Abstract

Excitation coefficients of the guided modes of a parabolic-index optical fiber by narrow input Gaussian beams are calculated. The effects of beam offset, tilt, width, and wave-front curvature are examined. A wave-optical procedure for optimizing the input Gaussian beamwidth (to excite as few mode groups as possible) as a function of beam offset is presented and shown to be in agreement with a simple Fourier-optics optimization.

© 1980 Optical Society of America

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References

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  1. L. Jeunhomme, J. P. Pocholle, Appl. Opt. 17, 463 (1978).
    [CrossRef] [PubMed]
  2. C. N. Kurz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
    [CrossRef]
  3. G. L. Yip, S. Nemoto, IEEE Trans. Microwave Theory Tech. MTT-23, 260 (1975).
    [CrossRef]
  4. M. Imai, E. H. Hara, Appl. Opt. 13, 1893 (1974).
    [CrossRef] [PubMed]
  5. M. Imai, E. H. Hara, Appl. Opt. 14, 169 (1975).
    [PubMed]
  6. D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972).
  7. A. Yariv, Quantum Electronics (Wiley, New York, 1975).

1978

1975

G. L. Yip, S. Nemoto, IEEE Trans. Microwave Theory Tech. MTT-23, 260 (1975).
[CrossRef]

M. Imai, E. H. Hara, Appl. Opt. 14, 169 (1975).
[PubMed]

1974

1969

C. N. Kurz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

Hara, E. H.

Imai, M.

Jeunhomme, L.

Kurz, C. N.

C. N. Kurz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972).

Nemoto, S.

G. L. Yip, S. Nemoto, IEEE Trans. Microwave Theory Tech. MTT-23, 260 (1975).
[CrossRef]

Pocholle, J. P.

Streifer, W.

C. N. Kurz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Yip, G. L.

G. L. Yip, S. Nemoto, IEEE Trans. Microwave Theory Tech. MTT-23, 260 (1975).
[CrossRef]

Appl. Opt.

IEEE Trans. Microwave Theory Tech.

C. N. Kurz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

G. L. Yip, S. Nemoto, IEEE Trans. Microwave Theory Tech. MTT-23, 260 (1975).
[CrossRef]

Other

D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972).

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

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Figures (15)

Fig. 1
Fig. 1

Geometry of the input Gaussian beam.

Fig. 2
Fig. 2

Geometry of excitation of the fiber. Tilt δ is assumed to be in the plane defined by the fiber axis and offset ξ.

Fig. 3
Fig. 3

Power excitation spectra among the degenerate mode groups of a parabolic-index fiber excited by an offset Gaussian beam matched to the fundamental mode. The beam waist is assumed to be located at the fiber input plane. The beam tilt is assumed zero. Identical curves are found for the excitation of the Gaussian-Laguerre and the Gaussian-Hermite modal functions. m and m′ are the group indices, respectively (w = w0 = 4.63 μm).

Fig. 4
Fig. 4

Power excitation coefficients of the HE1,q modes (EH1,q−2 modes) (Gaussian-Laguerre modes) excited by offset input Gaussian beams of zero tilt and wave-front curvature. The beamwidth is matched to the fundamental mode (w = w0 = 4.63 μm).

Fig. 5
Fig. 5

Power excitation coefficients of the HE2,q and HE3,q Gaussian-Laguerre modes excited by offset input Gaussian beams of zero tilt and wave-front curvature. The beamwidth is matched to the fundamental mode (w = w0 = 4.63 μm).

Fig. 6
Fig. 6

Power excitation coefficients of the HE1,q Gaussian-Laguerre modes excited by a tilted input Gaussian beam of zero offset and wave-front curvature. The beamwidth is matched to the fundamental mode (w = w0 = 4.63 μm, δ0 = 3.35°).

Fig. 7
Fig. 7

Power excitation coefficients of the HE2,q and HE3,q Gaussian-Laguerre modes excited by a tilted input Gaussian beam of zero offset and wave-front curvature. The beamwidth is matched to the fundamental mode (w = w0 = 4.63 μm, δ0 = 3.35°).

Fig. 8
Fig. 8

Relative power excitation coefficients of the HEp,1 Gaussian-Laguerre modes excited by an on-axis Gaussian beam of zero tilt as a function of beam wave-front curvature. The abscissa is scaled according to Δϕ = 2w0/R, the beam aperture. The beamwidth at the input plane is constant and matched to the fundamental mode (w = w0 = 4.63 μm).

