Abstract

Tilted-mirror facets have been introduced as a design alternative to antireflection coating for waveguide structures that require low facet reflectivity. We show that tilted-mirror facets with finite reflectivity can induce significant intermode coupling in semiconductor lasers and optical amplifiers. We describe the effect of tilt-induced intermode coupling on the cavity modes using a simple model and discuss the implications of our results on the design of tilted-mirror integrated-optical lasers and amplifiers.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
    [CrossRef]
  2. D. R. Scifres, W. Strelfer, R. D. Burnham, IEEE J. Quantum Electron. QE-14, 223 (1978).
    [CrossRef]
  3. J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
    [CrossRef]
  4. T. Mukai, Y. Yamamoto, T. Kimura, in Semiconductors and Semimetals, W. Beer, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 265–319.
    [CrossRef]
  5. J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).
  6. C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
    [CrossRef]
  7. T. Ikegami, IEEE J. Quantum Electron. QE-8, 470 (1972).
    [CrossRef]
  8. T. E. Rozzi, G. H. Veld, IEEE Trans. Microwave Theory Tech. MTT-28, 61 (1980).
    [CrossRef]
  9. P. Gelin, M. Petenzi, J. Citene, IEEE Trans. Microwave Theory Tech. MTT-29, 107 (1981).
    [CrossRef]
  10. E. Nishimura, N. Morita, N. Kumagai, IEEE Trans. Microwave Theory Tech. MTT-31, 923 (1983).
    [CrossRef]
  11. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 49–52 and 7–74.
  12. Intermode coupling was observed in gas lasers and analyzed in a geometrical-optics approximation by C. A. Hill, D. R. Hall, IEEE J. Quantum Electron. QE-22, 1078 (1986).
    [CrossRef]
  13. R. J. Lang, J. Salzman, A. Yariv, IEEE J. Quantum Electron. QE-22, 463 (1986).
    [CrossRef]
  14. J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
    [CrossRef]
  15. In writing Eq. (1) it is assumed that the waveguide is sufficiently long that leaky and radiation modes are strongly attenuated and can be ignored. The transverse y dependence of the field can be eliminated by an effective-index approximation.
  16. Equation (5) is a generalization to finite angles of incidence and intermode coupling of an earlier result due to J. McKenna, cited in F. K. Reinhart, I. Hayashi, M. B. Panish, J. Appl. Phys. 42, 4466 (1971).
    [CrossRef]
  17. R J. Hawkins, R. J. Deri, J. Opt. Soc. Am. A 4 (13), P97 (1987).
  18. The values of Kij depend on the modal profiles. For example, a cosh2(x) distribution may result in smaller values of Kij because their spectra of plane waves are narrower than the ones computed here. However, the ratios |Kij/K00| are expected to increase with tilt parameter in a way similar to that of the ratios computed from Eqs. (7a)–(7c).

1987 (2)

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

R J. Hawkins, R. J. Deri, J. Opt. Soc. Am. A 4 (13), P97 (1987).

1986 (3)

Intermode coupling was observed in gas lasers and analyzed in a geometrical-optics approximation by C. A. Hill, D. R. Hall, IEEE J. Quantum Electron. QE-22, 1078 (1986).
[CrossRef]

R. J. Lang, J. Salzman, A. Yariv, IEEE J. Quantum Electron. QE-22, 463 (1986).
[CrossRef]

J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).

1985 (2)

J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
[CrossRef]

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

1983 (1)

E. Nishimura, N. Morita, N. Kumagai, IEEE Trans. Microwave Theory Tech. MTT-31, 923 (1983).
[CrossRef]

1981 (1)

P. Gelin, M. Petenzi, J. Citene, IEEE Trans. Microwave Theory Tech. MTT-29, 107 (1981).
[CrossRef]

1980 (1)

T. E. Rozzi, G. H. Veld, IEEE Trans. Microwave Theory Tech. MTT-28, 61 (1980).
[CrossRef]

1978 (1)

D. R. Scifres, W. Strelfer, R. D. Burnham, IEEE J. Quantum Electron. QE-14, 223 (1978).
[CrossRef]

1977 (1)

B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
[CrossRef]

1972 (1)

T. Ikegami, IEEE J. Quantum Electron. QE-8, 470 (1972).
[CrossRef]

1971 (1)

Equation (5) is a generalization to finite angles of incidence and intermode coupling of an earlier result due to J. McKenna, cited in F. K. Reinhart, I. Hayashi, M. B. Panish, J. Appl. Phys. 42, 4466 (1971).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 49–52 and 7–74.

