Abstract

The mode coupling in double-sided tilted gratings between the LP01 and LP11 modes is analyzed for three different excitation conditions. It is found that a number of modes, including various LP11q modes with different polarization states and TE01, TM01, and HE21 modes, can be formed through the mode conversion process in the gratings and with a linear combination of four different LP11 modes. Efficient single-sided and double-sided tilted gratings are demonstrated. The tilted gratings presented exhibit LP01-to-LP11 mode coupling over a spectral width as wide as 160 nm. The transmission spectra of the tilted fiber gratings agree well with the spectrum predicted by the coupled-mode analysis.

© 2001 Optical Society of America

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References

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  1. Kyung S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photonics Technol. Lett. 11, 1286–1288 (1999).
    [CrossRef]
  2. T. Erdogan, J. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  3. K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fiber using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  4. H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
    [CrossRef]
  5. F. Ouellette, “Photorefractive intermodal exchangers in optical fiber,” 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 Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 348–349.
  7. Kyung S. Lee, T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394–1404 (2000).
    [CrossRef]
  8. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]
  9. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), pp. 7–88.
  10. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  11. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991).
  12. A. Yariv, “Frustration of Bragg reflection by cooperative dual-mode interference: a new mode of optical propagation,” Opt. Lett. 23, 1835–1836 (1998).
    [CrossRef]
  13. S. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
    [CrossRef]

2000 (1)

1999 (1)

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

1998 (1)

1997 (1)

1996 (1)

1991 (1)

F. Ouellette, “Photorefractive intermodal exchangers in optical fiber,” 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 fiber using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1989 (1)

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

1975 (1)

Andrejco, M.

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

Bilodo, F.

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

Case, S.

Erdogan, T.

Hill, K.

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

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), pp. 7–88.

Lee, Kyung S.

Kyung S. Lee, T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394–1404 (2000).
[CrossRef]

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

Malo, B.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fiber 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).

Ouellette, F.

F. Ouellette, “Photorefractive intermodal exchangers in optical fiber,” 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]

Pedrazzani, J. R.

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

Sipe, J.

Skinner, I.

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

Strasser, T.

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

Vineberg, K.

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

Yariv, A.

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 fiber 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 fiber,” IEEE J. Quantum Electron. 27, 796–803 (1991).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

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

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Other (4)

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

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), pp. 7–88.

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

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

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Figures (11)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Coupled power of the LP01 and LP11q modes as a function of γz in a double-sided tilted grating when an LP11c mode is incident.

Fig. 3
Fig. 3

Coupled power of the LP01 and LP11q modes as a function of γz in a double-sided tilted grating when an LP01 mode is incident.

Fig. 4
Fig. 4

Mode intensity patterns after the double-sided tilted gratings with different ϕ0 when an LP01 mode is incident (γz=π/2).

Fig. 5
Fig. 5

Coupled power of the LP01 and LP11q modes as a function of γz in a double-sided tilted grating with ϕ0=0 when the LP11c mode and LP11s modes are equally incident.

Fig. 6
Fig. 6

Various mode patterns formed by different combinations of four LP11 modes: LP11cx, LP11sx, LP11cy, and LP11sy.

Fig. 7
Fig. 7

Experimental setup of an interferometer system for writing transmissive tilted gratings (MF, mode filter; L1 and L2, cylindrical lenses; M1–M3, mirrors; BS, beam splitter; WLS, white-light source; OSA, optical spectrum analyzer).

Fig. 8
Fig. 8

Mode coupler configurations and mode intensity patterns: (a) x-tilted grating, (b) y-tilted grating, (c) double-sided tilted grating.

Fig. 9
Fig. 9

Transmission spectra of a single-sided tilted grating as the photosensitive fiber section between mode filters has been exposed for ∼1 min (short-dashed curve), ∼2 min (long-dashed curve), ∼2.5 min (dotted curves), and ∼4 min (jagged solid curve), and the transmission spectrum computed for a single-sided tilted grating (Gaussian apodized, grating length 1.2 mm, FWHM=0.6 mm) by using the coupled-mode analysis (parabo-loidal solid curve).

Fig. 10
Fig. 10

Transmission spectra of typical tilted gratings: (a) single-sided tilted grating and (b) double-sided tilted grating.

Fig. 11
Fig. 11

Grating periods required for the mode coupling from the LP01 core mode to the LP11 core mode in the wavelength range between 0.55 and 0.9 µm.

Equations (68)

