Abstract

A ray analysis of periodically segmented waveguides with parabolic-index variation in the high-index region is presented. We carried out the analysis using ray transfer matrices, which is convenient to implement and which can be extended to study different types of graded-index segmented waveguide. Results of this ray tracing approach clearly illustrate the waveguiding properties and the existence of stable and unstable regions of operation in segmented waveguides. We also illustrate the tapering action exhibited by segmented waveguides in which the duty cycle varies along the length of the waveguide. This analysis, although restricted to multimode structures, provides a clear visualization of the waveguiding properties in terms of ray propagation in segmented waveguides.

© 1998 Optical Society of America

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References

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  1. J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
    [CrossRef]
  2. C. J. Van der Poel, J. D. Bierlein, J. B. Brown, “Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
    [CrossRef]
  3. J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
    [CrossRef]
  4. F. Laurell, J. B. Brown, J. D. Bierlein, “Simultaneous generation of UV and visible light in segmented KTP waveguides,” Appl. Phys. Lett. 62, 1872–1874 (1993).
    [CrossRef]
  5. Z. Weissman, A. Hardy, M. Katz, M. Oron, D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
    [CrossRef] [PubMed]
  6. W. P. Risk, S. D. Lau, “Distributed-Bragg-reflection properties of segmented KTP waveguides,” Opt. Lett. 18, 272–274 (1993).
    [CrossRef] [PubMed]
  7. M. H. Chou, M. A. Arbore, M. M. Fejer, “Adiabatically tapered periodic segmentation of channel waveguides for mode-size transformation and fundamental mode excitation,” Opt. Lett. 21, 794–796 (1996).
    [CrossRef] [PubMed]
  8. Z. Weissman, A. Hardy, “Modes of periodically segmented waveguide,” IEEE J. Lightwave Technol. 11, 1831–1838 (1993).
    [CrossRef]
  9. L. Li, J. J. Burke, “Linear propagation characteristics of periodically segmented waveguides,” Opt. Lett. 17, 1195–1197 (1992).
    [CrossRef] [PubMed]
  10. K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Propagation characteristics of planar segmented waveguides with parabolic index segments,” Opt. Lett. 19, 2113–2115 (1994).
    [CrossRef] [PubMed]
  11. K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Equivalent waveguide model for parabolic index planar segmented waveguides,” Opt. Commun. 121, 27–30 (1995).
    [CrossRef]
  12. K. Thyagarajan, C. W. Chien, R. V. Ramaswamy, H. S. Kim, H. C. Cheng, “Proton exchanged periodically segmented waveguides in LiNbO3,” Opt. Lett. 19, 880–882 (1994).
    [CrossRef] [PubMed]
  13. P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
    [CrossRef]
  14. D. Nir, Z. Weissman, S. Ruschin, A. Hardy, “Periodically segmented waveguides in Ti:LiNbO3,” Opt. Lett. 19, 1732–1734 (1994).
    [CrossRef] [PubMed]
  15. D. Eger, M. Oron, M. Katz, “Optical characterisation of KTiOPO4 periodically segmented waveguides for second-harmonic generation of blue light,” J. Appl. Phys. 74, 4298–4302 (1993).
    [CrossRef]
  16. See, e.g., A. K. Ghatak, Optics, 2nd ed. (Tata McGraw-Hill, New Delhi, India, 1992), Chap. 2, p. 49.
  17. See, e.g., A. Yariv, Optical Electronics, 4th ed. (Holt, Rinehart & Winston, New York, 1991), Chap. 2, p. 19.
  18. See, e.g., A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), Chap. 2, p. 26.
  19. K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Mode propagation in planar segmented waveguides with parabolic index segments,” J. Indian Inst. Sci. 76, 175–182 (1996).

1996 (2)

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Mode propagation in planar segmented waveguides with parabolic index segments,” J. Indian Inst. Sci. 76, 175–182 (1996).