Fig. 9
Fig. 9

Relative power excitation coefficients of the HEp,1 Gaussian-Laguerre modes excited by an on-axis Gaussian beam of zero tilt and wave-front curvature as a function of beamwidth. The abscissa is scaled to the fundamental mode w0 = 4.63 μm.

Fig. 10
Fig. 10

Optimal input Gaussian beamwidth (to excite as few mode groups as possible) as a function of beam offset. The maximally excited mode group, calculated for matched Gaussian beams, is also shown on the abscissa. The solid line is calculated using wave optics, Eqs. (28) and (29); and the crosses are calculated using Fourier optics, Eqs. (40).

Fig. 11
Fig. 11

Sensitivity of the maximally excited mode group power on small variations of the input Gaussian beamwidth from optimal. m is the group index of the maximally excited mode group.

Fig. 12
Fig. 12

Power excitation spectrum of degenerate mode groups excited by an offset Gaussian beam of zero tilt and wave-front curvature. The excitation spectrum is computed for a matched input Gaussian beam (w = w0), for an optimal input Gaussian beam (w = 0.63w0), and for an input Gaussian beam of width near the optimum (w = 0.5w0). The beam offset ξ = 2.65w0. (The maximally excited mode groups are shown in the figure.)

Fig. 13
Fig. 13

Same as Fig. 12 but with an offset ξ = 3.33w0.

Fig. 14
Fig. 14

Same as Fig. 12 but with an offset ξ = 3.87w0.

Fig. 15
Fig. 15

Power excitation spectrum of the degenerate mode groups excited by an offset and tilted Gaussian beam of zero wave-front curvature. The spectrum is computed for various input beamwidth values. The offset ξ = 1.73w0 and the tilt δ = 4.58°.

Tables (1)

Tables Icon

Table I Maximally Excited Mode Groups

Equations (64)