Burnham, R. D.

D. R. Scifres, W. Strelfer, R. D. Burnham, IEEE J. Quantum Electron. QE-14, 223 (1978).
[CrossRef]

Caneau, C.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

Citene, J.

P. Gelin, M. Petenzi, J. Citene, IEEE Trans. Microwave Theory Tech. MTT-29, 107 (1981).
[CrossRef]

Deri, R. J.

R J. Hawkins, R. J. Deri, J. Opt. Soc. Am. A 4 (13), P97 (1987).

Frescura, B. L.

B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
[CrossRef]

Gelin, P.

P. Gelin, M. Petenzi, J. Citene, IEEE Trans. Microwave Theory Tech. MTT-29, 107 (1981).
[CrossRef]

Hall, D. R.

Intermode coupling was observed in gas lasers and analyzed in a geometrical-optics approximation by C. A. Hill, D. R. Hall, IEEE J. Quantum Electron. QE-22, 1078 (1986).
[CrossRef]

Hawkins, R J.

R J. Hawkins, R. J. Deri, J. Opt. Soc. Am. A 4 (13), P97 (1987).

Hayashi, I.

Equation (5) is a generalization to finite angles of incidence and intermode coupling of an earlier result due to J. McKenna, cited in F. K. Reinhart, I. Hayashi, M. B. Panish, J. Appl. Phys. 42, 4466 (1971).
[CrossRef]

Hill, C. A.

Intermode coupling was observed in gas lasers and analyzed in a geometrical-optics approximation by C. A. Hill, D. R. Hall, IEEE J. Quantum Electron. QE-22, 1078 (1986).
[CrossRef]

Hwang, C. J.

B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
[CrossRef]

Ikegami, T.

T. Ikegami, IEEE J. Quantum Electron. QE-8, 470 (1972).
[CrossRef]

Kimura, T.

T. Mukai, Y. Yamamoto, T. Kimura, in Semiconductors and Semimetals, W. Beer, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 265–319.
[CrossRef]

Kumagai, N.

E. Nishimura, N. Morita, N. Kumagai, IEEE Trans. Microwave Theory Tech. MTT-31, 923 (1983).
[CrossRef]

Lang, R.

J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
[CrossRef]

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

Lang, R. J.

R. J. Lang, J. Salzman, A. Yariv, IEEE J. Quantum Electron. QE-22, 463 (1986).
[CrossRef]

Lee, T. P.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

Luechinger, H.

B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
[CrossRef]

Margalit, S.

J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
[CrossRef]

Menocal, S. G.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

Mittelstein, M.

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

Morita, N.

E. Nishimura, N. Morita, N. Kumagai, IEEE Trans. Microwave Theory Tech. MTT-31, 923 (1983).
[CrossRef]

Morrison, C. B.

J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).

Mukai, T.

T. Mukai, Y. Yamamoto, T. Kimura, in Semiconductors and Semimetals, W. Beer, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 265–319.
[CrossRef]

Niesen, J. W.

J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).

Nishimura, E.

E. Nishimura, N. Morita, N. Kumagai, IEEE Trans. Microwave Theory Tech. MTT-31, 923 (1983).
[CrossRef]

Osinski, J. S.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

Panish, M. B.

Equation (5) is a generalization to finite angles of incidence and intermode coupling of an earlier result due to J. McKenna, cited in F. K. Reinhart, I. Hayashi, M. B. Panish, J. Appl. Phys. 42, 4466 (1971).
[CrossRef]

Payton, P. H.

J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).

Petenzi, M.

P. Gelin, M. Petenzi, J. Citene, IEEE Trans. Microwave Theory Tech. MTT-29, 107 (1981).
[CrossRef]

Reinhart, F. K.

