Abstract

The effective index method for calculating waveguide mode dispersion is reviewed and applied to uniform rectangular optical waveguides with both small and large index differences. The results are shown to be at least as accurate as other approximate techniques. The effective index method is then applied to channel waveguides assuming 1-D and 2-D diffusion. Channel waveguides without sideways diffusion are shown to be described by the method using a normalized notation and previously published universal dispersion curves. Two-dimensional diffusion theory is applied to treat the case of isotropic sideways diffusion. A new, normalized, 1-D universal chart is obtained which in conjunction with previous results defines waveguide mode dispersion in isotropically diffused 2-D channels.

© 1977 Optical Society of America

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  1. R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
    [CrossRef]
  2. W. Phillips, J. M. Hammer, J. Electron. Mater. 4, 549 (1975).
    [CrossRef]
  3. Y. Ohmachi, J. Noda, Appl. Phys. Lett. 27, 544 (1975) and references quoted therein.
    [CrossRef]
  4. R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976) and references quoted therein.
    [CrossRef]
  5. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  6. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
  7. C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
    [CrossRef]
  8. J. M. Hammer, Appl. Opt. 15, 319 (1976).
    [CrossRef] [PubMed]
  9. H. Kogelnik, V. Ramaswamy, Appl. Opt. 13, 1857 (1974).
    [CrossRef] [PubMed]
  10. G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
    [CrossRef]
  11. R. M. Knox, P. P. Toulios, Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1970).
  12. V. Ramaswamy, Bell Syst. Tech. J. 53, 697 (1974).
  13. G. B. Hocker, IEEE J. Quantum Electron. QE-12, 232 (1976).
    [CrossRef]
  14. I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
    [CrossRef]
  15. H. Kogelnik, Topics in Applied Physics (Springer-Verlag, New York, 1975), Vol. 7, Chap. 2.
    [CrossRef]
  16. H. Furuta, H. Noda, A. Ihaya, Appl. Opt. 13, 322 (1974).
    [CrossRef] [PubMed]
  17. J. Crank, The Mathematics of Diffusion (Oxford U. P., New York, 1975).
  18. The 1-D analog of this equation is derived in standard texts, for example, see Ref. 17.
  19. W. Phillips, J. M. Hammer, “Strip Waveguides and Coupling Modulators of Lithium Niobate–Tantalate,” Paper ThAs, OSA Topical Meeting in Integrated Optics, Salt Lake City, 12–14 January 1976.

1976

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976) and references quoted therein.
[CrossRef]

G. B. Hocker, IEEE J. Quantum Electron. QE-12, 232 (1976).
[CrossRef]

J. M. Hammer, Appl. Opt. 15, 319 (1976).
[CrossRef] [PubMed]

1975

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

W. Phillips, J. M. Hammer, J. Electron. Mater. 4, 549 (1975).
[CrossRef]

Y. Ohmachi, J. Noda, Appl. Phys. Lett. 27, 544 (1975) and references quoted therein.
[CrossRef]

1974

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

H. Furuta, H. Noda, A. Ihaya, Appl. Opt. 13, 322 (1974).
[CrossRef] [PubMed]

H. Kogelnik, V. Ramaswamy, Appl. Opt. 13, 1857 (1974).
[CrossRef] [PubMed]

V. Ramaswamy, Bell Syst. Tech. J. 53, 697 (1974).

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

1969

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

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).

Burns, W. K.

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

Crank, J.

J. Crank, The Mathematics of Diffusion (Oxford U. P., New York, 1975).

Dong, S. B.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Furuta, H.

Goell, J. E.

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).

Hammer, J. M.

J. M. Hammer, Appl. Opt. 15, 319 (1976).
[CrossRef] [PubMed]

W. Phillips, J. M. Hammer, J. Electron. Mater. 4, 549 (1975).
[CrossRef]

W. Phillips, J. M. Hammer, “Strip Waveguides and Coupling Modulators of Lithium Niobate–Tantalate,” Paper ThAs, OSA Topical Meeting in Integrated Optics, Salt Lake City, 12–14 January 1976.

Hocker, G. B.

G. B. Hocker, IEEE J. Quantum Electron. QE-12, 232 (1976).
[CrossRef]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

Ihaya, A.

Kaminow, I. P.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

Knox, R. M.

