Abstract

Coupling from the EH11 fundamental mode of a square, hollow bore waveguide laser to a square electrooptic modulator is treated. The misalignment loss that results from coupling of the laser fundamental mode into higher order modulator modes is calculated. Results are presented for loss from transverse displacement, angular tilt, and axial separation of the two elements. From these results, the alignment tolerances required to minimize loss for an intracavity modulator are found.

© 1976 Optical Society of America

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References

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  1. I. P. Kaminow, E. H. Turner, Appl. Opt. 5, 1612 (1966).
    [CrossRef] [PubMed]
  2. R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
    [CrossRef]
  3. R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
    [CrossRef]
  4. J. J. Degnan, D. R. Hall, IEEE J. Quantum Electron. QE-9, 901 (1973).
    [CrossRef]
  5. D. Marcuse, Bell Syst. Tech. J. 49, 1695 (1970).
  6. M. Imai, E. H. Hara, Appl. Opt. 13, 1893 (1974).
    [CrossRef] [PubMed]
  7. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  8. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]

1974 (1)

1973 (2)

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

J. J. Degnan, D. R. Hall, IEEE J. Quantum Electron. QE-9, 901 (1973).
[CrossRef]

1972 (1)

R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
[CrossRef]

1970 (1)

D. Marcuse, Bell Syst. Tech. J. 49, 1695 (1970).

1969 (1)

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

1966 (2)

Abrams, R. L.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
[CrossRef]

Bridges, W. B.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

Degnan, J. J.

J. J. Degnan, D. R. Hall, IEEE J. Quantum Electron. QE-9, 901 (1973).
[CrossRef]

Hall, D. R.

J. J. Degnan, D. R. Hall, IEEE J. Quantum Electron. QE-9, 901 (1973).
[CrossRef]

Hara, E. H.

Imai, M.

Kaminow, I. P.

Kogelnik, H.

Li, T.

Marcatili, E. A. J.

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

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 49, 1695 (1970).

Turner, E. H.

Appl. Opt. (3)

Bell Syst. Tech. J. (2)

D. Marcuse, Bell Syst. Tech. J. 49, 1695 (1970).

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

IEEE J. Quantum Electron. (3)

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
[CrossRef]

J. J. Degnan, D. R. Hall, IEEE J. Quantum Electron. QE-9, 901 (1973).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Misaligned x′,y′,z′ coordinate system.

Fig. 2
Fig. 2

Coupling loss from tilt.

Fig. 3
Fig. 3

Coupling loss from displacement.

Fig. 4
Fig. 4

Coupling loss from separation.

Tables (1)

Tables Icon

Table I First Few Expansion Coefficients for r = 1.422

Equations (29)

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c t = ( F 1 F 2 ) 1 / 2 2 P .
F 1 = d x d y ( E x H y * - E y H x * ) , F 2 = d x d y ( E x * H y - E y * H y ) .
P = 1 2 Re d x d y ( E x H y * - E y H x * )
E x = 0 , E y = E 1 cos ( π x / 2 a ) cos ( π y / 2 a ) exp [ - i ( k 1 z - ω t ) ] , H x = - ( n 1 / Z 0 ) E y , H y = 0 ,
P = ( n 1 a 2 E 1 2 ) / 2 Z 0 .
E y ( x , y ) = a E 0 m , n A m n ψ m n ( x , y )
ψ m n = 1 ( 2 m m ! ) 1 / 2 1 ( 2 n n ! ) 1 / 2 ( 2 π ) 1 / 2 1 w 0 · H m [ ( 2 ) 1 / 2 x / w 0 ] H n [ ( 2 ) 1 / 2 y / w 0 ] · exp [ - ( x 2 + y 2 ) / w 0 2 ] ,
- + d x - + d y ψ m n ( x , y ) ψ m n ( x , y ) = δ m m δ n n .
A m n = 1 ( 2 m m ! ) 1 / 2 1 ( 2 n n ! ) 1 / 2 ( 2 π ) 1 / 2 1 a w 0 · - a + a d x cos ( π x 2 a ) H m [ ( 2 ) 1 / 2 x w 0 ] exp ( - x 2 / w 0 2 ) · - a + a d y cos ( π y 2 a ) H n [ ( 2 ) 1 / 2 y w 0 ] exp ( - y 2 / w 0 2 ) .
I m = 2 0 1 d u cos ( π u / 2 ) H m [ ( 2 ) 1 / 2 r u ] exp ( - r 2 u 2 ) ,
A m n = 1 ( 2 m m ! ) 1 / 2 1 ( 2 n n ! ) 1 / 2 ( 2 π ) 1 / 2 r I m ( r ) I n ( r ) .
d A 00 d r I 0 ( I 0 + 2 r d I 0 d r ) = - I 0 I 2 2 .
I 0 = - 2 r d I 0 / d r
x = x , y = y cos ϕ + z sin ϕ , z = z cos ϕ - y sin ϕ + s .
τ = | m , n A m n 2 T n | 2 ,
T n ( Φ ) = 2 I n 0 1 d u cos ( u Φ ) cos ( π u 2 ) H n [ ( 2 ) 1 / 2 r u ] exp ( - r 2 u 2 )
L = 1 - τ ( Φ ) / τ ( Φ = 0 ) .
T 0 = 2 I 0 0 1 d u cos ( u Φ ) cos ( π u 2 ) exp ( - r 2 u 2 ) .
T 0 = 1 + Φ 2 4 r I 0 d I 0 d r = 1 - Φ 2 8 r 2 + O ( Φ 4 ) .
L 0 = 1 - τ 0 ( Φ ) / τ 0 ( Φ = O ) = ( k 0 a ϕ / 2 r ) 2 .
x = x , y = y + d , z = z + s .
τ = | m , n A m n 2 D n | 2 ,
D n ( Δ ) = 1 I n - 1 + 1 d u cos ( π u 2 ) H n [ ( 2 ) 1 / 2 ( r u + Δ ) ] exp [ - ( r u + Δ ) 2 ]
D 0 = 1 I 0 - 1 + 1 d u cos ( π u 2 ) exp [ - ( r u + Δ ) 2 ] .
D 0 = 1 - Δ 2 - r Δ 2 I 0 d I 0 d r = 1 - Δ 2 d r = 1 - Δ 2 2 + O ( Δ 4 ) .
ψ m n = 1 ( 2 m m ! ) 1 / 2 1 ( 2 n n ! ) 1 / 2 ( 2 π ) 1 / 2 1 w H m [ ( 2 ) 1 / 2 x w ] × H n [ ( 2 ) 1 / 2 y w ] · exp [ - ( x 2 + y 2 w 2 ) ( 1 + i z / b ) ] exp [ - i ( k z - ω t - Φ m n ) ] ,
w = w 0 [ 1 + ( z / b ) 2 ] 1 / 2 , Φ m n = ( m + n + 1 ) tan - 1 ( z / b ) , b = π w 0 2 / λ .
C = | α m , n A m n 2 exp ( i Φ m n ) S m S n | 2 ,
S m = 2 I m 0 1 d u cos ( π u 2 ) H m [ ( 2 ) 1 / 2 α r u ] exp [ ( - α 2 r 2 u 2 ( 1 + i s / b ) ) ] ,

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