Abstract

Mode conversion effects take place when a dielectric optical guide is bent and its curvature varies along the guide. This paper considers and compares such conversion losses of singly and doubly clad slabs (adopted as analogs of corresponding fibers) for single mode transmission. It is shown that the conversion loss of a W-type slab is much smaller than that of a singly clad slab, provided that the beamwidths of both guides are equal. This property is the consequence of the wider mode spacing of the W-guide.

© 1976 Optical Society of America

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References

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  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  2. D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).
  3. M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (February1975).
    [CrossRef]
  4. S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975); Appl. Opt. 15, 1681 (1976).
    [CrossRef] [PubMed]
  5. S. Kawakami, S. Nishida, Trans. Inst. Electron. Commun. Eng. Jpn. 57-C, 304 (September1974).
  6. S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (April1975).
    [CrossRef]
  7. Y. Ohtaka, S. Kawakami, S. Nishida, Trans. Inst. Electron. Comm. Eng. Jpn. 57-C, 187 (June1974).
  8. D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
  9. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 3.5 and 4.6.
  10. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Sec. 31, Chap. 8.
  11. K. Petermann, Electron. Lett. 12, No. 4, 107 (19February1976).
    [CrossRef]

1976 (1)

K. Petermann, Electron. Lett. 12, No. 4, 107 (19February1976).
[CrossRef]

1975 (3)

S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975); Appl. Opt. 15, 1681 (1976).
[CrossRef] [PubMed]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (April1975).
[CrossRef]

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (February1975).
[CrossRef]

1974 (2)

S. Kawakami, S. Nishida, Trans. Inst. Electron. Commun. Eng. Jpn. 57-C, 304 (September1974).

Y. Ohtaka, S. Kawakami, S. Nishida, Trans. Inst. Electron. Comm. Eng. Jpn. 57-C, 187 (June1974).

1971 (1)

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

1969 (2)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

Harris, J. H.

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (February1975).
[CrossRef]

Heiblum, M.

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (February1975).
[CrossRef]

Kawakami, S.

S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975); Appl. Opt. 15, 1681 (1976).
[CrossRef] [PubMed]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (April1975).
[CrossRef]

S. Kawakami, S. Nishida, Trans. Inst. Electron. Commun. Eng. Jpn. 57-C, 304 (September1974).

Y. Ohtaka, S. Kawakami, S. Nishida, Trans. Inst. Electron. Comm. Eng. Jpn. 57-C, 187 (June1974).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 3.5 and 4.6.

Miyagi, M.

Nishida, S.

S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975); Appl. Opt. 15, 1681 (1976).
[CrossRef] [PubMed]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (April1975).
[CrossRef]

S. Kawakami, S. Nishida, Trans. Inst. Electron. Commun. Eng. Jpn. 57-C, 304 (September1974).

Y. Ohtaka, S. Kawakami, S. Nishida, Trans. Inst. Electron. Comm. Eng. Jpn. 57-C, 187 (June1974).

Ohtaka, Y.

Y. Ohtaka, S. Kawakami, S. Nishida, Trans. Inst. Electron. Comm. Eng. Jpn. 57-C, 187 (June1974).

Petermann, K.

K. Petermann, Electron. Lett. 12, No. 4, 107 (19February1976).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Sec. 31, Chap. 8.

Appl. Opt. (1)

Bell Syst. Tech. J. (3)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

Electron. Lett. (1)

K. Petermann, Electron. Lett. 12, No. 4, 107 (19February1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (April1975).
[CrossRef]

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (February1975).
[CrossRef]

Trans. Inst. Electron. Comm. Eng. Jpn. (1)

Y. Ohtaka, S. Kawakami, S. Nishida, Trans. Inst. Electron. Comm. Eng. Jpn. 57-C, 187 (June1974).

Trans. Inst. Electron. Commun. Eng. Jpn. (1)

S. Kawakami, S. Nishida, Trans. Inst. Electron. Commun. Eng. Jpn. 57-C, 304 (September1974).

Other (2)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 3.5 and 4.6.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Sec. 31, Chap. 8.

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Figures (6)

Fig. 1
Fig. 1

Refractive index distribution of slab waveguides: (a) singly clad slab; (b) doubly clad (W-type) slab.

Fig. 2
Fig. 2

Schematic diagram of phase constants of guided and unguided modes.

Fig. 3
Fig. 3

Dependence of the sensitivity function on ν in the single mode and multimode regions. The dotted line in each (a)–(d) indicates the normalized phase constant of the TE0 mode, ν0 = β0/n0k0.