R. M. Knox, P. P. Toulios, Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1970).

Kogelnik, H.

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976) and references quoted therein.
[CrossRef]

H. Kogelnik, V. Ramaswamy, Appl. Opt. 13, 1857 (1974).
[CrossRef] [PubMed]

H. Kogelnik, Topics in Applied Physics (Springer-Verlag, New York, 1975), Vol. 7, Chap. 2.
[CrossRef]

Marcatili, E. A. J.

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

Noda, H.

Noda, J.

Y. Ohmachi, J. Noda, Appl. Phys. Lett. 27, 544 (1975) and references quoted therein.
[CrossRef]

Ohmachi, Y.

Y. Ohmachi, J. Noda, Appl. Phys. Lett. 27, 544 (1975) and references quoted therein.
[CrossRef]

Oliver, W.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Phillips, W.

W. Phillips, J. M. Hammer, J. Electron. Mater. 4, 549 (1975).
[CrossRef]

W. Phillips, J. M. Hammer, “Strip Waveguides and Coupling Modulators of Lithium Niobate–Tantalate,” Paper ThAs, OSA Topical Meeting in Integrated Optics, Salt Lake City, 12–14 January 1976.

Ramaswamy, V.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

V. Ramaswamy, Bell Syst. Tech. J. 53, 697 (1974).

H. Kogelnik, V. Ramaswamy, Appl. Opt. 13, 1857 (1974).
[CrossRef] [PubMed]

Schmidt, R. V.

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976) and references quoted therein.
[CrossRef]

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

Toulios, P. P.

R. M. Knox, P. P. Toulios, Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1970).

Turner, E. H.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

Yeh, C.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

Y. Ohmachi, J. Noda, Appl. Phys. Lett. 27, 544 (1975) and references quoted therein.
[CrossRef]

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976) and references quoted therein.
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

Bell Syst. Tech. J.

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

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).

V. Ramaswamy, Bell Syst. Tech. J. 53, 697 (1974).

IEEE J. Quantum Electron.

G. B. Hocker, IEEE J. Quantum Electron. QE-12, 232 (1976).
[CrossRef]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

J. Appl. Phys.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

J. Electron. Mater.

W. Phillips, J. M. Hammer, J. Electron. Mater. 4, 549 (1975).
[CrossRef]

Other

R. M. Knox, P. P. Toulios, Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1970).

H. Kogelnik, Topics in Applied Physics (Springer-Verlag, New York, 1975), Vol. 7, Chap. 2.
[CrossRef]

J. Crank, The Mathematics of Diffusion (Oxford U. P., New York, 1975).

The 1-D analog of this equation is derived in standard texts, for example, see Ref. 17.

W. Phillips, J. M. Hammer, “Strip Waveguides and Coupling Modulators of Lithium Niobate–Tantalate,” Paper ThAs, OSA Topical Meeting in Integrated Optics, Salt Lake City, 12–14 January 1976.

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Figures (6)

Fig. 1
Fig. 1

(a) Configuration of the rectangular dielectric waveguide. (b) A 1-D guide with equivalent confinement in the y direction. (c) An equivalent 1-D guide with confinement in the x direction. neff is the effective index of the mode in guide (b).

Fig. 2
Fig. 2

Mode dispersion curves for rectangular waveguides by several methods: — Goell’s computer solutions; - - - - - Marcatili’s analysis; — – — effective index method. In each case n2 = n3 = n4. (a) (n1n2) small, and a = b. (b) (n1n4) small, and a = 2b. (c) n1 = 1.5n4, and a = b.

Fig. 3
Fig. 3

(a) Diffused channel guide of width W, diffusion depth D, with no sideways diffusion. (b) Equivalent 1-D guide with confinement in the x direction. neff is the effective index of a mode in the planar diffused guide.

Fig. 4
Fig. 4

The shape in the x direction of the 2-D diffused index profile. g(2x/W) is defined by Eq. (10c) and is shown for various values of the ratio of strip width to diffusion depth W/D.

Fig. 5
Fig. 5

The shape of the index profile of the equivalent 1-D graded-index guide with confinement in the x direction. h(2x/W) is defined by Eq. (15b) and is plotted for various values of the ratio W/D.