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Ē t i ( r , ϕ , z ) = ( n 0 Y 0 ) 1 / 2 ( i û r û ϕ ) Ψ p , q i ( r ) × exp [ i ( q ϕ + β p , q i z ) ] ,
H ¯ t i ( r , ϕ , z ) = n 0 Y 0 û z × Ē t i .
n ( r ) 2 = n 0 2 [ 1 2 Δ ( r / a ) 2 ] ;
Ψ p , q i ( r ) = 1 w 0 [ 4 ( p 1 ) ! π e q ( p 1 + q 1 ) ! ] 1 / 2 ( 2 r / w 0 ) q 1 × L p 1 q 1 ( 2 r 2 / w 0 2 ) exp ( r 2 / w 0 2 ) ,
β p , q i = { n 0 2 k 0 2 [ q 1 + 2 ( p 1 ) + 1 ] / w 0 2 } 1 / 2 .
w 0 2 = 2 a / [ k 0 n 0 ( 2 Δ ) 1 / 2 ] .
e q = { 1 , q 1 2 , q = 1 , p , q = 1 , 2 , 3 , . . . ,
E r = + ( n 0 Y 0 ) 1 / 2 Ψ p , q 1 ( r ) cos ( q ϕ ) exp ( i β p , q 1 z ) ,
E ϕ = ( n 0 Y 0 ) 1 / 2 Ψ p , q 1 ( r ) sin ( q ϕ ) exp ( i β p , q 1 z ) ,
H r = ( n 0 Y 0 ) + 1 / 2 Ψ p , q 1 ( r ) sin ( q ϕ ) exp ( i β p , q 1 z ) ,
H ϕ = ( n 0 Y 0 ) + 1 / 2 Ψ p , q 1 ( r ) cos ( q ϕ ) exp ( i β p , q z ) .
E ϕ = ( n 0 Y 0 ) 1 / 2 Ψ p , 2 1 ( r ) exp ( i β p , 2 1 z ) ,
H r = ( n 0 Y 0 ) + 1 / 2 Ψ p , 2 1 ( r ) exp ( i β p , 2 1 z ) ,
E r = ( n 0 Y 0 ) 1 / 2 Ψ p , 2 1 ( r ) exp ( i β p , 2 1 z ) ,
H ϕ = ( n 0 Y 0 ) + 1 / 2 Ψ p , 2 1 ( r ) exp ( i β p , 2 1 z ) .
Ē t in ( r , ϕ , z 1 ) = ( 2 / w ) ( π Y 0 n ) 1 / 2 exp { [ 1 / w 2 + i k 0 / ( 2 R ) ] r 2 2 r ξ cos ϕ + ξ 2 } × exp [ i k 0 δ ( r cos ϕ ξ ) ] û x ,
H ¯ t in ( r , ϕ , z ) = Y 0 n û z × Ê t in , ( k 0 = k 0 n ) ,
w = w 1 { 1 + [ z 1 / ( k 0 w 1 2 ) ] 2 } 1 / 2 ,
R = ( k 0 w 1 2 ) 2 { 1 + [ z 1 / ( k 0 w 1 2 ) ] 2 } / z 1 .
Ē t in = E x û x = E x cos ϕ û r E x sin ϕ û ϕ = E r in û r + E ϕ in û ϕ ,
H ¯ t in = H y û y = H y sin ϕ û r + H y cos ϕ û ϕ = H r in û r + H ϕ in û ϕ .
Ē t in ( r , ϕ , 0 ) = p , q C p , q Ē t p , q ( r , ϕ , 0 ) + Ē R ,
H ¯ t in ( r , ϕ , 0 ) = p , q D p , q H ¯ t p , q ( r , ϕ , 0 ) + H ¯ R ,
C p , q = ( E r in H ϕ p , q * E ϕ in H r p , q * ) r d r d ϕ ,
D p , q = ( E r p , q H ϕ in * E ϕ p , q H r in * ) r d r d ϕ .
C p , q = ( n 0 Y 0 ) 1 / 2 E x Ψ p , q i cos [ ( q 1 ) ϕ ] r d r d ϕ ,
D p , q = ( n 0 Y 0 ) 1 / 2 E x * Ψ p , q i cos [ ( q 1 ) ϕ ] r d r d ϕ .
C p , q = a p , q + i b p , q ,
D p , q = a p , q i b p , q ,
P q , q = ( C p , q Ē p , q ) × ( D p , q H ¯ p , q ) * û z r d r d ϕ = Re ( C p , q D p , q ) = ( a p , q 2 + b p , q 2 ) .
Ē in = ( 2 / w ) ( π Y 0 n 0 ) 1 / 2 exp { [ ( x ξ ) 2 + y 2 ] / w 2 } × exp ( i k 0 δ y ) û x ,
H ¯ in = n 0 Y 0 û z × Ē in ,
C p , q = Ē in × H ¯ p , q û z d x d y = ( n 0 Y 0 ) 1 / 2 E x Ψ p , q ( x , y , 0 ) d x d y ,
D p , q = H ¯ in × Ē p , q û z d x d y = ( n 0 Y 0 ) 1 / 2 H y Ψ p , q ( x , y , 0 ) d x d y .