M. H. Chou, M. A. Arbore, M. M. Fejer, “Adiabatically tapered periodic segmentation of channel waveguides for mode-size transformation and fundamental mode excitation,” Opt. Lett. 21, 794–796 (1996).
[CrossRef] [PubMed]

1995 (2)

Z. Weissman, A. Hardy, M. Katz, M. Oron, D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
[CrossRef] [PubMed]

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Equivalent waveguide model for parabolic index planar segmented waveguides,” Opt. Commun. 121, 27–30 (1995).
[CrossRef]

1994 (4)

1993 (4)

W. P. Risk, S. D. Lau, “Distributed-Bragg-reflection properties of segmented KTP waveguides,” Opt. Lett. 18, 272–274 (1993).
[CrossRef] [PubMed]

D. Eger, M. Oron, M. Katz, “Optical characterisation of KTiOPO4 periodically segmented waveguides for second-harmonic generation of blue light,” J. Appl. Phys. 74, 4298–4302 (1993).
[CrossRef]

F. Laurell, J. B. Brown, J. D. Bierlein, “Simultaneous generation of UV and visible light in segmented KTP waveguides,” Appl. Phys. Lett. 62, 1872–1874 (1993).
[CrossRef]

Z. Weissman, A. Hardy, “Modes of periodically segmented waveguide,” IEEE J. Lightwave Technol. 11, 1831–1838 (1993).
[CrossRef]

1992 (1)

1990 (3)

J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

C. J. Van der Poel, J. D. Bierlein, J. B. Brown, “Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Arbore, M. A.

Baldi, P.

P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
[CrossRef]

Bhargava, R. N.

J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Bierlein, J. D.

F. Laurell, J. B. Brown, J. D. Bierlein, “Simultaneous generation of UV and visible light in segmented KTP waveguides,” Appl. Phys. Lett. 62, 1872–1874 (1993).
[CrossRef]

C. J. Van der Poel, J. D. Bierlein, J. B. Brown, “Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Brown, J. B.

F. Laurell, J. B. Brown, J. D. Bierlein, “Simultaneous generation of UV and visible light in segmented KTP waveguides,” Appl. Phys. Lett. 62, 1872–1874 (1993).
[CrossRef]

C. J. Van der Poel, J. D. Bierlein, J. B. Brown, “Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Burch, J. M.

See, e.g., A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), Chap. 2, p. 26.

Burke, J. J.

Cheng, H. C.

Chien, C. W.

Chou, M. H.

Colak, S.

J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

De Micheli, M. P.

P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
[CrossRef]

Eger, D.

Z. Weissman, A. Hardy, M. Katz, M. Oron, D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
[CrossRef] [PubMed]

D. Eger, M. Oron, M. Katz, “Optical characterisation of KTiOPO4 periodically segmented waveguides for second-harmonic generation of blue light,” J. Appl. Phys. 74, 4298–4302 (1993).
[CrossRef]

Fejer, M. M.

Gerrard, A.

See, e.g., A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), Chap. 2, p. 26.

Ghatak, A. K.

See, e.g., A. K. Ghatak, Optics, 2nd ed. (Tata McGraw-Hill, New Delhi, India, 1992), Chap. 2, p. 49.

Hardy, A.

Katz, M.

Z. Weissman, A. Hardy, M. Katz, M. Oron, D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
[CrossRef] [PubMed]

D. Eger, M. Oron, M. Katz, “Optical characterisation of KTiOPO4 periodically segmented waveguides for second-harmonic generation of blue light,” J. Appl. Phys. 74, 4298–4302 (1993).
[CrossRef]

Khurgin, J.

J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Kim, H. S.

Lau, S. D.

Laubacher, D. B.

J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Laurell, F.

F. Laurell, J. B. Brown, J. D. Bierlein, “Simultaneous generation of UV and visible light in segmented KTP waveguides,” Appl. Phys. Lett. 62, 1872–1874 (1993).
[CrossRef]

Li, L.

Mahalakshmi, V.

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Mode propagation in planar segmented waveguides with parabolic index segments,” J. Indian Inst. Sci. 76, 175–182 (1996).