Fig. 4
Fig. 4

Mode conversion loss of singly and doubly clad slabs. The abscissa denotes the correlation length of the curvature [given by Eq. (27)].

Fig. 5
Fig. 5

Junction of curved and straight slabs.

Fig. 6
Fig. 6

Gradually bent slab waveguide.

Tables (1)

Tables Icon

Table I Relevant Parameters for Slab Waveguides

Equations (49)

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C Γ ( s - s ) = Γ ( s ) Γ ( s ) ,
P Γ ( k ) = - C Γ ( s ) exp ( - j k s ) d s ,
exp { - [ α n ( Γ ) + j β n ( Γ ) ] d s } ,
V = ( 1 - a 2 ) 1 / 2 n 0 k 0 T ,
α c , r = 8 ( n 0 k 0 T ) 2 × 0 P Γ [ β 0 - β ( ρ ) ] ( U 0 V sin σ T ) 2 d ( ρ T ) π ( 1 + 1 / W 0 ) [ sin 2 σ T + ( σ / ρ ) 2 cos 2 σ T ] ( W 0 2 + ρ 2 T 2 ) 4 ,
β 0 = [ ( n 0 k 0 ) 2 - ( U 0 / T ) 2 ] 1 / 2 ~ n 0 k 0 - U 0 2 / 2 n 0 k 0 T 2 ,
V = U 0 / cos U 0 , W 0 = ( V 2 - U 0 2 ) 1 / 2 , }
β ( ρ ) = [ ( a n 0 k 0 ) 2 - ρ 2 ] 1 / 2 ~ a n 0 k 0 - ρ 2 / 2 n 0 k 0 ,
σ = [ ρ 2 + ( V / T ) 2 ] 1 / 2 .
α c , n = 8 ( n 0 k 0 T ) 2 ( U 0 U n ) 2 P Γ ( β 0 - β n ) ( 1 + 1 / W 0 ) ( 1 + 1 / W n ) ( U n 2 - U 0 2 ) 4 ,
β n = [ ( n 0 k 0 ) 2 - ( U n / T ) 2 ] 1 / 2 ~ n 0 k 0 - U n 2 / 2 n 0 k 0 T 2 ,
V = U n / sin U n , W n = ( V 2 - U n 2 ) 1 / 2 . }
α c , r / 8 ( n 0 k 0 T ) 2 = ( n 0 k 0 T ) 2 0 a P Γ ( β 0 - β ) S r ( ν ) d ν ,
S r ( ν ) = ( U 0 V sin σ T ) 2 / ρ T π ( 1 + 1 / W 0 ) [ sin 2 σ T + ( σ / ρ ) 2 cos 2 σ T ] ( W 0 2 + ρ 2 T 2 ) 4 ,
ν max = a 1 - V 2 / 2 ( n 0 k 0 T ) 2 .
S r ( ν ) [ K r / ( n 0 k 0 T ) 2 ] δ ( ν - ν p ) ,
K r = ( n 0 k 0 T ) 2 0 a S r ( ν ) d ν .
S g , n ( ν ) = [ K n / ( n 0 k 0 T ) 2 ] δ ( ν - ν n ) ,
K n = ( U 0 U n ) 2 ( 1 + 1 / W 0 ) ( 1 + 1 / W n ) ( U n 2 - U 0 2 ) 4 .
v = ( 1 - b 2 ) 1 / 2 n 0 k 0 T .
β 0 > b n 0 k 0 > β 1
α n 8 ( n 0 k 0 T ) 2 ( U 0 U n ) 2 P Γ ( β 0 - β n ) ( 1 + 1 / W 0 ) ( 1 + 1 / W n ) ( U n 2 - U 0 2 ) 4 ,
β < a n 0 k 0
( b 2 - a 2 ) 1 / 2 ( 1 - b 2 ) - 1 / 2 tanh [ ( b 2 - a 2 ) 1 / 2 × ( 1 - b 2 ) - 1 / 2 δ · v ] = - cot v .
2 x 0 = π T / U 0 .
α s c = 8 ( 4 x 0 n 0 U 0 / λ ) 4 0 a P Γ [ π 2 8 n 0 k 0 x 0 2 W 0 2 + ( ρ T ) 2 U 0 2 ] × S r ( ν ) d ν
α s c 8 ( 4 x 0 n 0 U 0 / λ ) 2 K r P Γ ( π 2 8 n 0 k 0 x 0 2 ν 0 - ν p 1 - ν 0 )
