Abstract

Propagating beam solutions for optical waveguides can be made to generate such mode-related properties as propagation constants, relative mode powers, and group delays with high precision and considerable flexibility. These quantities are needed in the analysis of optical fiber dispersion. The technique requires the generation of correlation functions from the numerical solutions of a wave equation. These correlation functions are in turn Fourier-transformed with respect to axial distance z. The resulting spectra display sharp resonances corresponding to mode groups, and the positions and heights of these resonances determine the previously mentioned mode properties. The spectral analysis is made highly accurate by the use of line-shape fitting techniques. With this method, mode group delays can be determined to a precision of ±0.12 psec/km using a computation covering a 5-cm propagation path.

© 1980 Optical Society of America

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  1. M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [CrossRef] [PubMed]
  2. M. D. Feit, J. A. Fleck, Appl. Opt. 18, 2843 (1979).
    [CrossRef] [PubMed]
  3. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  4. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  5. J. J. Ramskov Hansen, E. Nicolaisen, Appl. Opt. 17, 2831 (1978).
    [CrossRef]
  6. J. J. Ramskov Hansen, Opt. Quantum Electron. 10, 521 (1978).
    [CrossRef]
  7. R. Olshansky, Appl. Opt. 15, 782 (1976).
    [CrossRef] [PubMed]
  8. S. Choudhary, L. B. Felsen, J. Opt. Soc. Am. 67, 1192 (1977).
    [CrossRef]
  9. J. A. Arnaud, W. Mammel, Electron. Lett. 12, 6 (1976).
    [CrossRef]
  10. J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973).
    [CrossRef]
  11. A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
    [CrossRef]
  12. T. Tanaka, Y. Suematsu, Trans. IECE Jpn. E59, 11 (1976).
  13. D. Marcuse, Appl. Opt. 18, 2078 (1979).
  14. A different form of least-squares fit from the one employed in this paper has been used by G. N. Kamm, J. Appl. Phys. 49, 5951 (1978).
    [CrossRef]
  15. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  16. J. A. Fleck, J. R. Morris, E. S. Bliss, IEEE J. Quantum Electron. QE-14353 (1978).
    [CrossRef]
  17. D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972).
  18. M. D. Feit, Appl. Opt. 18, 2927 (1979).
    [CrossRef] [PubMed]

1979 (3)

1978 (6)

J. J. Ramskov Hansen, Opt. Quantum Electron. 10, 521 (1978).
[CrossRef]

J. J. Ramskov Hansen, E. Nicolaisen, Appl. Opt. 17, 2831 (1978).
[CrossRef]

M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
[CrossRef] [PubMed]

A different form of least-squares fit from the one employed in this paper has been used by G. N. Kamm, J. Appl. Phys. 49, 5951 (1978).
[CrossRef]

J. A. Fleck, J. R. Morris, E. S. Bliss, IEEE J. Quantum Electron. QE-14353 (1978).
[CrossRef]

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[CrossRef]

1977 (1)

1976 (5)

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

R. Olshansky, Appl. Opt. 15, 782 (1976).
[CrossRef] [PubMed]

T. Tanaka, Y. Suematsu, Trans. IECE Jpn. E59, 11 (1976).

J. A. Arnaud, W. Mammel, Electron. Lett. 12, 6 (1976).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

1973 (2)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, W. Mammel, Electron. Lett. 12, 6 (1976).
[CrossRef]

Bliss, E. S.

J. A. Fleck, J. R. Morris, E. S. Bliss, IEEE J. Quantum Electron. QE-14353 (1978).
[CrossRef]

Blok, H.

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[CrossRef]

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973).
[CrossRef]

Choudhary, S.

Dil, J. G.

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973).
[CrossRef]

Feit, M. D.

Felsen, L. B.

Fleck, J. A.

M. D. Feit, J. A. Fleck, Appl. Opt. 18, 2843 (1979).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, E. S. Bliss, IEEE J. Quantum Electron. QE-14353 (1978).
[CrossRef]

M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Gronthoud, A. G.

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[CrossRef]

Kamm, G. N.

A different form of least-squares fit from the one employed in this paper has been used by G. N. Kamm, J. Appl. Phys. 49, 5951 (1978).
[CrossRef]

Keck, D. B.

Mammel, W.

J. A. Arnaud, W. Mammel, Electron. Lett. 12, 6 (1976).
[CrossRef]

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Marcuse, D.

D. Marcuse, Appl. Opt. 18, 2078 (1979).

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

Morris, J. R.