Fig. 6
Fig. 6

The solid lines present a universal chart of the normalized parameters b′ and V′ describing the modes of a channel waveguide with 2-D diffusion. The normalized guide width V′ is defined by Eq. (16b), and the normalized mode effective index b′ is defined by Eq. (16a). bo is the normalized parameter describing a 1-D guide diffused in the same manner. The dashed curves related b′ and V′ with m = 0,1,2 for a guide with no sideways diffusion and where the phase shifts at the reflecting sides are functions of b′. The dot–dash curve for m = 0 represents a guide with no sideways diffusion but with a constant phase shift of −π/2 at the sides, representative of a diffused boundary.

Equations (35)

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V = k D ( n s 2 - n b 2 ) 1 / 2 ,
b = ( n eff 2 - n b 2 ) / ( n s 2 - n b 2 ) ,
V = k W ( n eff 2 - n b 2 ) 1 / 2 = V b 1 / 2 W / D ,
b = ( n eff 2 - n b 2 ) / ( n eff 2 - n b 2 ) .
n eff = [ n b 2 + b b ( n s 2 - n b 2 ) ] 1 / 2 ,
n eff n b + Δ n b b .
k x = k ( n eff 2 - n eff 2 ) 1 / 2 ,
k x b = k ( n eff 2 - n b ) 1 / 2 ,
k x k [ 2 Δ n n b b ( 1 - b ) ] 1 / 2 ,
k x b k ( 2 Δ n n b b b ) 1 / 2 .
C ( x , y ) = M 4 π D t exp [ - x 2 + y 2 / 4 D t ] ,
D = 2 ( D t ) 1 / 2 .
C ( x , y ) = C 1 exp ( - y 2 / D 2 ) - W / 2 W / 2 exp [ - ( x - x ) 2 / D 2 ] d x ,
C ( x , y ) = ½ C 2 exp ( - y 2 / D 2 ) [ erf ( W + 2 x 2 D ) + erf ( W - 2 x 2 D ) ] .
n 2 ( x , y ) = n b 2 + ( n s 2 - n b 2 ) f ( y / D ) g ( 2 x / W ) ,
f ( y / D ) = exp ( - y 2 / D 2 )
g ( 2 x / W ) = ½ { erf [ W 2 D ( 1 + 2 x W ) ] + erf [ W 2 D ( 1 - 2 x W ) ] } .
V ( x ) = k D [ ( n s 2 - n b 2 ) g ( 2 x / W ] 1 / 2 = V o [ g ( 2 x / W ) ] 1 / 2 ,
b ( x ) = n eff ( x ) 2 - n b 2 ( n s 2 - n b 2 ) g ( 2 x / W )
b o = ( n o 2 - n b 2 ) / ( n s 2 - n b 2 ) ,
n eff ( x ) 2 = n b 2 + ( n s 2 - n b 2 ) g ( 2 x / W ) b ( x ) = n b 2 + [ ( n o 2 - n b 2 ) g ( 2 x / W ) b ( x ) ] / b o ,
n eff ( x ) 2 = n b 2 + ( n o 2 - n b 2 ) h ( 2 x / W ) ,
h ( 2 x / W ) = g ( 2 x / W ) b ( x ) / b o .
b = ( n eff 2 - n b 2 ) / ( n o 2 - n b 2 )
V = k W ( n o 2 - n b 2 ) 1 / 2 = V o b o 1 / 2 W / D .
2 k - x t x t [ n eff 2 ( x ) - n eff 2 ] 1 / 2 d x - π 2 - π 2 = 2 m π ,
n eff ( x t ) = n eff .
V 0 u t [ h ( u ) - b ] 1 / 2 d u = ( m + ½ ) π .
V ( 1 - b ) 1 / 2 = m π + 2 tan - 1 [ b / ( 1 - b ) ] 1 / 2 .
V ( 1 - b ) 1 / 2 = ( m + ½ ) π .
n eff = n b + Δ n b o b
k x = k [ n eff 2 ( 0 ) - n eff 2 ] 1 / 2 ,
k x b = k ( n eff 2 - n b 2 ) 1 / 2 .
k x = k { 2 Δ n n b b o [ g ( o ) b ( o ) b o - b ] } 1 / 2 ,
k x b = k ( 2 Δ n n b b o b ) 1 / 2 .

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