Abstract

We discuss the possibility of observing and measuring coupling among the modes of a multimode fiber by using a fiber resonator. If both the injected and spurious modes are simultaneously at resonance, both modes appear at the resonator output and their power ratio is proportional to the power coupling coefficient. This result is based on perturbation theory whose limits of validity are discussed. We also discuss the problem of mode identification and the difficulties of power measurements that are caused by overlapping of modes in the far-field radiation pattern of the resonator.

© 1977 Optical Society of America

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References

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  1. S D Personick, “Time Dispersion in Dielectric Waveguides,” B. S. T. J. 50, 843–859 (1971).
  2. D. Gloge, “Optical Power Flow in Multimode Fibers,” B. S. T. J.,  51, 1767–1783 (1972).
  3. W. A. Gambling, D. N. Payen, and H. Matsumura, “Mode Conversion Coefficients in Optical Fibers,” Appl. Opt. 14, 1538–1542 (1975).
    [Crossref] [PubMed]
  4. J. A. Young and D. Marcuse, “Waveguide Measurements in Multimode Cavities,” in Millimeter Waves, Microwave Research Institute Simposia Series, Vol. 9 (Polytechnic, New York, 1960), pp. 513–533.
  5. R. H. Dicke, “General Microwave Circuit Theorems,” in Principles of Microwave Circuits, Radiation Laboratory Series (McGraw-Hill, New York, 1948), p. 146.
  6. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), p. 104.
  7. Reference 6, p. 179.
  8. Reference 6, p. 63.
  9. D. Marcuse, Light Transmission Optics (Van Nostrand–Rein–hold, New York, 1972), p. 270.
  10. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  11. Reference 6, p. 229.
  12. D. Marcuse, “Excitation of Parabolic-Index Fibers with Incoherent Sources,” B. S. T. J. 54, 1507–1530 (1975).

1975 (2)

W. A. Gambling, D. N. Payen, and H. Matsumura, “Mode Conversion Coefficients in Optical Fibers,” Appl. Opt. 14, 1538–1542 (1975).
[Crossref] [PubMed]

D. Marcuse, “Excitation of Parabolic-Index Fibers with Incoherent Sources,” B. S. T. J. 54, 1507–1530 (1975).

1972 (1)

D. Gloge, “Optical Power Flow in Multimode Fibers,” B. S. T. J.,  51, 1767–1783 (1972).

1971 (2)

S D Personick, “Time Dispersion in Dielectric Waveguides,” B. S. T. J. 50, 843–859 (1971).

D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252–2258 (1971).
[Crossref] [PubMed]

Dicke, R. H.

R. H. Dicke, “General Microwave Circuit Theorems,” in Principles of Microwave Circuits, Radiation Laboratory Series (McGraw-Hill, New York, 1948), p. 146.

Gambling, W. A.

Gloge, D.

D. Gloge, “Optical Power Flow in Multimode Fibers,” B. S. T. J.,  51, 1767–1783 (1972).

D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252–2258 (1971).
[Crossref] [PubMed]

Marcuse, D.

D. Marcuse, “Excitation of Parabolic-Index Fibers with Incoherent Sources,” B. S. T. J. 54, 1507–1530 (1975).

J. A. Young and D. Marcuse, “Waveguide Measurements in Multimode Cavities,” in Millimeter Waves, Microwave Research Institute Simposia Series, Vol. 9 (Polytechnic, New York, 1960), pp. 513–533.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), p. 104.

D. Marcuse, Light Transmission Optics (Van Nostrand–Rein–hold, New York, 1972), p. 270.

Matsumura, H.

Payen, D. N.

Personick, S D

S D Personick, “Time Dispersion in Dielectric Waveguides,” B. S. T. J. 50, 843–859 (1971).

Young, J. A.

J. A. Young and D. Marcuse, “Waveguide Measurements in Multimode Cavities,” in Millimeter Waves, Microwave Research Institute Simposia Series, Vol. 9 (Polytechnic, New York, 1960), pp. 513–533.

Appl. Opt. (2)

B. S. T. J. (3)

S D Personick, “Time Dispersion in Dielectric Waveguides,” B. S. T. J. 50, 843–859 (1971).

D. Gloge, “Optical Power Flow in Multimode Fibers,” B. S. T. J.,  51, 1767–1783 (1972).

D. Marcuse, “Excitation of Parabolic-Index Fibers with Incoherent Sources,” B. S. T. J. 54, 1507–1530 (1975).

Other (7)

Reference 6, p. 229.

J. A. Young and D. Marcuse, “Waveguide Measurements in Multimode Cavities,” in Millimeter Waves, Microwave Research Institute Simposia Series, Vol. 9 (Polytechnic, New York, 1960), pp. 513–533.

R. H. Dicke, “General Microwave Circuit Theorems,” in Principles of Microwave Circuits, Radiation Laboratory Series (McGraw-Hill, New York, 1948), p. 146.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), p. 104.

Reference 6, p. 179.

Reference 6, p. 63.

D. Marcuse, Light Transmission Optics (Van Nostrand–Rein–hold, New York, 1972), p. 270.

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Figures (10)

FIG. 1
FIG. 1

Schematic diagram, of the fiber resonator.

FIG. 2
FIG. 2

Comparison of the perturbation solution with the exact solution. The curves are independent of the mode labels, the fiber length, and the power transmission coefficients of the resonator mirrors.

FIG. 3
FIG. 3

Far-field radiation patterns of step-index fiber modes with μ = 1 for V = 40, n1 = 1.457, and Δ = 0.01.

FIG. 4
FIG. 4

Far-field radiation pattern of step-index fiber modes for μ = 2.