J. A. Fleck, J. R. Morris, E. S. Bliss, IEEE J. Quantum Electron. QE-14353 (1978).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Nicolaisen, E.

Olshansky, R.

Ramskov Hansen, J. J.

J. J. Ramskov Hansen, Opt. Quantum Electron. 10, 521 (1978).
[CrossRef]

J. J. Ramskov Hansen, E. Nicolaisen, Appl. Opt. 17, 2831 (1978).
[CrossRef]

Suematsu, Y.

T. Tanaka, Y. Suematsu, Trans. IECE Jpn. E59, 11 (1976).

Tanaka, T.

T. Tanaka, Y. Suematsu, Trans. IECE Jpn. E59, 11 (1976).

Appl. Opt. (7)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Electron. Lett. (1)

J. A. Arnaud, W. Mammel, Electron. Lett. 12, 6 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. A. Fleck, J. R. Morris, E. S. Bliss, IEEE J. Quantum Electron. QE-14353 (1978).
[CrossRef]

J. Appl. Phys. (1)

A different form of least-squares fit from the one employed in this paper has been used by G. N. Kamm, J. Appl. Phys. 49, 5951 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Quantum Electron. (2)

J. J. Ramskov Hansen, Opt. Quantum Electron. 10, 521 (1978).
[CrossRef]

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[CrossRef]

Opto-electronics (1)

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973).
[CrossRef]

Trans. IECE Jpn. (1)

T. Tanaka, Y. Suematsu, Trans. IECE Jpn. E59, 11 (1976).

Other (1)

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

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Figures (9)

Fig. 1
Fig. 1

Complex field-amplitude spectrum P 1 ( β ) for 1-D square-law refractive medium illuminated by Gaussian beam. Heights of resonant peaks are proportional to power in the corresponding modes. Positions of peaks locate the guide's propagation constants.

Fig. 2
Fig. 2

One-dimensional power-law refractive medium with α = 1.85 illuminated by off-axis Gaussian beam: (a) complex field-amplitude spectrum P 1 ( β ); (b) field frequency derivative spectrum P 2 ( β ). The latter closely resembles the former except for scale factor. The similarity in shape is due to the small variation in β n / ω over the mode set.

Fig. 3
Fig. 3

Group delays vs −β for 1-D power-law refractive medium with α = 1.85, off-axis Gaussian illumination (see Fig. 2). Straight line represents locus of exact values for profile of infinite extent, obtainable either by WKB theory or virial theorem.

Fig. 4
Fig. 4

One-dimensional power-law waveguide with α = 1.85 illuminated by incoherent source. Cladding begins 31.25 μm from center. (a) Plot of complex field-amplitude spectrum P 1 ( β ) vs −β showing guided and leaky modes. (b) Group delays vs −β showing departure from WKB theory for modes near cutoff.

Fig. 5
Fig. 5

Mode spectrum excited in 2-D square-law refractive medium (large-core fiber) by on-axis Gaussian beam.

Fig. 6
Fig. 6

Large-core radius 2-D α = 1.85 fiber illuminated by on-axis Gaussian beam: (a) modal power spectrum P 1 ( β ) vs −β; (b) mode-group delays vs −β showing WKB solution (the straight line).

Fig. 7
Fig. 7

Small-core radius 2-D α = 1.85 fiber illuminated by on-axis Gaussian beam: (a) modal power spectrum P 1 ( β ) vs −β; (b) mode-group delays vs −β showing deviation from WKB solution.

Fig. 8
Fig. 8

Small-core radius 2-D α = 1.85 fiber illuminated by off-axis Gaussian: (a) modal spectrum; (b) mode-group delays vs −β. Heights of spectral peaks determined by least-squares fit.

Fig. 9
Fig. 9

Two-dimensional square-law medium illuminated by on-axis Gaussian (see Fig. 5 for spectrum). Calculations of mode-group delays vs −β with least-squares fit are shown for two separate propagation distances. These enable determination of correction term that scales inversely with propagation distance. Corrected values agree with analytical value to within ±0.12 psec/km.

