Abstract

For a general class of graded-index fiber profiles analytical expressions for propagation constants of guided modes have been evaluated by the evanescent field theory to the order 0(k−9) in asymptotic form. The results are compared with the analytical result of the third order perturbation theory. Analytical asymptotic expressions for group delays are also derived. The asymptotic expressions have been applied for a numerical comparison with the WKB theory and the perturbation theory to the third order. Herefrom, we have estimated the accuracy of the WKB calculations of group delays to be approximately 10 psec/km for near parabolic profiles. We show that the evanescent field theory gives a very accurate determination of the propagation constants and group delays of the lower order modes in a near parabolic profile. For profiles which are far from a parabolic shape, we find that the perturbation theory gives wrong results and that the asymptotic error in the evanescent field theory increases. The accuracy of the WKB theory is found to be within the asymptotic error for higher order modes of near-parabolic profiles and all modes of profiles which are far from a parabolic shape.

© 1979 Optical Society of America

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References

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  1. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1562 (1973).
  2. J. J. Ramskov Hansen, E. Nicolaisen, Appl. Opt. 17, 2831 (1978).
    [CrossRef]
  3. S. Choudhary, L. B. Felsen, J. Opt. Soc. Am. 67, 1192 (1977).
    [CrossRef]
  4. J. J. Ramskov Hansen, G. Jacobsen, Appl. Opt. 18, 3 (1979).
    [CrossRef]
  5. R. Tobey, PL/1—Formac Interpreter User’s Reference Manual, IBM 1967/69, 360D-03.3.004.
  6. K. Petermann, Arch. Elektron. Ubertragungstech. 29, 345 (1975).
  7. W. Streifer, C. N. Kurtz, J. Opt. Soc. Am. 57, 775 (1967).
    [CrossRef]
  8. L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958).
  9. L. B. Felsen, E. Navon, J. Opt. Soc. Am. 68, 1408A (1978).
  10. E. Navon, “Guided wave propagation in bounded inhomogeneous media,” Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Polytechnic Institute of N.Y. (June1978).

1979

1978

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

L. B. Felsen, E. Navon, J. Opt. Soc. Am. 68, 1408A (1978).

1977

1975

K. Petermann, Arch. Elektron. Ubertragungstech. 29, 345 (1975).

1973

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1562 (1973).

1967

W. Streifer, C. N. Kurtz, J. Opt. Soc. Am. 57, 775 (1967).
[CrossRef]

Choudhary, S.

Felsen, L. B.

L. B. Felsen, E. Navon, J. Opt. Soc. Am. 68, 1408A (1978).

S. Choudhary, L. B. Felsen, J. Opt. Soc. Am. 67, 1192 (1977).
[CrossRef]

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1562 (1973).

Jacobsen, G.

Kurtz, C. N.

W. Streifer, C. N. Kurtz, J. Opt. Soc. Am. 57, 775 (1967).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958).

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1562 (1973).

Navon, E.

L. B. Felsen, E. Navon, J. Opt. Soc. Am. 68, 1408A (1978).

E. Navon, “Guided wave propagation in bounded inhomogeneous media,” Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Polytechnic Institute of N.Y. (June1978).

Nicolaisen, E.

Petermann, K.

K. Petermann, Arch. Elektron. Ubertragungstech. 29, 345 (1975).

Ramskov Hansen, J. J.

Streifer, W.

W. Streifer, C. N. Kurtz, J. Opt. Soc. Am. 57, 775 (1967).
[CrossRef]

Tobey, R.

R. Tobey, PL/1—Formac Interpreter User’s Reference Manual, IBM 1967/69, 360D-03.3.004.

Appl. Opt.

Arch. Elektron. Ubertragungstech.

K. Petermann, Arch. Elektron. Ubertragungstech. 29, 345 (1975).

Bell Syst. Tech. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1562 (1973).

