Abstract

A new method for characterization of uniaxial planar waveguides from their measured effective mode indices is presented. The theory is outlined and expressions for efficient computer analysis are given. Uniaxial waveguide samples have been made in c-cut LiNbO3 by proton exchange with and without post annealing in order to test the method on both steplike and graded-index profiles. The resulting characterization of the samples is discussed in relation to the inverse WKB method. Finally, the importance of incorporating the effects of material birefringence in the characterization of these kinds of waveguides is investigated.

© 1996 Optical Society of America

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References

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  1. R. R. A. Syms, “Silica-on-silicon integrated optics,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 121–150.
    [CrossRef]
  2. J. M. White, P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
    [CrossRef] [PubMed]
  3. K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. LT-3, 385–391 (1985).
    [CrossRef]
  4. J. Albert, G. L. Yip, “Refractive-index profiles of planar waveguides made by ion-exchange in glass,” Appl. Opt. 24, 3692–3693 (1985).
    [CrossRef] [PubMed]
  5. S. Pelli, G. C. Righini, “Introduction to integrated optics: characterization and modelling of optical waveguides,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 1–20.
    [CrossRef]
  6. X. Mu, X. Yue, J. Chen, J. Wang, Z. Shao, “Planar waveguide refractive index distribution functions determined precisely from mode indices,” Appl. Opt. 33, 3227–3230 (1994).
    [CrossRef] [PubMed]
  7. J. F. Offersgaard, “Waveguides formed by multiple layers of dielectric, semiconductor, or metallic media with optical loss and anisotropy,” J. Opt. Soc. Am. A 12, 2122–2128 (1995).
    [CrossRef]
  8. W. Charczenko, I. Januar, A. R. Mickelson, “Modelling of proton-exchanged and annealed channel waveguides and directional couplers,” J. Appl. Phys. 73, 3139–3148 (1993).
    [CrossRef]
  9. J. Chiwell, I. Hodgkinson, “Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  10. S. T. Vohra, A. R. Mickelson, S. E. Asher, “Diffusion characteristics and waveguiding properties of proton-exchanged and annealed LiNbO3 channel waveguides,” Appl. Phys. 66, 5161–5174 (1989).
  11. X. F. Cao, R. V. Ramaswamy, R. Srivastava, “Characterization of annealed proton exchanged LiNbO3 waveguides for nonlinear frequency conversion,” J. Lightwave Technol. 10, 1302–1313 (1992).
    [CrossRef]
  12. J. Olivares, J. M. Cabera, “Guided modes with ordinary refractive index in proton exchanged LiNbO3 waveguides,” Appl. Phys. Lett. 62, 2468–2470 (1993).
    [CrossRef]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  14. W. Kaplan, Advanced Mathematics for Engineers (Addison-Wesley, Reading, Mass.1981).
  15. P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
    [CrossRef]
  16. C. E. Rice, “The structure and properties of Li1−xHxNbO3,” J. Solid State Chem. 64, 188–199 (1986).
    [CrossRef]
  17. W. L. Bond, “Measurement of the refractive index of several crystals,” Appl. Phys. 36, 1674–1677 (1965).
  18. M. R. Shenoy, R. M. de la Rue, “On the refractive index of rutile,” Proc. Inst. Electr. Eng. Part J 139, 163–165 (1991).

1995 (1)

1994 (1)

1993 (2)

W. Charczenko, I. Januar, A. R. Mickelson, “Modelling of proton-exchanged and annealed channel waveguides and directional couplers,” J. Appl. Phys. 73, 3139–3148 (1993).
[CrossRef]

J. Olivares, J. M. Cabera, “Guided modes with ordinary refractive index in proton exchanged LiNbO3 waveguides,” Appl. Phys. Lett. 62, 2468–2470 (1993).
[CrossRef]

1992 (1)

X. F. Cao, R. V. Ramaswamy, R. Srivastava, “Characterization of annealed proton exchanged LiNbO3 waveguides for nonlinear frequency conversion,” J. Lightwave Technol. 10, 1302–1313 (1992).
[CrossRef]

1991 (1)

M. R. Shenoy, R. M. de la Rue, “On the refractive index of rutile,” Proc. Inst. Electr. Eng. Part J 139, 163–165 (1991).

1989 (1)

S. T. Vohra, A. R. Mickelson, S. E. Asher, “Diffusion characteristics and waveguiding properties of proton-exchanged and annealed LiNbO3 channel waveguides,” Appl. Phys. 66, 5161–5174 (1989).