Tables (3)

Tables Icon

Table I Comparison of Numerically and Analytically Determined Propagation Constants, Relative Mode Powers, and Mode Delays for 1-D Square-Law Fiber Illuminated by Gaussian Beam

Tables Icon

Table II Comparison of Numerically and Analytically Determined Propagation Constants and Mode Delays for a 2-D Square-Law Fiber Illuminated by Gaussian Beam

Tables Icon

Table III Comparison of Single-Resonance and Multiple-Resonance Least-Squares Fit Computations of Mode-Group Delays with Analytical Values for 2-D Square-Law Fibera

Equations (82)

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h ( t , z ) = n W n δ ( t z / υ n ) ,
1 / υ n = ( β n / ω ) ,
2 E x 2 + 2 E y 2 + 2 E z 2 + ω 2 c 2 n 2 ( ω , x , y ) E = 0 ,
E ( ω , x , y , z ) = ( ω , x , y , z ) exp ( ikz ) ,
k = ( n 0 ω ) / c ,
2 z 2 + 2 ik z = 2 + k 2 { [ n ( x , y ) n 0 ] 2 1 } ,
2 ik z = 2 + k 2 { [ n ( x , y ) n 0 ] 2 1 } .
( x , y , z ) = n A n u n ( x ) exp ( i β n z ) ,
( x , y , z ) = n A n u n ( x ) exp ( i β n z ) ,
u n ( x ) u n ( x ) ,
A n = A n ,
β n = ( β n 2 + 2 k β n ) / 2 k ,
β n = k [ 1 ( 1 + 2 β n / k ) 1 / 2 ] .
1 υ n = β n ω = n 0 c { ( 1 + β n k + c n 0 β n ω ) [ 1 + ( 2 β n k ) ] 1 / 2 1 } .
P 1 ( z ) = * ( x , y , 0 ) ( x , y , z ) dxdy = * ( x , y , 0 ) ( x , y , z ) .
( x , y , z ) = n , j A nj u nj ( x , y ) exp ( i β n z ) ,
P 1 ( z ) = n , j | A nj | 2 exp ( i β n z ) .
P 1 ( β ) = n , j | A nj | 2 δ ( β β n ) ,
W n = j | A nj | 2 .
Δ β n = 1 2 Δ β = π / Z ,
ω ( z ) = 1 cos 2 π z Z ,
1 ( β β n ) = 1 Z 0 Z exp [ i ( β β n ) z ] ω ( z ) dz = exp [ i ( β β n ) Z ] 1 i ( β β n ) Z 1 2 ( exp { i [ ( β β n ) Z + 2 π ] } 1 i [ ( β β n ) Z + 2 π ] + exp { i [ β β n ) Z 2 π ] } 1 i [ ( β β n ) Z 2 π ] ) .
P 1 ( β ) n W n 1 ( β β n ) .
P 1 ( β ) W n 1 ( β β n ) ,
W n = P 1 ( β m ) / 1 ( β m β n ) .
P 2 ( z ) = * ( x , y , 0 ) ω ( ω , x , y , z ) dxdy = * ( x , y , 0 ) ω ( ω , x , y , z ) ,
P 1 ( z ) ω = ω * ( x , y , 0 ) ( ω , x , y , z ) = * ( x , y , 0 ) ω ( ω , x , y , z ) .
* ( x , y , 0 ) ω ( ω , x , y , z ) = ω n , j | A nj ( ω ) | 2 exp [ i β n ( ω ) z ] = n , j ω | A nj ( ω ) | 2 exp [ i β n ( ω ) z ] iz n , j β n ω | A nj | 2 exp [ i β n ( ω ) z ] ,
A nj ( ω ) = u nj ( ω , x , y ) ( x , y , 0 ) dxdy .
P 2 ( β ) = 1 Z 0 Z P 2 ( z ) exp ( i β z ) ω ( z ) dz = n W n ω 1 ( β β n ) iZ n β n ω W n 2 ( β β n ) ,