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Equivalent waveguide model for parabolic index planar segmented waveguides,” Opt. Commun. 121, 27–30 (1995).
[CrossRef]

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Propagation characteristics of planar segmented waveguides with parabolic index segments,” Opt. Lett. 19, 2113–2115 (1994).
[CrossRef] [PubMed]

Nir, D.

Nouh, S.

P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
[CrossRef]

Oron, M.

Z. Weissman, A. Hardy, M. Katz, M. Oron, D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
[CrossRef] [PubMed]

D. Eger, M. Oron, M. Katz, “Optical characterisation of KTiOPO4 periodically segmented waveguides for second-harmonic generation of blue light,” J. Appl. Phys. 74, 4298–4302 (1993).
[CrossRef]

Ostrowsky, D. B.

P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
[CrossRef]

Ramaswamy, R. V.

Risk, W. P.

Ruschin, S.

Shenoy, M. R.

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Mode propagation in planar segmented waveguides with parabolic index segments,” J. Indian Inst. Sci. 76, 175–182 (1996).

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Equivalent waveguide model for parabolic index planar segmented waveguides,” Opt. Commun. 121, 27–30 (1995).
[CrossRef]

P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
[CrossRef]

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Propagation characteristics of planar segmented waveguides with parabolic index segments,” Opt. Lett. 19, 2113–2115 (1994).
[CrossRef] [PubMed]

Stolzenberger, R.

J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Thyagarajan, K.

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Mode propagation in planar segmented waveguides with parabolic index segments,” J. Indian Inst. Sci. 76, 175–182 (1996).

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Equivalent waveguide model for parabolic index planar segmented waveguides,” Opt. Commun. 121, 27–30 (1995).
[CrossRef]

K. Thyagarajan, C. W. Chien, R. V. Ramaswamy, H. S. Kim, H. C. Cheng, “Proton exchanged periodically segmented waveguides in LiNbO3,” Opt. Lett. 19, 880–882 (1994).
[CrossRef] [PubMed]

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Propagation characteristics of planar segmented waveguides with parabolic index segments,” Opt. Lett. 19, 2113–2115 (1994).
[CrossRef] [PubMed]

Van der Poel, C. J.

C. J. Van der Poel, J. D. Bierlein, J. B. Brown, “Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Weissman, Z.

Yariv, A.

See, e.g., A. Yariv, Optical Electronics, 4th ed. (Holt, Rinehart & Winston, New York, 1991), Chap. 2, p. 19.

Appl. Phys. Lett. (4)

J. D. Bierlein, D. B. Laubacher, J. B. Brown, C. J. Van der Poel, “Balanced phase matching in segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

C. J. Van der Poel, J. D. Bierlein, J. B. Brown, “Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. Khurgin, S. Colak, R. Stolzenberger, R. N. Bhargava, “Mechanism of efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

F. Laurell, J. B. Brown, J. D. Bierlein, “Simultaneous generation of UV and visible light in segmented KTP waveguides,” Appl. Phys. Lett. 62, 1872–1874 (1993).
[CrossRef]

IEEE J. Lightwave Technol. (1)

Z. Weissman, A. Hardy, “Modes of periodically segmented waveguide,” IEEE J. Lightwave Technol. 11, 1831–1838 (1993).
[CrossRef]

J. Appl. Phys. (1)

D. Eger, M. Oron, M. Katz, “Optical characterisation of KTiOPO4 periodically segmented waveguides for second-harmonic generation of blue light,” J. Appl. Phys. 74, 4298–4302 (1993).
[CrossRef]

J. Indian Inst. Sci. (1)

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Mode propagation in planar segmented waveguides with parabolic index segments,” J. Indian Inst. Sci. 76, 175–182 (1996).