1986 (1)

C. E. Rice, “The structure and properties of Li1−xHxNbO3,” J. Solid State Chem. 64, 188–199 (1986).
[CrossRef]

1985 (2)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. LT-3, 385–391 (1985).
[CrossRef]

J. Albert, G. L. Yip, “Refractive-index profiles of planar waveguides made by ion-exchange in glass,” Appl. Opt. 24, 3692–3693 (1985).
[CrossRef] [PubMed]

1984 (1)

1976 (1)

1969 (1)

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

1965 (1)

W. L. Bond, “Measurement of the refractive index of several crystals,” Appl. Phys. 36, 1674–1677 (1965).

Albert, J.

Asher, S. E.

S. T. Vohra, A. R. Mickelson, S. E. Asher, “Diffusion characteristics and waveguiding properties of proton-exchanged and annealed LiNbO3 channel waveguides,” Appl. Phys. 66, 5161–5174 (1989).

Bond, W. L.

W. L. Bond, “Measurement of the refractive index of several crystals,” Appl. Phys. 36, 1674–1677 (1965).

Cabera, J. M.

J. Olivares, J. M. Cabera, “Guided modes with ordinary refractive index in proton exchanged LiNbO3 waveguides,” Appl. Phys. Lett. 62, 2468–2470 (1993).
[CrossRef]

Cao, X. F.

X. F. Cao, R. V. Ramaswamy, R. Srivastava, “Characterization of annealed proton exchanged LiNbO3 waveguides for nonlinear frequency conversion,” J. Lightwave Technol. 10, 1302–1313 (1992).
[CrossRef]

Charczenko, W.

W. Charczenko, I. Januar, A. R. Mickelson, “Modelling of proton-exchanged and annealed channel waveguides and directional couplers,” J. Appl. Phys. 73, 3139–3148 (1993).
[CrossRef]

Chen, J.

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. LT-3, 385–391 (1985).
[CrossRef]

Chiwell, J.

de la Rue, R. M.

M. R. Shenoy, R. M. de la Rue, “On the refractive index of rutile,” Proc. Inst. Electr. Eng. Part J 139, 163–165 (1991).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Heidrich, P. F.

Hodgkinson, I.

Januar, I.

W. Charczenko, I. Januar, A. R. Mickelson, “Modelling of proton-exchanged and annealed channel waveguides and directional couplers,” J. Appl. Phys. 73, 3139–3148 (1993).
[CrossRef]

Kaplan, W.

W. Kaplan, Advanced Mathematics for Engineers (Addison-Wesley, Reading, Mass.1981).

Martin, R. J.

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

Mickelson, A. R.

W. Charczenko, I. Januar, A. R. Mickelson, “Modelling of proton-exchanged and annealed channel waveguides and directional couplers,” J. Appl. Phys. 73, 3139–3148 (1993).
[CrossRef]

S. T. Vohra, A. R. Mickelson, S. E. Asher, “Diffusion characteristics and waveguiding properties of proton-exchanged and annealed LiNbO3 channel waveguides,” Appl. Phys. 66, 5161–5174 (1989).

Mu, X.

Offersgaard, J. F.

Olivares, J.

J. Olivares, J. M. Cabera, “Guided modes with ordinary refractive index in proton exchanged LiNbO3 waveguides,” Appl. Phys. Lett. 62, 2468–2470 (1993).
[CrossRef]

Pelli, S.

S. Pelli, G. C. Righini, “Introduction to integrated optics: characterization and modelling of optical waveguides,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 1–20.
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Ramaswamy, R. V.

X. F. Cao, R. V. Ramaswamy, R. Srivastava, “Characterization of annealed proton exchanged LiNbO3 waveguides for nonlinear frequency conversion,” J. Lightwave Technol. 10, 1302–1313 (1992).
[CrossRef]

Rice, C. E.

C. E. Rice, “The structure and properties of Li1−xHxNbO3,” J. Solid State Chem. 64, 188–199 (1986).
[CrossRef]

Righini, G. C.

S. Pelli, G. C. Righini, “Introduction to integrated optics: characterization and modelling of optical waveguides,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 1–20.
[CrossRef]

Shao, Z.

Shenoy, M. R.

M. R. Shenoy, R. M. de la Rue, “On the refractive index of rutile,” Proc. Inst. Electr. Eng. Part J 139, 163–165 (1991).

Srivastava, R.

X. F. Cao, R. V. Ramaswamy, R. Srivastava, “Characterization of annealed proton exchanged LiNbO3 waveguides for nonlinear frequency conversion,” J. Lightwave Technol. 10, 1302–1313 (1992).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, “Silica-on-silicon integrated optics,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 121–150.
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Tien, P. K.

