), p. 85.

Lee, K. S.

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photon. Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

Lemiare, P.

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

Malo, B.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991).

Meltz, G.

Morey, W. W.

Ortega, B.

Ouellette, F.

F. Ouellette, “Photorefractive intermodal exchangers in optical fibers,” IEEE J. Quantum Electron. 27, 796–803 (1991).
[CrossRef]

Park, H.

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

Patrick, H.

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

Pedrazzani, J. R.

T. Strasser, J. R. Pedrazzani, M. Andrejco, “Reflective-mode conversion with UV-induced phase gratings in two-mode fiber,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper FB3, pp. 348–349.

Reekie, L.

Sipe, J.

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

T. Erdogan, J. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

Skinner, I.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Stegall, D.

T. Erdogan, D. Stegall, “Impact of dispersion on the bandwidth of long-period fiber-grating filters,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 280–282.

Strasser, T.

T. Strasser, J. R. Pedrazzani, M. Andrejco, “Reflective-mode conversion with UV-induced phase gratings in two-mode fiber,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper FB3, pp. 348–349.

Tsao, C.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, Oxford, 1992).

Vengsarkar, A.

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

Vineberg, K.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Appl. Opt. (1)

Electron. Lett. (2)

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

F. Ouellette, “Photorefractive intermodal exchangers in optical fibers,” IEEE J. Quantum Electron. 27, 796–803 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photon. Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

J. Lightwave Technol. (1)

A. Vengsarkar, P. Lemiare, J. Judkins, V. Bhatia, T. Erdogan, J. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–64 (1996).
[CrossRef]

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

Opt. Lett. (1)

Other (7)

H. Patrick, A. Kersey, F. Bucholtz, K. Ewing, J. Judkins, A. Vengsarkar, “Chemical sensor based on long-period fiber grating response to index of refraction,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 420–421.

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), p. 85.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991).

C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, Oxford, 1992).

T. Erdogan, D. Stegall, “Impact of dispersion on the bandwidth of long-period fiber-grating filters,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 280–282.

T. Strasser, J. R. Pedrazzani, M. Andrejco, “Reflective-mode conversion with UV-induced phase gratings in two-mode fiber,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper FB3, pp. 348–349.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Configuration of a single-sided tilted grating: (a) for an x-tilted grating, (b) for a y-tilted grating.

Fig. 2
Fig. 2

Coupling coefficients for cladding modes with (a) l = 1, (b) l = 2, (c) l = 3 in a transmissive tilted grating (θ = 88°).

Fig. 3
Fig. 3

Coupling coefficients as a function of tilt angles for cladding modes with (a) l = 1, (b) l = 2, (c) l = 3 in a transmissive tilted grating.

Fig. 4
Fig. 4

Coupling coefficients for cladding modes with (a) l = 1, θ = 0°, (b) l = 2, θ = 5°, (c) l = 3, θ = 7.5° in a reflective tilted grating.

Fig. 5
Fig. 5

Coupling coefficients as a function of tilt angles for cladding modes with (a) l = 1, (b) l = 2, (c) l = 3 in a reflective tilted grating.

Fig. 6
Fig. 6

Coupling coefficients for two core modes in a transmissive tilted grating (θ = 88°, a = 25 µm).

Fig. 7
Fig. 7

Coupling coefficients of LP1m and LP2m core modes in a transmissive tilted grating as a function of tilt angles for two cases: (a) a = 25 µm, λ = 1.55 µm, Δ = 0.005 and (b) a = 2.5 µm, λ = 0.633 µm, Δ = 0.006.

Fig. 8
Fig. 8

Coupling coefficients of LP1m core modes in a reflective tilted grating as a function of tilt angles (a = 25 µm, λ = 1.55 µm).

Fig. 9
Fig. 9

Transmission spectrum of a transmissive tilted grating. Near 630 nm a LP01-to-LP11-core-mode coupling occurs. The LP11 core mode pattern was measured at 632.8 nm. Many cladding mode couplings appear above 700 nm.