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κνμ(z)=(ω/4)02πdϕ0r dγ Δε(r, ϕ, z)×Eν(r, ϕ)Eμ*(r, ϕ),
Elm, q=Elmelm, q exp[i(βz-ωt)]=jˆElmJl(μlmr)cos lϕsin lϕ×exp[i(βz-ωt)](q=c, s),
Elm=4Z0blmelπna2|Jl-1(ulma)Jl+1(ulma)|1/2,
el=2for(l=0)1for(l0).
Δϵ(x, y, z, θ)=2ϵ0n1Δn(x, y, z, θ),
Δn(x, y, z, θ)=n1σ(z)[1+v cos(2Kgz)],
z=z cos θ-x sin θforx-tiltedgratingz cos θ-y sin θfory-tiltedgrating.
2K=2Kg cos θ,
E(r, ϕ, z, t)=A01(z)E01(r)exp(iβ01z)+q=c,sA11q(z)E11q(r, ϕ)exp(iβ11qz)×exp(-iωt),
κνμ(z)=gμν+ exp(2iKgz cos θ)+gμν- exp(-2iKgz cos θ),
gμν±=ω4ϵ0n12σ(z)v02πdϕ0ar dr exp(2iKgr cos ϕ sin θ)EνEμ*forx-tiltedgratingω4ϵ0n12σ(z)v02πdϕ0ar dr exp(2iKgr sin ϕ sin θ)EνEμ*fory-tiltedgrating.
g11c01±x=g11s01±y=Ag±0ar dr J1(2rKg sin θ)J1(u11r)J0(u0r),
g11s01±x=g11c01±y=0,
Δn(x, y, z, θ)=n1σ(z)v[cos(2Kgz)+cos(2Kgz+ϕ0)],
κ1101(z)=g11c01+x exp(2iKzz cos θ)+g11c01-x exp(-2iKgz cos θ)+g11s01+y exp(2iKgz cos θ+iϕ0)+g11s01-y exp(-2iKgz cos θ-iϕ0).
dA01dz=iA11cg11c01+x exp(-2iδz)+iA11sg11s01+y exp(-2iδz+iϕ0),
dA11cdz=iA01g0111c-x exp(2iδz),
dA11sdz=±iA01g0111s-y exp(2iδz-iϕ0),
2δβ01β11s-2Kg cos θ=β01β11c-2Kg cos θ.
dRdz=iδR+iGcSc+iGsSs,
dScdz=-iδSc+iGc*R,
dSsdz=-iδSs+iGs*R,
Gc=g11c01+x,
Gs=g11s01+y exp(iϕ0).
R(z)=(iGc/γ)sin(γz),
Sc(z)=|Gs|2γ2-δ2 exp(-iδz)+|Gc|2γ2-δ2 cos(γz)-i|Gc|2δ(γ2-δ2)γ sin(γz),
Ss(z)=-Gs*Gcγ2-δ2 exp(-jδz)+Gs*Gcγ2-δ2 cos(γz)-iGs*Gcδ(γ2-δ2)γ sin(γz),
γ2=δ2+|Gc|2+|Gs|2.
|R(z)|2=sin2(γz)/2,
|Sc(z)|2=cos4(γz/2),
|Ss(z)|2=sin4(γz/2),
R(z)=cos(γz)+(iδ/γ)sin(γz),
Sq(z)=(iGq*/γ)sin(γz),q=c, s.
|R(z)|2=cos2(γz)+(δ/γ)2 sin2(γz),
|Sq(z)|2=(|Gq|/γ)2 sin2(γz),
|R(z)|2+|Sc(z)|2+|Sc(z)|2=1.
E(r, ϕ)=(i/γ)[Gc*J1(u11r)cos ϕ+Gs*J1(u11r)sin ϕ]×exp(iβ11z-ωt),
I(r, ϕ)[1+sin(2ϕ)cos ϕ0]J12(u11r)/2,
I(r, ϕ)cos2(ϕ-π/4)J12(u11r)forϕ0=2πm(m=integer)J12(u11r)/2forϕ0=(m±1/2)πsin2(ϕ-π/4)J12(u11r)forϕ0=(2m+1)π,
R(z)=i(Gc+Gs)2γ sin(γz),
Ss(z)=|Gs|2-Gc*Gs2(γ2-δ2) exp(-iδz)+|Gc|2+Gc*Gs2(γ2-δ2) cos(γz)-i(|Gc|2+Gc*Gs)δ2(γ2-δ2)γ sin(γz),
Ss(z)=|Gc|2-Gs*Gc2(γ2-δ2) exp(-iδz)+|Gs|2+Gs*Gc2(γ2-δ2) cos(γz)-i(|Gs|2+Gs*Gc)δ2(γ2-δ2)γ sin(γz).
|R(z)|2=(1+cos ϕ0)sin2(γz)/2,
|Sc(z)|2={1+cos2(γz)-[1-cos2(γz)]cos ϕ0}/4,
R(z)=R(0)cos(γz)+(i/γ)[Sc(0)Gc+Ss(0)Gs]sin(γz),
Sc(z)=[Sc(0)-[1-cos(γz)]Gc*[Sc(0)Gc+Ss(0)Gs]/γ2]+i[R(0)Gc*/γ]sin(γz),
Ss(z)={Ss(0)-[1-cos(γz)]Gs*[Sc(0)Gc+Ss(0)Gs]/γ2}+i[R(0)Gs*/γ]sin(γz),
Et=acxe11cx+asxe11sx+acye11cy+asye11sy,
Et=(xˆ+yˆ)J1(u11r)(cos ϕ+sin ϕ)/2,
I1cos2(ϕ-π/4)J12(u11r);
Et=(xˆ+yˆ)J1(u11r)(cos ϕ-sin ϕ)/2,
Itsin2(ϕ-π/4)J12(u11r);
Et=(xˆ-yˆ)J1(u11r)(cos ϕ-sin ϕ)/2,
Itsin2(ϕ-π/4)J12(u11r);
Et=xˆJ1(u11r)(cos ϕ+sin ϕ)/2,
Itcos2(ϕ-π/4)J12(u11r);
Et=yˆJ1(u11r)(cos ϕ-sin ϕ)/2,
Itsin2(ϕ-π/4)J12(u11r);
Et=(xˆ+yˆ)J1(u11r)(cos ϕ)/2,
ItJ12(u11r)cos2 ϕ;
Et=(xˆ cos ϕ+yˆ sin ϕ)J1(u11r)/2,
ItJ12(u11r)/2;
Et=(-xˆ cos ϕ+yˆ sin ϕ)J1(u11r)/2,
ItJ12(u11r)/2;
Et=(xˆ sin ϕ+yˆ cos ϕ)J1(u11r)/2,
ItJ12(u11r)/2;
Et=(xˆ sin ϕ-yˆ cos ϕ)J1(u11r)/2,
ItJ12(u11r)/2.

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