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Vohra, S. T.

S. T. Vohra, A. R. Mickelson, S. E. Asher, “Diffusion characteristics and waveguiding properties of proton-exchanged and annealed LiNbO3 channel waveguides,” Appl. Phys. 66, 5161–5174 (1989).

Wang, J.

White, J. M.

Yip, G. L.

Yue, X.

Appl. Opt. (3)

Appl. Phys. (2)

S. T. Vohra, A. R. Mickelson, S. E. Asher, “Diffusion characteristics and waveguiding properties of proton-exchanged and annealed LiNbO3 channel waveguides,” Appl. Phys. 66, 5161–5174 (1989).

W. L. Bond, “Measurement of the refractive index of several crystals,” Appl. Phys. 36, 1674–1677 (1965).

Appl. Phys. Lett. (2)

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

J. Olivares, J. M. Cabera, “Guided modes with ordinary refractive index in proton exchanged LiNbO3 waveguides,” Appl. Phys. Lett. 62, 2468–2470 (1993).
[CrossRef]

J. Appl. Phys. (1)

W. Charczenko, I. Januar, A. R. Mickelson, “Modelling of proton-exchanged and annealed channel waveguides and directional couplers,” J. Appl. Phys. 73, 3139–3148 (1993).
[CrossRef]

J. Lightwave Technol. (2)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. LT-3, 385–391 (1985).
[CrossRef]

X. F. Cao, R. V. Ramaswamy, R. Srivastava, “Characterization of annealed proton exchanged LiNbO3 waveguides for nonlinear frequency conversion,” J. Lightwave Technol. 10, 1302–1313 (1992).
[CrossRef]

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

J. Solid State Chem. (1)

C. E. Rice, “The structure and properties of Li1−xHxNbO3,” J. Solid State Chem. 64, 188–199 (1986).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

M. R. Shenoy, R. M. de la Rue, “On the refractive index of rutile,” Proc. Inst. Electr. Eng. Part J 139, 163–165 (1991).

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

W. Kaplan, Advanced Mathematics for Engineers (Addison-Wesley, Reading, Mass.1981).

S. Pelli, G. C. Righini, “Introduction to integrated optics: characterization and modelling of optical waveguides,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 1–20.
[CrossRef]

R. R. A. Syms, “Silica-on-silicon integrated optics,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 121–150.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Planar waveguide consisting of N uniaxial layers between the uniaxial substrate s and ambient a. The optical axis of the uniaxial media is along the x axis, and the waveguide-propagation direction is along the z axis.

Fig. 2
Fig. 2

Calculated effective index versus the number of subdividing step-index layers: (a) As-exchanged guide (sample S3 in Table 2). (b) Annealed guide (sample S1 in Table 2). From top to bottom the curves represent the mode numbers 0, 1, and 2.

Fig. 3
Fig. 3

Extraordinary refractive-index profiles of as-exchanged guides before (solid curves) and after (dashed curves) storage fitted by the profile model Eq. (7). The exchange times for samples S1, S2, S3, and S4 are 0 h 30 min, 2 h 00 min, 4 h 10 min, and 8 h 00 min, respectively.

Fig. 4
Fig. 4

Extraordinary refractive index profile of annealed waveguides S1–S4 fitted by the profile model of Eq. (7) (solid curves). The inverse WKB results based on the method in Ref. 2 are shown by the filled circles.

Tables (2)

Tables Icon

Table 1 Measured Modes in Proton-Exchanged LiNbO3 Waveguides Immediately after Proton Exchange (PE), after 3 Weeks Storage, and after Annealing

Tables Icon

Table 2 Fitted Model Parameters of As-Exchanged Waveguides before and after Storage, and after Annealing

Equations (29)

