Abstract

We compute the radiation losses of a rectangular dielectric waveguide (integrated optics channel waveguide) that is tapered so that its wider cross-sectional dimension increases by roughly a factor of three while its narrow dimension remains constant. As the waveguide widens its refractive index decreases to ensure that the waveguide supports only one guided mode. The taper is approximated by a discontinuous staircase curve. A rectangular waveguide taper of 2-μm thickness, tapering from 3- to 10-μm width through fourteen steps of 0.25-μm height, has a minimum loss (at 0.6328-μm wavelength) of 0.13 dB for a 200-μm taper length.

© 1980 Optical Society of America

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References

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  1. H. P. Hsu, A. F. Milton, Electron. Lett. 12, 404 (1976).
    [Crossref]
  2. S. K. Sheem, T. G. Giallorenzi, Opt. Lett. 3, 73 (1978).
    [Crossref] [PubMed]
  3. W. K. Burns, Appl. Opt. 18, 2536 (1979).
    [Crossref] [PubMed]
  4. M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49, 3693 (1978).
    [Crossref]
  5. T. Kimura, M. Saruwatari, J. Yamada, S. Uehara, T. Miyashita, Appl. Opt. 17, 2420 (1978).
    [Crossref] [PubMed]
  6. V. Ramaswamy, R. Alferness, BTL; private communication.
  7. R. D. Standley, V. Ramaswamy, Appl. Phys. Lett. 25, 711 (1974).
    [Crossref]
  8. D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
    [Crossref]
  9. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  10. A. W. Snyder, W. R. Young, J. Opt. Soc. Am. 68, 297 (1978).
    [Crossref]
  11. D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 918.
  12. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  13. See Ref. 12, p. 110.
  14. D. Marcuse, Appl. Opt. 17, 763 (1978).
    [Crossref] [PubMed]

1979 (1)

1978 (5)

1976 (1)

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 404 (1976).
[Crossref]

1975 (1)

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
[Crossref]

1974 (1)

R. D. Standley, V. Ramaswamy, Appl. Phys. Lett. 25, 711 (1974).
[Crossref]

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Alferness, R.

V. Ramaswamy, R. Alferness, BTL; private communication.

Burns, W. K.

Collier, R. J.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
[Crossref]

David, S. A.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
[Crossref]

Fukuma, M.

M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49, 3693 (1978).
[Crossref]

Giallorenzi, T. G.

Gregory, R. T.

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 918.

Herriot, D. R.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
[Crossref]

Hsu, H. P.

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 404 (1976).
[Crossref]

Iwasaki, H.

M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49, 3693 (1978).
[Crossref]

Kimura, T.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Marcuse, D.

D. Marcuse, Appl. Opt. 17, 763 (1978).
[Crossref] [PubMed]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Milton, A. F.

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 404 (1976).
[Crossref]

Miyashita, T.

Noda, J.

M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49, 3693 (1978).
[Crossref]

Ramaswamy, V.

R. D. Standley, V. Ramaswamy, Appl. Phys. Lett. 25, 711 (1974).
[Crossref]

V. Ramaswamy, R. Alferness, BTL; private communication.

Saruwatari, M.

Sheem, S. K.

Snyder, A. W.

Stafford, J. W.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
[Crossref]

Standley, R. D.

R. D. Standley, V. Ramaswamy, Appl. Phys. Lett. 25, 711 (1974).
[Crossref]

Uehara, S.

Yamada, J.

Young, D. M.

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 918.

Young, W. R.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. D. Standley, V. Ramaswamy, Appl. Phys. Lett. 25, 711 (1974).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Electron. Lett. (1)

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 404 (1976).
[Crossref]

IEEE Trans. Electron. Devices (1)

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, IEEE Trans. Electron. Devices, ED-22, 385 (1975).
[Crossref]

J. Appl. Phys. (1)

M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49, 3693 (1978).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (4)

V. Ramaswamy, R. Alferness, BTL; private communication.

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 918.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

See Ref. 12, p. 110.

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Figures (9)

Fig. 1
Fig. 1

Rectangular channel waveguide embedded in a substrate and bordering on air.

Fig. 2
Fig. 2

For computational purposes the dielectric waveguide is enclosed in a metal box.

Fig. 3
Fig. 3

Field distribution along the y direction at x = 0 for two different widths of the waveguide and different core index values.

Fig. 4
Fig. 4

Field distributions in x direction for y = 1.2 μm for two different waveguides.

Fig. 5
Fig. 5

Geometry of the step-tapered waveguide.

Fig. 6
Fig. 6

Radiation loss in dB of the step-tapered channel waveguide as a function of taper length L. The dotted portion of the curve is of doubtful accuracy because its high loss exceeds the limits of perturbation theory.

Fig. 7
Fig. 7

Radiation far-field pattern of the channel waveguide averaged over the azimuth angle ψ for different taper lengths L.

Fig. 8
Fig. 8

Radiation far-field pattern of the channel waveguide averaged over the azimuth angle ψ for different taper lengths L.

Fig. 9
Fig. 9

Radiation far-field pattern of the channel waveguide averaged over the azimuth angle ψ for L = 6000 μm.

Equations (19)

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2 ϕ x 2 + 2 ϕ y 2 + { [ n ( x , y ) k ] 2 β g 2 } ϕ = 0 ,
ϕ = n , m = 1 A n m cos [ ( 2 n 1 ) π a x ] sin ( m π b y ) .
( β g 2 n 2 2 k 2 ) A n m = n , m = 1 { 4 a b K n n , m m π 2 [ ( 2 n 1 ) 2 a 2 + m 2 b 2 ] δ n n δ m m } A n m .
K n n , m m = k 2 c / 2 c / 2 d x 0 d d y { [ n 2 ( x , y ) n 2 2 ] × cos [ ( 2 n 1 ) π a x ] cos [ ( 2 n 1 ) π a x ] × sin ( m π b y ) sin ( m π b y ) } .
K A j = x j A j ,
( K y j I ) 1 A j = 1 x j y j A j ,
B = i c i A i ,
( K y j I ) n B = i c i ( x i y j ) n A i c j ( x j y j ) n A j .
E x = 1 n 2 π [ 2 ( n 2 2 k 2 κ 2 ) ω 0 β P ] 1 / 2 sin ( σ y ) exp [ i ( κ x + β z ) ] ,
κ 2 + σ 2 + β 2 = n 2 2 k 2 .
κ = n 2 k sin θ cos ψ ,
σ = n 2 k sin θ sin ψ ,
β = n 2 k cos θ .
Δ P P = | c ( κ , σ ) | 2 d κ d σ = ( n 2 k ) 2 0 π d θ 0 π d ψ | c ( κ , σ ) | 2 cos θ sin θ ,
c ( κ , σ ) = 0 L R exp [ i 0 z ( β g β ) d z ] d z .
R = ω 0 4 ( β β g ) P ( n 2 ) z E * x ϕ d x d y .
( n 2 ) z = Δ ( n 2 ) δ ( z z i ) .
n 1 ( z ) = n 2 + [ n 1 ( o ) n 2 ] c ( o ) c ( z ) .
c ( z i ) = c ( o ) + [ c ( L ) c ( o ) ] ( z i L ) 2

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