J. Opt. Soc. Am.

W. Streifer, C. N. Kurtz, J. Opt. Soc. Am. 57, 775 (1967).
[CrossRef]

L. B. Felsen, E. Navon, J. Opt. Soc. Am. 68, 1408A (1978).

S. Choudhary, L. B. Felsen, J. Opt. Soc. Am. 67, 1192 (1977).
[CrossRef]

Other

R. Tobey, PL/1—Formac Interpreter User’s Reference Manual, IBM 1967/69, 360D-03.3.004.

E. Navon, “Guided wave propagation in bounded inhomogeneous media,” Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Polytechnic Institute of N.Y. (June1978).

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958).

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Figures (5)

Fig. 1
Fig. 1

Index profiles. Profiles 1 and 2 are given by Eq. (1) with the values of n0, a0, and a1 stated in the text.

Fig. 2
Fig. 2

Normalized propagation constant of mode LP00 for an index profile with α = 2.5. The abscissa axis gives the order in 1/k in the calculations by the evanescent field theory and the order of the perturbation plus one for calculations by the perturbation theory.

Fig. 3
Fig. 3

Normalized propagation constant of mode LP50 for an index profile with α = 2.5. Abscissa axis as in Fig. 2.

Fig. 4
Fig. 4

Normalized propagation constant of mode LP00 for an index profile with α = 4.1. Abscissa axis as in Fig. 2.

Fig. 5
Fig. 5

Normalized propagation constant of mode LP50 for an index profile with α = 4.1. Abscissa axis as in Fig. 2.

Tables (4)

Tables Icon

Table I Normalized Propagation Constant and Group Delay of Mode LP00 for an Index Profile with α = 2.5

Tables Icon

Table II Normalized Propagation Constant and Group Delay of Mode LP50 for an Index Profile with α = 2.5

Tables Icon

Table III Normalized Propagation Constant and Group Delay of Mode LP00 for an Index Profile with α = 4.1

Tables Icon

Table IV Normalized Propagation Constant and Group Delay of Mode LP50 for an Index Profile with α = 4.1

Equations (30)