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m 0 [ ω , β ˜ ; n ( j ) , d ( j ) ] = 0 ,
1 2 ξ ( j ) + 1 2 η [ ξ ( j ) ] = j - ½ N - 2 ,             j = 1 , 2 , , N - 1 ,
ξ ( j ) x ( j ) - x ( 0 ) x ( N ) - x ( 0 ) ,
η [ ξ ( j ) ] n e [ x ( j ) ] - n e [ x ( 0 ) ] n e [ x ( N ) ] - n e [ x ( 0 ) ] .
n e [ x ( N ) ] = n e ( s ) + δ ,
n ( j ) = 2 3 n [ x ( j - 1 ) + x ( j ) 2 ] + 1 6 n [ x ( j - 1 ) ] + 1 6 n [ x ( j ) ] ,
Δ n e [ x ] = Δ n e s erf [ h - x d x ] + erf [ h + x d x ] 2 erf [ h d x ] ,
Δ n o [ x ] = κ Δ n e [ x ] ,
β ˜ a k = β ˜ b l b l a k .
d { m 0 [ ω , β ˜ ; n ( j ) , d ( j ) ] } = 0
β ˜ b l = - m 0 b l / m 0 β ˜ ,             b l { n o ( j ) , n e ( j ) , d ( j ) } ,
M 0 = G ( a ) · G ( 1 ) · · G ( N ) · G ( s ) ,
G ( j ) = [ cos [ k 0 k ˜ x ( j ) d ( j ) ] - i p t ( j ) q t ( j ) sin [ k 0 k ˜ x ( j ) d ( j ) ] - i q t ( j ) p t ( j ) sin [ k 0 k ˜ x ( j ) d ( j ) ] cos [ k 0 k ˜ x ( j ) d ( j ) ] ] ,
k ˜ x ( j ) = { [ n o ( j ) ] 2 - β ˜ 2 } 1 , 2 ,             q t ( j ) p t ( j ) = 0 μ 0 k ˜ x ( j )
k ˜ x ( j ) = n o ( j ) n e ( j ) { [ n e ( j ) ] 2 - β ˜ 2 } 1 , 2 ,             q t ( j ) p t ( j ) = - 0 μ 0 [ n o ( j ) ] 2 k ˜ x ( j )
G ( j ) β ˜ = - β ˜ k ˜ x ( j ) G ( j ) k ˜ x ( j )
G ( j ) β ˜ = - β ˜ k ˜ x ( j ) ( n z ( j ) n x ( j ) ) 2 G ( j ) k ˜ x ( j )
G ( j ) β ˜ = β ˜ [ k ˜ x ( j ) ] 2 [ k 0 k ˜ x ( j ) d ( j ) sin [ k 0 k ˜ x ( j ) d ( j ) ] - i p t ( j ) q t ( j ) { ± sin [ k 0 k ˜ x ( j ) d ( j ) ] - k 0 k ˜ x ( j ) d ( j ) cos [ k 0 k ˜ x ( j ) d ( j ) ] } i q t ( j ) p t ( j ) { ± sin [ k 0 k ˜ x ( j ) d ( j ) ] + k 0 k ˜ x ( j ) d ( j ) cos [ k 0 k ˜ x ( j ) d ( j ) ] } k 0 k ˜ x ( j ) d ( j ) sin [ k 0 k ˜ x ( j ) d ( j ) ] ] ,
G ( j ) d ( j ) = - k 0 k ˜ x ( j ) [ sin [ k 0 k ˜ x ( j ) d ( j ) ] i p t ( j ) q t ( j ) cos [ k 0 k ˜ x ( j ) d ( j ) ] i q t ( j ) p t ( j ) cos [ k 0 k ˜ x ( j ) d ( j ) ] sin [ k 0 k ˜ x ( j ) d ( j ) ] ] ,
G ( j ) n o ( j ) = - n o ( j ) β ˜ G ( j ) β ˜ .
G ( j ) n e ( j ) = - n e ( j ) β ˜ G ( j ) β ˜ ,
G ( j ) n o ( j ) = β ˜ n o ( j ) [ 1 - ( n e ( j ) ) 2 β ˜ 2 ] G ( j ) β ˜ + 2 n o ( j ) [ 0 - G 12 ( j ) G 21 ( j ) 0 ] .
G ( a ) = [ - 0 μ 0 k ˜ x ( a ) - 1 - 0 μ 0 k ˜ x ( a ) 1 ]
G ( s ) = [ 1 1 0 μ 0 k ˜ x ( s ) - 0 μ 0 k ˜ x ( s ) ] .
G ( a ) = [ - 0 μ 0 [ n o ( a ) ] 2 k ˜ x ( a ) 0 μ 0 [ n o ( a ) ] 2 k ˜ x ( a ) ] ,
G ( s ) = [ k ˜ x ( s ) - k x ( s ) - 0 μ 0 [ n o ( s ) ] 2 - 0 μ 0 [ n z ( s ) ] 2 ] .
M 0 b l = G ( 0 ) · · G ( j - 1 ) · G ( j ) b l · G ( j + 1 ) · · G ( N + 1 ) .
T ( j ) = G ( j ) · T ( j + 1 ) ,
d T ( j ) = G ( j ) β ˜ · T ( j + 1 ) + G ( j ) · d T ( j + 1 ) ,

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