Abstract

The power coupling efficiency of an elliptical-spot-size Gaussian beam into a multimode step-index fiber is derived using a full-wave analysis. Numerical results indicate that the power coupling efficiency increases not only with decreasing index mismatch but also with increasing spot size of the Gaussian beam at its waist and the relative core–cladding refractive-index difference of the fiber. The coupling loss due to beam spread is found to be large when the spot size at the beam waist is small. The coupling loss is also found to increase with increasing wavelength of the Gaussian beam. A slight difference of power coupling efficiency is found with the different direction of electric polarization for a Gaussian beam of large spot size at its waist.

© 1983 Optical Society of America

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References

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  1. D. Marcuse, Bell Syst. Tech. J. 49, 1695 (1970).
  2. J. R. Stern, R. B. Dyott, Electron. Lett. 7, 52 (1971).
    [CrossRef]
  3. M. Imai, E. H. Hara, Appl. Opt. 13, 1893 (1974);Appl. Opt.14, 169 (1975).
    [CrossRef] [PubMed]
  4. M. Mostafavi, T. Itoh, R. Mittra, Appl. Opt. 14, 2190 (1975).
    [CrossRef] [PubMed]
  5. L. G. Cohen, Bell Syst. Tech. J. 51, 573 (1972).
  6. C. A. Brackett, J. Appl. Phys. 45, 2636 (1974).
    [CrossRef]
  7. D. Kato, J. Appl. Phys. 44, 2756 (1973).
    [CrossRef]
  8. D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), pp. 367–371.
  9. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1964), pp. 887–888, 916–919.
  10. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 230–233.

1975

1974

1973

D. Kato, J. Appl. Phys. 44, 2756 (1973).
[CrossRef]

1972

L. G. Cohen, Bell Syst. Tech. J. 51, 573 (1972).

1971

J. R. Stern, R. B. Dyott, Electron. Lett. 7, 52 (1971).
[CrossRef]

1970

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

Brackett, C. A.

C. A. Brackett, J. Appl. Phys. 45, 2636 (1974).
[CrossRef]

Cohen, L. G.

L. G. Cohen, Bell Syst. Tech. J. 51, 573 (1972).

Dyott, R. B.

J. R. Stern, R. B. Dyott, Electron. Lett. 7, 52 (1971).
[CrossRef]

Hara, E. H.

Imai, M.

Itoh, T.

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), pp. 367–371.

Kato, D.

D. Kato, J. Appl. Phys. 44, 2756 (1973).
[CrossRef]

Marcuse, D.

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 230–233.

Mittra, R.

Mostafavi, M.

Stern, J. R.

J. R. Stern, R. B. Dyott, Electron. Lett. 7, 52 (1971).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

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

L. G. Cohen, Bell Syst. Tech. J. 51, 573 (1972).

Electron. Lett.

J. R. Stern, R. B. Dyott, Electron. Lett. 7, 52 (1971).
[CrossRef]

J. Appl. Phys.

C. A. Brackett, J. Appl. Phys. 45, 2636 (1974).
[CrossRef]

D. Kato, J. Appl. Phys. 44, 2756 (1973).
[CrossRef]

Other

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), pp. 367–371.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1964), pp. 887–888, 916–919.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 230–233.

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Figures (5)

Fig. 1
Fig. 1

Geometry of a Gaussian beam coupled into a multimode step-index fiber of core radius a, with w2x as the beam halfwidth at the beam waist along the x axis; only the x-z dimension is shown for convenient illustration.

Fig. 2
Fig. 2

Power coupling efficiency η is shown as a function of fiber relative core–cladding refractive-index difference Δ in percent for different index mismatch Δc. The wavelength of the beam is λ = 0.85 μm, with spot size w2x = 0.4 μm, w2y = 1.2 μm at the beam waist, the fiber core refractive index is n1 = 1.46.

Fig. 3
Fig. 3

Power coupling efficiency η is shown as a function of fiber relative core–cladding refractive-index difference Δ in percent for three different spot sizes of the Gaussian beam at its waist. The wavelength λ of the beam is 0.86 μm, the refractive index n1 of the fiber core is 1.45.

Fig. 4
Fig. 4

Power coupling efficiency η is shown as a function of beam halfwidth w2y at the beam waist for different beam halfwidth w2x at the beam waist. The wavelength of the Gaussian beam is 0.84 μm. The fiber parameters are n1 = 1.45, Δ = 1%, N.A. = 0.205.