Fig. 10
Fig. 10

Transmission spectrum of a transmissive tilted grating. An LP01-to-LP11-core-mode coupling appears near 600 nm, and many LP01-to-cladding-mode couplings occur above 670 nm. A 2m cladding-mode pattern was measured at 773 nm.

Equations (32)

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

κνμz=ω/402πdϕ 0 rdrΔεr, ϕ, z×Eνr, ϕEμ*r, ϕ,
Elm,jco=ElmcoJlulmrcos lϕsin lϕexpiβz-ωtj=x, y,
Elmco=4Z0blmelπna2|Jl-1ulmaJl+1ulma|1/2,
el=2, l=01, l0.
Elm,rcl=iElmclulm2Jl+1ulmr+Jl-1ulmr-σζ0ln12Jl+1ulmr-Jl-1ulmr×expilϕexpiβz-ωt  ra,
Elm,ϕcl=iElmclulm2Jl+1ulmr-Jl-1ulmr-σζ0ln12Jl+1ulmr+Jl-1ulmr×expilϕexpiβz-ωt  ra,
Δεx, y, z, θ=2ε0n1Δn1x, y, z, θ,
Δn1x, y, z, θ=n1σz1+νcos2Kgz,
z=z cos θ-x sin θ, for x-tilted gratingz cos θ-y sin θ, for y-tilted grating.
2K=2Kg cos θ,
dAμdz=iν Aνκνμ expiβν-βμz+i ν Bνκνμ exp-iβν+βμz,
dBμdz=-i ν Aνκνμ expiβν+βμz-i ν Bνκνμ exp-iβν-βμz,
κνμz=gνμ+ exp2iKgz cos θ+gνμ-×exp-2iKgz cos θ+fνμ,
gνμ±=ω4 ε0n12σzν02πdϕ 0a rdr exp2iKgr cos ϕ sin θEνEμ*, for x-tilted grating,ω4 ε0n12σzν02πdϕ 0a rdr exp2iKgr sin ϕ sin θEνEμ*, for y-tilted grating,
fνμ=ω2 ε0n12σz02πdϕ 0a rdrEνEμ*.
dA01dz=if01-01A01+i l,m glm-01j+Almj×exp-2iδ01-lmj-tz  j=co, cl,
l,mdAlmjdz=iAlmjflm-lmj+ig01-lmj-A01×exp2iδ01-lmj-tz  j=co, cl,
2δ01-lmj-t=β01-βlmj-2Kg cos θ,
dA01dz=if01-01A01+ig01-01+B01 exp-2iδ01-01rz+il,m Blmjglm-01j+ exp-2iδ01-lmj-rzj=co, cl,
dB01dz=-iA01g01-01- exp2iδ01-01rz-iB01f01-01,
l,mdBlmjdz=-ig01-lmj-A01 exp2iδ01-lmj-rz-iBlmjflm-lmj  j=co, cl,
2δ01-lmj-r=β01+βlmj-2Kg cos θ,
δ01-01r=β01-Kg cos θ,
glm-01cl±x=Agcl±x0a rdrJl-12rKg sin θ×Jl-1ulmrJ0u0r, for x-tilted grating,
glm-01cl±y=Agcl±y0a rdrJl-12rKg sin θ×Jl-1ulmrJ0u0r, for y-tilted grating,
Agcl±x=il-1π2n1Δnulm×1+σζ0/n12lE01coElmcl/2λZ0,
Agcl±y=±1l-1π2n1Δnulm×1+σζ0/n12lE01coElmcl/2λZ0,
ulm22π/λ2n12-neff lm2.
glm-01co±x=Agco±0a rdrJl2rKg sin θJlulmrJ0u01rfor cos lϕ0for sin lϕ,
Agco±=ilπ2n1ΔnE0coElmco/2λZ0,
glm-01co±y=Agco± coslπ/20a rdrJl2rKg sin θJlulmrJ0u01rfor cos lϕAgco± sinlπ/20a rdrJl2rKg sin θJlulmrJ0u01rfor sin lϕ,
glm-01co±y=0for cos lϕAgco±il-10a rdrJl2rKg sin θJlulmrJ0u01rfor sin lϕ.

Metrics