Opt. Commun. (2)

P. Baldi, M. R. Shenoy, S. Nouh, M. P. De Micheli, D. B. Ostrowsky, “Estimation of the extent and influence of longitudinal diffusion on LiNbO3 proton exchanged segmented strip waveguides,” Opt. Commun. 104, 308–312 (1994).
[CrossRef]

K. Thyagarajan, V. Mahalakshmi, M. R. Shenoy, “Equivalent waveguide model for parabolic index planar segmented waveguides,” Opt. Commun. 121, 27–30 (1995).
[CrossRef]

Opt. Lett. (7)

Other (3)

See, e.g., A. K. Ghatak, Optics, 2nd ed. (Tata McGraw-Hill, New Delhi, India, 1992), Chap. 2, p. 49.

See, e.g., A. Yariv, Optical Electronics, 4th ed. (Holt, Rinehart & Winston, New York, 1991), Chap. 2, p. 19.

See, e.g., A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), Chap. 2, p. 26.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a buried planar segmented waveguide with parabolic-index variation in the transverse direction. Λ is the period of segmentation; 2a is the depth of the waveguide.

Fig. 2
Fig. 2

Periodic focusing and defocusing of rays in a segmented waveguide with parabolic-index variation for γ = 0.5 and Λ = 50 μm.

Fig. 3
Fig. 3

Ray paths in a segmented waveguide with γ = 0.8 and Λ = 5 μm, corresponding to the stable region of operation for (a) on-axis launching and (b) off-axis launching. For an angle θ exceeding θmax, the ray refracts out of the waveguide.

Fig. 4
Fig. 4

Variation of the maximum angle of guidance θmax with the duty cycle of segmentation for Λ = 5 μm. The dashed curve shows the corresponding variation for an equivalent (uniform) waveguide.

Fig. 5
Fig. 5

Ray paths in a segmented waveguide with γ = 0.5 and Λ = 100 μm, corresponding to the unstable region of operation for (a) on-axis launching and (b) off-axis launching. Note that all the rays refract out of the waveguide.

Fig. 6
Fig. 6

Ray paths in a segmented waveguide taper with period of segmentation Λ = 5 μm and a duty cycle that increases from 0.1 to 0.9 along the length of the waveguide.

Fig. 7
Fig. 7

Comparison between paraxial and nonparaxial ray paths in a segmented waveguide with γ = 0.8 and Λ = 5 μm.

Tables (1)

Tables Icon

Table 1 Comparison between Periods of Focusing in the Parabolic-Index Segmented Waveguide with Λ = 5 μm and in the Corresponding Equivalent (Uniform) Waveguide for Different Values of Duty Cycle

Equations (17)

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

n 2 x = n 1 2 1 - α 2 x 2 ,   | x | < a = n 2 2 ,   | x | a ,
M z = u z v z ,
R HM = 1 Λ - d 0 1 ,   R PM = cos   α d sin   α d / α - α   sin   α d cos   α d ,
R PH = 1 0 0 n x / n 2 ,   R HP = 1 0 0 n 2 / n x ,
A B C D = R HP R HM R PH R PM = cos   α d - n x n 2   α Λ - d sin   α d sin   α d α + n x n 2 Λ - d cos   α d - α   sin   α d cos   α d .
u m v m = A B C D m u 0 v 0 ,
A B C D = cos   α d - n 1 n 2   α Λ - d sin   α d sin   α d α + n 1 n 2 Λ - d cos   α d - α   sin   α d cos   α d .
p = A + D 2 ± A + D 2 2 - 1 1 / 2 .
A B C D m
- 1 < A + D 2 < 1 ,
- 1 < cos   α d - n 1 2 n 2   α Λ - d sin   α d < 1 ,
n 1 = 1.875 ,   n 2 = 1.85 ,   α = 0.045   μ m - 1 , λ 0 = 0.85   μ m .
d 2 x d z 2 + ρ 2 x z = 0 ,
β ˜ = n x 0 cos   θ 0 ,
x z = x 0 2 + 1 ρ 2 tan 2   θ 0 1 / 2 sin ρ z + sin - 1 × x 0 x 0 2 + 1 ρ 2 tan 2   θ 0 1 / 2 1 / 2 .
x z = x d + zx d ,
tan   θ z = x z .

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