FIG. 5
FIG. 5

Far-field radiation pattern of a mode of higher order, ν = 10, μ = 1. Note that the verticalaxis corresponds to 10P(Θ).

FIG. 6
FIG. 6

Similar to Fig. 5 with ν = 0, μ = 5.

FIG. 7
FIG. 7

Far-field radiation patterns of parabolic-index fiber modes with p = 0. It is again V = 40, n1 = 1.457, and Δ = 0.01.

FIG. 8
FIG. 8

Far-field radiation patterns of parabolic-index fiber modes with p = 1.

FIG. 9
FIG. 9

Far-field radiation pattern of a mode of higher order, p = 4, ν = 0.

FIG. 10
FIG. 10

Far-field radiation pattern of a mode of higher order, p = 0, μ = 10.

Equations (48)

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a j ( ) = r a 1 ( + ) δ 1 j + i t c j ( ) ( 0 ) ,
c j ( + ) ( 0 ) = r c j ( ) ( 0 ) + i t a 1 ( + ) δ 1 j ,
c j ( ) ( L ) = r c j ( + ) ( L ) ,
d j ( + ) = i t c j ( + ) ( L ) .
r 2 = R and t 2 = T ,
R + T = 1 .
d c j ( + ) d z = i β j c j ( + ) + m = 1 N K j m ( + ) c m ( + ) ,
d c j ( ) d z = i β j c j ( ) + m = 1 N K j m ( ) c m ( ) ,
K j m ( ) = K j m ( + ) = K j m ( z )
K j m * = K m j
c j ( + ) ( L ) = e i β j L ( c j ( + ) ( 0 ) + m = 1 N κ j m ( + ) c m ( + ) ( 0 ) ) ,
c j ( ) ( 0 ) = e i β j L ( c j ( ) ( L ) + m = 1 N κ j m ( ) c m ( ) ( L ) ) .
κ j m ( + ) = 0 L K j m ( z ) e i ( β j β m ) z d z
κ j m ( ) = 0 L K j m ( z ) e i ( β j β m ) ( L z ) d z .
( 1 r 2 e 2 i β j L ) d j ( + ) r 2 m = 1 N g j m e i ( β j + β m ) L d m ( + ) = t 2 ( κ j 1 ( + ) + δ j 1 ) a 1 ( + ) e i β j L .
g j m = 2 0 L K j m ( z ) cos [ ( β j β m ) z ] d z
g j m = g m j * .
κ j m ( + ) = 1 2 g j m .
a j ( ) = r [ a 1 ( + ) δ j 1 + e i β j L ( d j ( + ) + m = 1 N κ j m ( ) d m ( + ) ) ] .
κ j m ( ) = 1 2 e i ( β j β m ) L g j m .
d 1 ( + ) = a 1 ( + ) e i β 1 L .
P j P 1 = | d j ( + ) a 1 ( + ) | 2 = ( R | g j 1 | T ) 2 .
h j 1 = 1 L | 0 L K j 1 ( z ) e i ( β 1 β j ) z d z | 2 = 1 L [ ( 0 L K j 1 ( z ) cos [ ( β j β 1 ) z ] d z ) 2 + ( 0 L K j 1 ( z ) sin [ ( β j β 1 ) z ] d z ) 2 ] .
| g j 1 | 2 = 2 L h j 1 .
| g j 1 | / T 1 .
β j = [ ( n 1 k ) 2 ( j π / 2 a ) 2 ] 1 / 2 ,
g j m = ( j m ) q / | j m | 3 ,
| g j 1 | / T < 0.1 .
β j = [ ( n 1 k ) 2 U j 2 / a 2 ] 1 / 2
β p , ν = [ ( n 1 k ) 2 2 ( V / a 2 ) ( 2 p + ν + 1 ) ] 1 / 2
V = n 1 k a ( 2 Δ ) 1 / 2 ,
β j L = m j π ,
( β j β m ) L = ( m j m m ) π .
Q = f d f = ω L T ω / β n 1 k L T .
Δ f = c / 2 n 1 L ,
Δ f / d f = π / T .
Δ f / f = Δ λ / λ 4 π 2 n 1 a 2 / ( | U j 2 U m 2 | λ L ) .
U j = j ( 1 2 π ) .
Δ λ / λ = 16 n 1 a 2 / ( | j 2 m 2 | λ L ) .
Δ λ λ = 2 π 2 n 1 a 2 ( 1 n 2 / n 1 ) [ ( 2 p + ν + 1 ) 2 ( 2 p + ν + 1 ) 2 ] λ L .
P ( Θ ) = 2 U 01 2 ( V 2 U ν μ 2 ) U ν μ 2 ( 1 + δ ν 0 ) ( V 2 U 01 2 ) J ν 2 ( k a Θ ) [ ( k a Θ ) 2 U ν μ 2 ] 2 cos 2 ν ϕ .
k = 2 π / λ free - space propagation constant ,
δ ν μ = { 1 ν = μ , 0 ν μ .
J ν ( U ν μ ) = 0
Θ p = U ν μ / k a = n 1 U ν μ ( 2 Δ ) 1 / 2 / V .
P ( Θ ) = 2 p ! ( 1 + δ ν 0 ) ( p + ν ) ! ( K Θ 2 ) ν × e K Θ 2 [ L p ( ν ) ( K Θ 2 ) ] 2 cos 2 ν ϕ ,
K = n 0 2 k a / n 1 ( 2 Δ ) 1 / 2 = n 0 2 V / 2 n 1 2 Δ ,
n ( r ) = n 1 [ 1 ( r / a ) 2 Δ ] .