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n 2 ( r ) = n 0 2 - a 0 2 r 2 ( 1 + a 1 r 2 ) 2 ;             0 r ,
U μ ν ( r , θ , z ) = [ m = 0 A m ( r ) k m ] exp { i k [ j = 0 p j z k i + i I ( r ) ] } exp ( i ν θ )
β μ ν = k i = 0 p i k i .
β μ ν 2 = k 2 i = 0 B i k i ,
B i j = 0 i p j p i - j ,
τ = 1 c ( n 0 2 + j = 1 2 - j 2 B j / k j ) ( n 0 2 + j = 1 B j k j ) - 1 / 2 ,
β μ ν 2 = n 0 2 k 2 - 2 a 0 k ( 2 μ + ν + 1 ) .
β ( N ) 2 k 2 i = 0 N B i k i .
r 1 r 2 [ n 2 ( r ) k 2 - β μ ν 2 - ν 2 r 2 ] 1 / 2 d r = ( μ + ½ ) π ,
τ μ ν = 1 c r 1 r 2 k n 2 ( r ) / V ( r ) d r r 1 r 2 β μ ν / V ( r ) d r ,
V ( r ) = [ n 2 ( r ) k 2 - β μ ν 2 - ν 2 r 2 ] 1 / 2 .
U μ ν ( r , θ , z ) = ( k a 0 r 2 ) ν / 2 L μ ν ( k a 0 r 2 ) × exp { i [ β μ ν ( 0 ) z + i k a 0 r 2 / 2 ] } exp ( i ν θ ) ,
β μ ν ( 0 ) 2 = n 0 2 k 2 - 2 a 0 k ( 2 μ + ν + 1 ) ,
n 2 ( r ) = n 0 2 - a 0 2 r 2 + δ [ n 2 ( r ) ] ,
β μ ν ( M ) 2 = β μ ν ( 0 ) 2 + i = 1 M Δ β μ ν ( i ) 2 .
Δ β μ ν ( 1 ) 2 = H μ , μ , ν ;
Δ β μ ν ( 2 ) 2 = m μ H μ , m , ν 2 4 a 0 k ( m - μ ) ;
Δ β μ ν ( 3 ) 2 = i μ m μ H μ , i , ν H μ , m , ν H i , m , ν 16 a 0 2 k 2 ( m - μ ) ( i - μ ) - H μ , μ , ν m μ H μ , m , ν 2 16 a 0 2 k 2 ( m - μ ) 2 ,
H μ , m , ν = r = 0 θ = 0 2 π U μ ν δ [ n 2 ( r ) ] k 2 U m ν * r d r d θ r = 0 θ = 0 2 π U μ ν U m ν * r d r d θ ,
δ [ n 2 ( r ) ] = - 2 a 1 a 0 2 r 4 ,
Δ β μ ν ( 1 ) 2 = - a 1 ( 4 + 12 μ + 12 μ ν + 6 ν + 12 μ 2 + 2 ν 2 ) ,
Δ β μ ν ( 2 ) 2 = 4 a 1 2 a 0 k ( 18 μ + 9 ¾ ν + 25 ½ μ ν + 10 ½ μ ν + 25 ½ μ 2 ν + 25 ½ μ 2 + 17 μ 3 + 5 ¼ ν 2 + ν 3 + 4 ½ ) ;
Δ β μ ν ( 3 ) 2 = - 2 a 1 3 a 0 2 k 2 ( 417 μ + 208 ½ ν + 792 μ ν + 442 μ ν 2 + 123 μ ν 3 + 1125 μ 2 ν + 442 μ 2 ν 2 + 750 μ 3 ν + 792 μ 2 + 750 μ 3 + 375 μ 4 + 173 ν 2 + 61 ½ ν 3 + 8 ν 4 + 79 ) .
Profile 1 : n 0 = 1.5 , a 0 = 6.207 × 10 - 3 μ m - 1 , and a 1 = 5.215 × 10 - 4 μ m - 2 ; Profile 2 : n 0 = 1.5 , a 0 = 2.5 × 10 - 3 μ m - 1 ,             and a 1 = 5.0 × 10 - 3 μ m - 2 .
B 0 = n 0 2 ,
B 1 = - 2 a 0 ( 2 μ + ν + 1 ) ,
B 2 = - a 1 ( 4 + 12 μ + 12 μ ν + 6 ν + 12 μ 2 + 2 ν 2 ) ,
B 3 = 3 a 1 2 a 0 ( 16 μ + 8 ν + 24 μ ν + 10 μ ν 2 + 24 μ 2 ν + 24 μ 2 + 16 μ 3 + 5 ν 2 + ν 3 + 4 ) ,
B 4 = - 2 a 1 3 a 0 2 ( 216 μ + 108 ν + 426 μ ν + 282 μ ν 2 + 72 μ ν 3 + 630 μ 2 ν + 282 μ 2 ν 2 + 420 μ 3 ν + 426 μ 2 + 420 μ 3 + 210 μ 4 + 95 ν 2 + 36 ν 3 + 5 ν 4 + 44 ) ,
B 5 = 3 a 1 4 4 a 0 3 ( 7072 μ + 3536 ν + 17040 μ ν + 15280 μ ν 2 + 6240 μ ν 3 + 1126 μ ν 4 + 33120 μ 2 ν + 22320 μ 2 ν 2 + 14880 μ 3 ν 2 + 16080 μ 4 ν + 17040 μ 2 + 22080 μ 3 + 16080 μ 4 + 6432 μ 5 + 3920 ν 2 + 2120 ν 3 + 563 ν 4 + 59 ν 5 + 1232 + 6240 μ 2 ν 3 + 32160 μ 3 ν ) .

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