Equation (5) is a generalization to finite angles of incidence and intermode coupling of an earlier result due to J. McKenna, cited in F. K. Reinhart, I. Hayashi, M. B. Panish, J. Appl. Phys. 42, 4466 (1971).
[CrossRef]

Reith, L. A.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

Ripper, J. E.

B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
[CrossRef]

Rozzi, T. E.

T. E. Rozzi, G. H. Veld, IEEE Trans. Microwave Theory Tech. MTT-28, 61 (1980).
[CrossRef]

Salzman, J.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

R. J. Lang, J. Salzman, A. Yariv, IEEE J. Quantum Electron. QE-22, 463 (1986).
[CrossRef]

J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
[CrossRef]

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

Scifres, D. R.

D. R. Scifres, W. Strelfer, R. D. Burnham, IEEE J. Quantum Electron. QE-14, 223 (1978).
[CrossRef]

Strelfer, W.

D. R. Scifres, W. Strelfer, R. D. Burnham, IEEE J. Quantum Electron. QE-14, 223 (1978).
[CrossRef]

Veld, G. H.

T. E. Rozzi, G. H. Veld, IEEE Trans. Microwave Theory Tech. MTT-28, 61 (1980).
[CrossRef]

Venkatesan, T.

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

Yamamoto, Y.

T. Mukai, Y. Yamamoto, T. Kimura, in Semiconductors and Semimetals, W. Beer, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 265–319.
[CrossRef]

Yariv, A.

R. J. Lang, J. Salzman, A. Yariv, IEEE J. Quantum Electron. QE-22, 463 (1986).
[CrossRef]

J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
[CrossRef]

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

Zah, C. E.

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

Zinkiewicz, L. M.

J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).

Appl. Phys. Lett. (3)

B. L. Frescura, C. J. Hwang, H. Luechinger, J. E. Ripper, Appl. Phys. Lett. 31, 770 (1977).
[CrossRef]

J. Salzman, R. Lang, S. Margalit, A. Yariv, Appl. Phys. Lett. 47, 9 (1985).
[CrossRef]

J. Salzman, R. Lang, T. Venkatesan, M. Mittelstein, A. Yariv, Appl. Phys. Lett. 47, 445 (1985).
[CrossRef]

Electron. Lett. (1)

C. E. Zah, J. S. Osinski, C. Caneau, S. G. Menocal, L. A. Reith, J. Salzman, T. P. Lee, Electron. Lett. 23, 990 (1987).
[CrossRef]

IEEE J. Quantum Electron. (4)

T. Ikegami, IEEE J. Quantum Electron. QE-8, 470 (1972).
[CrossRef]

D. R. Scifres, W. Strelfer, R. D. Burnham, IEEE J. Quantum Electron. QE-14, 223 (1978).
[CrossRef]

Intermode coupling was observed in gas lasers and analyzed in a geometrical-optics approximation by C. A. Hill, D. R. Hall, IEEE J. Quantum Electron. QE-22, 1078 (1986).
[CrossRef]

R. J. Lang, J. Salzman, A. Yariv, IEEE J. Quantum Electron. QE-22, 463 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

T. E. Rozzi, G. H. Veld, IEEE Trans. Microwave Theory Tech. MTT-28, 61 (1980).
[CrossRef]

P. Gelin, M. Petenzi, J. Citene, IEEE Trans. Microwave Theory Tech. MTT-29, 107 (1981).
[CrossRef]

E. Nishimura, N. Morita, N. Kumagai, IEEE Trans. Microwave Theory Tech. MTT-31, 923 (1983).
[CrossRef]

J. Appl. Phys. (1)

Equation (5) is a generalization to finite angles of incidence and intermode coupling of an earlier result due to J. McKenna, cited in F. K. Reinhart, I. Hayashi, M. B. Panish, J. Appl. Phys. 42, 4466 (1971).
[CrossRef]

J. Opt. Soc. Am. A (1)

R J. Hawkins, R. J. Deri, J. Opt. Soc. Am. A 4 (13), P97 (1987).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. W. Niesen, C. B. Morrison, P. H. Payton, L. M. Zinkiewicz, Proc. Soc. Photo-Opt. Instrum. Eng. 719, 208 (1986).