α w = 8 ( 4 x 0 n 0 U 0 / λ ) 2 K 1 P Γ ( π 2 8 n 0 k 0 x 0 2 ν 0 - ν 1 1 - ν 0 ) ,
C Γ ( l ) = ( 1 / R 2 ) exp ( - l 2 / 2 l 0 2 ) .
P Γ ( k ) = ( 2 π ) 1 / 2 l 0 R - 2 exp ( - k 2 l 0 2 / 2 ) .
d 2 ψ d x 2 + 2 ( n 0 k 0 ) 2 R x ψ + [ n 2 ( x ) k 0 2 - β c 2 ] ψ = 0 ,
d 2 E y d x 2 + [ n 2 ( x ) k 0 2 - β 2 ] E y = 0.
ψ = E 0 ( x ) + [ odd a n E n ( x ) + 0 U odd ( ρ ) E odd ( ρ , x ) d ρ ] / R ,
E j ( x ) = [ T ( 1 + 1 / W j ) ] - 1 / 2 × { [ cos ( U j x / T ) sin ( U j x / T ) ] , x < T ( cos U j sin U j ) exp [ - W j ( x / T - 1 ) ] , x > T ,
W j = ( U j tan U j - U j cot U j ) ,
E odd ( ρ , x ) = [ sin 2 σ T + ( σ / ρ ) 2 cos 2 σ T ] - 1 / 2 × { sin σ x ,             x < T [ sin σ T - j ( σ / ρ ) cos σ T ] exp j ρ ( x - T ) + c . c .             x > T ,
σ T = [ ( ρ T ) 2 + V 2 ] 1 / 2 = [ ρ 2 + ( 1 - a 2 ) n 0 2 k 0 2 ] 1 / 2 T .
- E m ( x ) E n ( x ) d x = δ m n ,
- E odd ( ρ , x ) E odd ( ρ , x ) d x = δ ( ρ - ρ ) .
a n = 2 n 0 2 k 0 2 β 0 2 - β n 2 - x E 0 ( x ) E n ( x ) d x = ± 8 ( n 0 k 0 ) 2 T 3 U 0 U n [ ( 1 + 1 / W 0 ) ( 1 + 1 / W n ) ] 1 / 2 ( U n 2 - U 0 2 ) 3
U odd ( ρ ) = 2 n 0 2 k 0 2 β 0 2 - β 2 ( ρ ) - x E 0 ( x ) E odd ( ρ , x ) d x = 8 ( n 0 k 0 ) 2 T 3 U 0 V sin σ T { π ( 1 + 1 / W 0 ) [ sin 2 σ T + ( σ / ρ ) 2 cos 2 σ T ] } 1 / 2 ( W 0 2 + ρ 2 T 2 ) 3 .
r ( x , s 2 ) = s 1 s 2 d s 0 d ρ E odd [ ρ - a n 0 k 0 sin ϕ ( s , s 2 ) , x - d ( s , s 2 ) ] × U odd ( ρ ) ζ ( s ) exp ( - j β 0 s ) exp [ - j β ( ρ ) ( s 2 - s ) ] ,
ϕ ( s , s 2 ) 1 , ρ d ( s , s 2 ) 1
    α β 0 - β ( ρ ) ,
    α ( s 2 - s 1 ) 1.
r ( x , s 2 ) exp ( - j β 0 s 2 ) s 1 s 2 d s × 0 d ρ E odd ( ρ , x ) U odd ( ρ ) ζ ( s ) × exp { - α ( ρ ) + j [ β 0 - β ( ρ ) ] } ( s 2 - s ) .
Δ P = Δ s s 1 s 2 d s s 1 s 2 d s ζ ( s ) ζ ( s ) 2 α ( ρ ) U odd 2 ( ρ ) × exp [ - α ( ρ ) ( 2 s 2 - s - s ) ] exp { j [ β 0 - β ( ρ ) ] ( s - s ) }
P ζ ( k ) = k 2 P Γ ( k ) ,
Δ P / Δ s 0 d ρ - d q U odd 2 ( ρ ) C ζ ( q ) × exp { - α ( ρ ) + j [ β 0 - β ( ρ ) ] } q 0 U odd 2 ( ρ ) P ζ [ β 0 - β ( ρ ) ] d ρ = 0 U odd 2 ( ρ ) [ β 0 - β ( ρ ) ] 2 P Γ [ β 0 - β ( ρ ) ] d ρ ,

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