Fig. 5
Fig. 5

Power coupling efficiency η is shown as a function of beam wavelength λ for two different beam spot sizes at the beam waist. The fiber parameters are n1 = 1.45, Δ = 1%, N.A. = 0.205.

Equations (31)

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E i t ( x , y , o ) = x ̂ E ( 0 ) exp ( x 2 w 2 x 2 y 2 w 2 y 2 ) ,
E i x ( x , y , z ) = d α E 0 ( α , γ ) exp ( i β 3 z ) exp [ i ( α x + γ y ) ] d γ ,
β 3 = ( k 3 2 α 2 γ 2 ) 1 / 2 , k 3 2 = n 3 2 k 0 2 = ω 2 μ 0 3 , k 0 = ( 2 π ) / λ ;
E 0 ( α , γ ) = 1 4 π 2 d x E i x ( x , y , o ) exp [ i ( α x + γ y ) ] d y .
E 0 ( α , γ ) = E ( 0 ) 4 π w 2 x w 2 y exp [ 1 4 ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) ] .
E i = ( π i e ) + i ω μ 0 π i m × + k 3 2 π i e ,
H i = ( π i m ) i ω 3 π i e × + k 3 2 π i m ,
π i j = d α A j ( α , γ ) exp ( i β 3 z ) exp [ i ( α x + γ y ) ] d γ
E i x = d α [ α β 3 A e ( α , γ ) + ω μ 0 γ A m ( α , γ ) ] × exp [ i ( β 3 z + α x + γ y ) ] d γ ,
E i y = d α [ γ β 3 A e ( α , γ ) + ω μ 0 α A m ( α , γ ) ] × exp [ i ( β 3 z + α x + γ y ) ] d γ .
A e ( α , γ ) = α E 0 ( α , γ ) β 3 ( α 2 + γ 2 ) ,
A m ( α , γ ) = γ E 0 ( α , γ ) ω μ 0 ( α 2 + γ 2 ) .
H i y ( x , y , o ) = d α ( γ β 3 A m + α ω 3 A e ) exp [ i ( α x + γ y ) ] d y = E ( 0 ) w 2 x w 2 y 4 π d α ( γ 2 β 3 ω μ 0 + ω 3 α 2 β 3 ) × exp [ 1 4 ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) + i ( α x + γ y ) ] α 2 + γ 2 d γ ,
π 3 j = d α A j [ exp ( i β 3 z ) + R j exp ( i β 3 z ) ] × exp [ i ( α x + γ y ) ] d γ ;
π 1 j = d α A j T j exp ( i β 1 z ) exp [ i ( α x + γ y ) ] d γ ,
[ π 3 e z ] z = z 0 = [ π 1 e z ] z = z 0 , k 3 2 [ π 3 e ] z = z 0 = k 1 2 [ π 1 e ] z = z 0 .
[ π 3 m z ] z = z 0 = [ π 1 m z ] z = z 0 , [ π 3 m ] z = z 0 = [ π 1 m ] z = z 0 .
T e = 2 β 3 exp [ i ( β 3 β 1 ) z 0 ] n 1 2 β 3 / n 3 2 + β 1 ,
T m = 2 β 3 exp [ i ( β 3 β 1 ) z 0 ] β 3 + β 1 .
E 1 x = E w α A α A d α γ A γ A ( α 2 β 1 n 1 2 β 3 / n 3 2 + β 1 + γ 2 β 3 β 3 + β 1 ) × exp [ g ( α , γ ) ] α 2 + γ 2 d γ ,
E 1 y = E w α A α A d α γ A γ A ( β 1 n 1 2 β 3 / n 3 2 + β 1 β 3 β 3 + β 1 ) × α γ exp [ g ( α , γ ) ] α 2 + γ 2 d γ ,
H 1 x = E w α A α A d α γ A γ A [ β 3 β 1 ω μ 0 ( β 3 + β 1 ) ω 1 n 1 2 β 3 / n 3 2 + β 1 ] × α γ exp [ g ( α , γ ) ] α 2 + γ 2 d γ ,