Ψ p , q ( x , y , z ) = [ w 0 2 π 2 ( p + q ) p ! q ] 1 / 2 H p ( 2 w 0 x ) H q ( 2 w 0 y ) × exp [ ( x 2 + y 2 ) / w 0 2 ] exp ( i β p , q z )
β p , q = [ n 0 2 k 0 2 2 ( 2 p + 2 q + 2 ) / w 0 2 ] 1 / 2 p , q = 0 , 1 , 2 , . . . } .
C p , q = 2 ( w w 0 ) 1 1 + ( w w 0 ) 2 [ ( w 0 2 w 2 ) 2 ( w 0 2 + w 2 ) ] ( p + q ) / 2 ( p ! q ! ) 1 / 2 × H p [ 2 w 0 ξ ( w 0 4 w 4 ) ] H q [ k 0 n 0 δ w 0 w 2 2 ( w 0 4 w 4 ) ] × exp [ ξ 2 ( w 0 2 + w 2 ) k 0 2 n 0 2 δ 2 w 0 2 w 2 4 ( w 0 2 + w 2 ) ] ,
D p , q = C p , q .
P m = p = 0 m 1 C p , q 2 δ q , m p 1 = ( 2 w 0 w ) 2 ( 1 + w 0 2 w 2 ) 2 [ w 0 2 w 2 2 ( w 0 2 + w 2 ) ] m 1 exp [ 2 ξ 2 / ( w 2 + w 0 2 ) ] × p = 0 m 1 [ p ! ( m p 1 ) ! ] 1 H p × [ 2 ξ w 0 ( w 0 4 w 4 ) 1 / 2 ] 2 H m p 1 ( 0 ) 2 .
m max = ξ 2 / w 0 2 + 1 ,
( w 0 2 w 2 ) m 1 H p [ 2 ξ w 0 ( w 0 4 w 4 ) 1 / 2 ] 2 .
β max = n ( ξ ) cos ( δ ) k 0 .
m max = ( n 0 k 0 w 0 / 2 ) 2 ( 1 cos 2 δ ) + ( ξ / w 0 ) 2 cos 2 δ .
I NF ( r , ϕ , z ) = exp [ 2 r 2 / w ( z ) 2 ] ,
I FF ( θ ) = exp [ 2 θ 2 / θ 0 2 ] ,
D 1 ( ξ ) = m / ξ = ( 2 / w 0 ) ξ cos 2 δ ( 2 / w 0 ) ξ ,
D 2 ( δ ) = m / δ = ( n 0 2 k 0 2 w 0 2 / 4 ξ 2 / w 0 2 ) 2 cos δ sin δ ( n 0 2 k 0 2 w 0 2 / 4 ξ 2 / w 0 2 ) 2 δ .
m high = ( n 0 k 0 w 0 / 2 ) 2 ( 1 cos 2 θ 0 ) + ( ξ + w ) 2 / w 0 2 cos 2 θ 0 ( n 0 k 0 w 0 / 2 ) 2 θ 0 2 + ( ξ + w ) 2 / w 0 2 . .
m low = { ( ξ w ) 2 / w 0 2 0 } ξ > w , ξ < w .
M = m high m low = ( n 0 k 0 w 0 θ 0 / 2 ) 2 + { 4 ξ w / w 0 2 ( ξ + w ) 2 / w 0 2 } ξ > w , ξ < w .
θ 0 = 2 / ( k w ) = 2 / ( k 0 n 0 w ) .
M ( ξ , w ) = ( w 0 / w ) 2 + { 4 ξ w / w 0 2 ( w + ξ ) 2 / w 0 2 } ξ > w , ξ < w .
M ( ξ , w ) / w = 2 w 0 2 / w 3 + { 4 ξ / w 0 2 2 ( w + ξ ) / w 0 2 } = 0 ξ > w , ξ < w .
w opt = { w 0 ( w 0 / 2 ξ ) 1 / 3 w 0 ( w 0 / ( ξ + w 0 ) ) 1 / 3 } ξ > w , ξ < w .
Ψ p , q 1 ( r , ϕ , z ) = ( 2 p ! w 0 2 π e q + 1 ( p + q ) ! ) 1 / 2 ( 2 r / w 0 ) q L p q ( 2 r 2 / w 0 2 ) × exp ( r 2 / w 0 2 ) [ cos ( q ϕ ) sin ( q ϕ ) ] exp ( i β p , q 1 z ) ,
β p , q 1 = [ n 0 2 k 0 2 4 ( 2 p + q + 1 ) / w 0 2 ] 1 / 2 ,
p = p 1 q = q 1 .
Ψ 0 , 2 i = 2 w 0 1 / π exp ( r 2 / w 0 2 ) [ cos ( 2 ϕ ) sin ( 2 ϕ ) ] = 2 w 0 1 / π exp ( r 2 / w 0 2 ) { x 2 / w 0 2 y 2 / w 0 2 2 x y / w 0 2 } i = 1 , i = 2 ,
Ψ 1 , 0 = w 0 1 / π exp ( r 2 / w 0 2 ) ( 1 2 x 2 + 2 y 2 w 0 2 ) ,
Ψ 2 , 0 = w 0 1 / 2 π ( 4 x 2 / w 0 2 ) exp ( r 2 / w 0 2 ) ,
Ψ 0 , 2 = w 0 1 / 2 π ( 4 y 2 / w 0 2 1 ) exp ( r 2 / w 0 2 ) ,
Ψ 1 , 1 = w 0 1 / π 4 x y / w 0 2 exp ( r 2 / w 0 2 ) .
( Ψ 2 , 0 Ψ 0 , 2 Ψ 1 , 1 ) = ( 2 / 2 2 / 2 0 2 / 2 2 / 2 0 0 0 1 ) ( Ψ 1 , 0 Ψ 0 , 2 1 Ψ 0 , 2 2 ) .
( Ψ 1 , 0 Ψ 0 , 1 ) = ( 0 1 1 0 ) ( Ψ 0 , 1 1 Ψ 0 , 1 2 ) .

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