Abstract

Three different grating tilting formulas for predicting the optimum grating tilt angle for strong mode coupling in planar waveguide gratings are derived. The optimum tilt angles obtained by the ray-optics approach deviate ∼1° for transmissive mode coupling and ∼1.5° for reflective mode coupling from those computed by the coupled-mode approach. The coupled-mode analysis and ray-optics analysis of the tilted planar waveguide gratings show that the transmissive planar waveguide gratings should be tilted more than 84° for strong TE0-to-TEμ mode coupling near 1550 nm.

© 2000 Optical Society of America

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References

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  1. T. Erdogan, J. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  2. C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
    [CrossRef]
  3. K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photon. Technol. Lett. 11, 1286–1288 (1999).
    [CrossRef]
  4. K. S. Lee, T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394–1404 (2000).
    [CrossRef]
  5. K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in optical fiber,” J. Opt. Soc. Am. A, submitted for publication (A7734).
  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. L. Dong, B. Ortega, L. Reekie, “Coupling characteristics of cladding modes in tilted optical fiber Bragg gratings,” Appl. Opt. 37, 5099–5105 (1998).
    [CrossRef]
  8. K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
    [CrossRef]
  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. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
    [CrossRef]

2000 (1)

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 (2)

L. Dong, B. Ortega, L. Reekie, “Coupling characteristics of cladding modes in tilted optical fiber Bragg gratings,” Appl. Opt. 37, 5099–5105 (1998).
[CrossRef]

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

1997 (1)

1996 (1)

1979 (1)

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

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.

Dong, L.

Erdogan, T.

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

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]

K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in optical fiber,” J. Opt. Soc. Am. A, submitted for publication (A7734).

Haggans, C.

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

Kogelnik, H.

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

Lee, K. S.

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

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

K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in optical fiber,” J. Opt. Soc. Am. A, submitted for publication (A7734).

Li, Y.

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

Ortega, B.

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.

Saito, S.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Sakaki, H.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Singh, H.

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

Sipe, J.

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.

Varner, W.

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Zippin, M.

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

C. Haggans, H. Singh, W. Varner, Y. Li, M. Zippin, “Narrow-band rejection using tilted photoinduced gratings in single-mode fibers,” IEEE Photon. Technol. Lett. 10, 690–692 (1998).
[CrossRef]

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

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

Other (3)

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

K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in optical fiber,” J. Opt. Soc. Am. A, submitted for publication (A7734).

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.

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Figures (5)

Fig. 1
Fig. 1

Ray picture for mode coupling between an incident TE0 mode and a TEμ mode diffracted at ϕμ in a tilted planar waveguide grating.

Fig. 2
Fig. 2

Coupling coefficients as a function of tilt angle for TEμ modes in (a) a transmissive planar waveguide grating and (b) a reflective planar waveguide grating.

Fig. 3
Fig. 3

Grating tilt angles for maximum TE mode coupling at different wavelengths in a transmissive planar waveguide grating.

Fig. 4
Fig. 4

Grating tilt angles for maximum TE mode coupling at different wavelengths in a reflective planar waveguide grating.

Fig. 5
Fig. 5

Ray picture for reflective mode coupling between an incident TE0 mode and a TEμ mode diffracted at ϕμ in a planar waveguide grating.

Tables (2)

Tables Icon

Table 1 Comparison of Optimum Grating Tilt Angles θTEμ Calculated for Transmissive Mode Coupling

Tables Icon

Table 2 Comparison of Optimum Grating Tilt Angles θTEμ Calculated for Reflective Mode Coupling

Equations (32)

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κνμz=ω/4- Δx, zEyνxEyμ*xdx,
Eyνμ=Efνμ×coskfνμh-ϕsνμexp-γcνμx-hcoskfνμx-ϕsνμcosϕsνμexpγsνμx,,,xh 0xh, x0
kfνμ=knf2-neffνμ21/2,
ϕsνμ=tan-1γsνμ/kfνμ.
γsνμ=kneffνμ2-ns21/2,
γcνμ=kneffνμ2-nc21/2,
kfνh-ϕcν-ϕsν=νπ,
Efνμ=4μ0/0neffνμheffνμ1/2
Δx, z, θ=20nfΔn1x, z, θ,0xh0,all other x,
Δn1x, z, θ=nfσz1+νi cos2Kgz,
2K=2Kg cos θ,
κνμz=gνμ+ exp2iKgz cos θ+gνμ- exp-2iKgz cos θ+fνμ.
dAνdz=ifννAν+igμν+Aμ exp-2iδνμtz,
dAμdz=ifμμAμ+igνμ-Aν exp2iδνμtz,
2δνμt=βν-βμ-2Kg cos θ.
dAνdz=ifννAν+igμν+Bμ exp-2iδνμrz,
dBμdz=-igνμ-Aν exp2iδνμrz-iBμfμμ,
2δνμr=βν+βμ-2Kg cos θ.
Kνμz=2Agsin S1/S1cosS1+2Kgz cos θ-ϕsν+ϕsμ+sin S2/S2cosS2-2Kgz cos θ-ϕsν+ϕsμ+sin S3/S3cosS3+2Kgz cos θ-ϕsν-ϕsμ+sin S4/S4cosS4-2Kgz cos θ-ϕsν-ϕsμ,
S1=kfν+kfμ-2Kg sin θh/2,S2=kfν+kfμ+2Kg sin θh/2,S3=kfν-kfμ-2Kg sin θh/2,S4=kfν-kfμ+2Kg sin θh/2,
Ag=iν+μω0nfΔnhEfνEfμ/32=iν+μπnfΔnh/4λneffνneffμheffνheffμ1/2.
gνμ+=Ag exp-iKgh sin θsin S1S1+-1ν+μsin S2S2+-1μsin S3S3+-1νsin S4S4,
gνμ-=Ag expiKgh sin θsin S2S2+-1ν+μsin S1S1+-1μsin S4S4+-1νsin S3S3.
g0μ±=Ag±1μ expiKgh sin θ×sin S1S1+sin S4S4+-1μsin S2S2+sin S3S3.
glm-01±  0a rdrJl2rKg sin θJlulmrJ0u0r
glm-01±  0a rdrJl-12rKg sin θJl-1ulmrJ0u0r
θ=tan-1n12-nefflm21/2/neff01 ± nefflm
Λgnfcosθ-ϕ0-cosθ+ϕμ=λ
sin θnf2-neff02+nf2-neffμ2+cos θneff0-neffμ=λ/Λg.
Λgnfcosθ+ϕ0+cosθ-ϕμ=λ
sin θ-nf2-neff02+nf2-neffμ2+cos θneff0+neffμ=λ/Λg.
θ=ϕμ-ϕ0/2.

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