Other (4)

The values of Kij depend on the modal profiles. For example, a cosh2(x) distribution may result in smaller values of Kij because their spectra of plane waves are narrower than the ones computed here. However, the ratios |Kij/K00| are expected to increase with tilt parameter in a way similar to that of the ratios computed from Eqs. (7a)–(7c).

In writing Eq. (1) it is assumed that the waveguide is sufficiently long that leaky and radiation modes are strongly attenuated and can be ignored. The transverse y dependence of the field can be eliminated by an effective-index approximation.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 49–52 and 7–74.

T. Mukai, Y. Yamamoto, T. Kimura, in Semiconductors and Semimetals, W. Beer, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 265–319.
[CrossRef]

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 (3)

Fig. 1
Fig. 1

Self-coupling and intermode coupling of the fundamental and first-order TE modes as a function of the tilt parameter t. The solid lines were calculated using Eqs. (7a)(7c) with parameters w0 = 2.4 μm, λ0 = 1.55 μm, and neff = 3.5. The dashed lines were calculated using Eq. (5) with mode distributions that are numerical results of a finite-difference treatment of the Helmholtz equation with a symmetric slab waveguide with the following parameters: guide width, 4.8 μm; nguide = 3.5; and ncladding = 3.49.

Fig. 2
Fig. 2

The normalized effective reflectivity of the cavity modes γ ˆ ± and that of the fundamental mode as a function of the tilt parameter t [= w0kneff sin(θ)] for |f| = 0.7 at various values of the phase of f, where f = exp[−i(β1β0)L] calculated using Eqs. (7a)(7c) and (9) for TE modes. Solid line, fundamental waveguide mode; dashed lines, γ ˆ +; dotted lines, γ ˆ (The dashed–dotted line is the degenerate solution γ ˆ + = γ ˆ for real values of f.)

Fig. 3
Fig. 3

The normalized effective reflectivity of the cavity modes γ ˆ ± and that of the fundamental mode as a function of the tilt parameter t [= w0kneff sin(θ)] for |f| = 0.7 at various values of the phase of f, where f = exp[−i(β1β0)L] calculated using Eqs. (7a)(7c) and (9) with K01K10 replaced by −K01K10. Solid line, fundamental waveguide mode; dashed lines, γ ˆ +; dotted lines, γ ˆ .

Equations (13)

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

E ( x , z ) = m = 0 N α m m ( x ) exp ( i β m z ) ,
m ( x ) = F m ( l ) exp ( i β m l x ) d ( β m l ) ,
m r ( x ) = r ( l ) F m ( l ) exp ( i β m l x ) d ( β m l ) .
K m j = j ( x ) m r ( x ) d x ,
K m j = F j ( l 0 l ) F m ( l 0 + l ) r ( l ) d l ,
r ( l ) = r 0 ( 1 ± n eff l 2 ) ,
K 00 = r 0 ( 1 ± p ) exp ( 1 2 t 2 ) ,
K 10 = K 01 = i t r 0 ( 1 ± p ) exp ( 1 2 t 2 ) ,
K 11 = r 0 [ ( 1 t 2 ) ± p ( 3 t 2 ) ] exp ( 1 2 t 2 ) ,
[ P ( L ) K ( l 0 ) P ( L ) K ( l 0 ) ] A = γ 2 A ,
P ( L ) = [ exp ( i ( β 0 L ) 0 0 exp ( i β l L ) ] , K = [ K 00 K 10 K 01 K 11 ] , A = [ α 0 α 1 ] ,
γ ± = K 00 exp ( i β 0 L ) + K 11 exp ( i β 1 L ) 2 ± { [ K 00 exp ( i β 0 L ) K 11 exp ( i β 1 L ) 2 ] 2 + K 01 K 10 exp [ i ( β 0 + β 1 ) L ] } 1 / 2 .
γ = r 0 exp ( Im β 0 L ) γ ˆ ( t , Δ β L ) ,

Metrics