H 1 y = E w α A α A d α γ A γ A [ γ 2 β 3 β 1 ω μ 0 ( β 3 + β 1 ) + α 2 ω 1 n 1 2 β 3 / n 3 2 + β 1 ] × exp [ g ( α , γ ) ] α 2 + γ 2 d γ ,
E w = E ( 0 ) w 2 x w 2 y 2 π , α A k 1 sin θ 1 , γ A ( α A 2 α 2 ) 1 / 2 , sin θ 1 = ( 1 n 2 2 / n 1 2 ) 1 / 2 ( 2 Δ ) 1 / 2 , Δ 1 n 2 / n 1 , Δ c 1 n 3 / n 1 , n 3 = ( 1 Δ c ) n 1 , g ( α , γ ) ( w 2 2 α 2 + w 2 y 2 γ 2 ) / 4 + i [ α x + γ y + β 1 ( z z 0 ) + β 3 z 0 ] .
P s = ½ R E dx E ix ( x , y , o ) H i y * ( x , y , o ) dy = 2 π 2 | E w | 2 0 k 3 d α 0 ( k 3 2 α 2 ) 1 / 2 ( γ 3 β 3 ω μ 0 + ω 3 α 2 β 3 ) × exp [ ½ ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) ] α 2 + γ 2 d γ ,
P f = ½ R E [ a a d x ( a 2 x 2 ) 1 / 2 ( a 2 x 2 ) 1 / 2 ( E 1 x H 1 y * E 1 y H 1 x * ) d y ] = ½ | E w | 2 R E { α A α A d α γ A γ A ( α 2 β 1 n 1 2 β 3 / n 3 2 + β 1 + γ 2 β 3 β 3 + β 1 ) × exp [ f ( α , γ ) ] α 2 + γ 2 d γ α A α A d α γ A γ A [ γ 2 β 3 β 1 ω μ 0 ( β 3 + β 1 ) + α 2 ω 1 n 1 2 β 3 / n 3 2 + β 1 ] exp [ f * ( α , γ ) ] α 2 + γ 2 d γ α A α A d α γ A γ A × ( β 1 n 1 2 β 3 / n 3 2 + β 1 β 3 β 3 + β 1 ) α γ exp [ f ( α , γ ) ] α 2 + γ 2 d γ × α A α A d α γ A γ A [ β 3 β 1 ω μ 0 ( β 3 + β 1 ) ω 1 n 1 2 β 3 / n 3 2 + β 1 ] × α γ exp [ f * ( α , γ ) ] α 2 + γ 2 d γ } a a d x ( a 2 x 2 ) 1 / 2 ( a 2 x 2 ) 1 / 2 × exp [ i ( α α ) x + i ( γ γ ) y ] d y , ( 19 )
f ( α , γ ) ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) / 4 + i [ β 1 ( z z 0 ) + β 3 z 0 ] , f * ( α , γ ) = g * ( α , γ ) + i ( α x + γ y ) , β j = ( k j 2 α 2 γ 2 ) 1 / 2 , j = 1 , 3 , γ A = ( α A 2 α 2 ) 1 / 2 ,
d x exp [ i ( α α ) x + i ( γ γ ) y ] d y = 4 π 2 δ ( α α ) δ ( γ γ ) ;
P f = 8 π 2 | E w | 2 0 α A d α 0 γ A [ γ 2 β 3 3 β 1 ω μ 0 ( β 3 + β 1 ) 2 + ω 1 β 1 α 2 ( n 1 2 β 3 / n 3 2 + β 1 ) 2 ] × exp [ ½ ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) ] α 2 + γ 2 d γ .
η = P f / P s = 4 0 α A d α 0 γ A [ γ 2 β 3 2 β 1 ω μ 0 ( β 3 + β 1 ) 2 + ω 1 β 1 α 2 ( n 1 2 β 3 / n 3 2 + β 1 ) 2 ] × exp [ ½ ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) ] α 2 + γ 2 d γ / 0 k 3 d α 0 ( k 3 2 α 2 ) 1 / 2 × ( γ 2 β 3 ω μ 0 + ω 3 α 2 β 3 ) exp [ ½ ( w 2 x 2 α 2 + w 2 y 2 γ 2 ) ] α 2 + γ 2 d γ .
w 2 j ( z 0 ) = w 2 j ( 0 ) [ 1 + λ 2 z 0 2 π 2 w 2 j 4 ( 0 ) ] 1 / 2 , j = x , y .

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