2 ( β β n = exp [ i ( β β n ) Z ] [ 1 i ( β β n ) Z ] 1 ( β β n ) 2 Z 2 1 2 ( exp { i [ ( β β n ) Z + 2 π ] } { 1 i [ ( β β n ) Z + 2 π ] } 1 [ ( β β n ) Z + 2 π ] 2 + exp { i [ ( β β n ) Z 2 π ] } { 1 i [ ( β β n ) Z 2 π ] } 1 [ ( β β n ) Z 2 π ] 2 ) .
P 2 ( β ) = iZ n β n ω W n 2 ( β β n ) .
P 2 ( β ) = iZ β n ω W n 2 ( β β n ) .
β n ω = i P 2 ( β m ) ZW n 2 ( β m β n ) .
P 2 ( β ) = A ( β ) + B ( β ) Z ,
β n ω A Z + B .
( x , y , z + Δ z ) = exp ( ic Δ z 4 n 0 ω 2 ) exp ( i n 0 ω Δ z 2 c { [ n ( x , y n 0 ] 2 1 } ) × exp ( ic Δ z 4 n 0 ω 2 ) ( x , y , z ) + O ( Δ z 3 ) ,
( x , y , z ) = m = N / 2 + 1 N / 2 n = N / 2 + 1 N / 2 mn ( z ) exp [ 2 π i L ( mx + ny ) ] ,
mn ( z + Δ z ) = mn ( z ) exp [ ic Δ z 4 n 0 ω ( 2 π L ) 2 ( m 2 + n 2 ) ] .
T = c Δ z 2 n 0 ω 2 ,
V = n 0 ω Δ z 2 c { [ n ( x , y ) n 0 ] 2 1 } ,
( T ) / ( ω ) = ( T / ω ) ,
( V ) / ( ω ) = V / ω .
( z + Δ z ) ω = exp ( i T 2 ) exp ( iV ) exp ( i T 2 ) [ ( z ) ω + i T 2 ω ( z ) ] i exp ( i T 2 ) V ω exp ( i T 2 ) ( z ) + iT 2 ω ( z + Δ z ) + O ( Δ z ) 3 ,
( z ) ω + iT 2 ω ( z ) .
2 z ω = ic 2 ω n 0 2 ω + i ω n 0 2 c { [ n ( x , y ) n 0 ] 2 1 } ω ic 2 ω 2 n 0 2 + i n 0 2 c { [ n ( x , y ) n 0 ] 2 1 } .
n 2 = { n 1 2 [ 1 2 Δ ( x a ) α ] x a , n 0 2 = ( 1 2 Δ ) n 1 2 x a ,
u n ( x ) = ( π 1 / 2 2 n n ! ) 1 / 2 H n ( x / σ a ) exp ( x 2 / 2 σ a 2 ) ,
σ a = [ a k ( 2 Δ ) 1 / 2 ] 1 / 2 .
β n = Δ n 1 2 n 0 ω c n 1 n 0 ( 2 Δ ) 1 / 2 a ( n + ½ ) ,
A 2 n = [ 2 n ! π 2 2 n 1 ( 1 + b 2 ) ] 1 / 2 ( 1 b 2 1 + b 2 ) n 1 n ! ,
β ω = 2 n 0 c ( α 2 + α ) + β ω ( 2 α 2 + α ) ,
β ( m , n ) = Δ n 1 2 n 0 ω c n 1 n 0 ( 2 Δ ) 1 / 2 a ( m + n + 1 ) .
β n ( cm 1 )
Δ β n ( cm 1 )
A n 2 / A 0 2
β n / ω ( nsec / km )
β n ( cm 1 )
A n 2 / A 0 2
β n / ω
Δ β n = 80.9499
β n ( cm 1 )
β n / ω ( nsec / km )
β n ( cm 1 )
β n / ω ( nsec / km )
δ = ( β β n ) Z / 2 π ,
1 ( δ ) = 1 ( δ ) 1 2 [ 1 ( δ + 1 ) + 1 ( δ 1 ) ] ,
1 ( δ ) = exp ( 2 π i δ ) 1 2 π i δ ,
1 ( 0 ) = 1 .
P 1 ( β ) = W n 1 ( β ) ,
R ( δ ) = P 1 [ ( m + 1 ) Δ β ] P 1 [ ( m 1 ) Δ β ] = 1 ( δ + 1 ) 1 ( δ 1 ) ,
δ = ( β m β n ) Z / 2 π .
δ 2 3 δ + 2 δ 2 + 3 δ + 2 = R .
δ 2 3 r δ + 2 = 0 ,
r = ( 1 + R ) / ( 1 R ) .
δ = [ 3 r + ( 9 r 2 8 ) 1 / 2 2 , R < 1 , 3 r ( 9 r 2 8 ) 1 / 2 2 , R > 1 .
β n = β m 2 π δ Z ,
W n = P 1 ( m Δ β ) / 1 ( δ ) .
2 ( δ ) = 2 ( δ ) ½ [ 2 ( δ + 1 ) + 2 ( δ 1 ) ] ,
2 ( δ ) = exp ( 2 π i δ ) ( 1 2 π i δ ) 1 4 π 2 δ 2 2 ( 0 ) = 1 2 }
P 2 ( β ) = iZ β n ω W n 2 ( δ ) .
β n ω = i P 2 ( m Δ β ) ZW n 2 ( δ ) .

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