Abstract

An accurate numerical method is described for solving the Helmholtz equation for a general class of optical fibers. The method yields detailed information about the spatial and angular properties of the propagating beam as well as the modal propagation constants for the fiber. The method is applied to a practical graded-index fiber under the assumptions of both coherent and incoherent illumination. A spectral analysis of the calculated field shows that leaky modes are lost and steady-state propagating conditions are established over a propagation distance of a fraction of a meter.

© 1978 Optical Society of America

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  1. For a comprehensive survey of research in this field see, for example, Optical Fiber Technology, D. Gloge, Ed. (IEEE Press, New York, 1976).
  2. See also J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  3. Examples of modal theory can be found in D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973); D. Gloge, IEEE Trans. Microwave Theory Tech. MIT-23, 106 (1975); D. Marcuse, Theory of Dielectrical Optical Waveguides (Academic, New York, 1964); and R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  4. EVA Buffered Corguide Fibers Product Bulletin No. 2 (Telecommunication Products Dept., Corning Glass Works, Corning, N.Y. 14830, 1May1976).
  5. See, for example, J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  6. Operator splitting results in a separation of the propagation and phase updating parts of the calculation. Since the operators inside the bracket in Eq. (7) do not commute, splitting must involve an approximation that holds for limited propagation distances Δz. By symmetrizing the splitting, the commutation error is further reduced. For a more complete discussion of this question see the Appendix of Ref. 5.
  7. For a comprehensive review of the application of the Fresnel approximation to problems in nonlinear optics see, for example, J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]
  8. This follows from the sampling theorem; see, for example, E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Engle-wood Cliffs, N.J., 1974), pp. 99–102.
  9. It is more usual to write n as n = n0[1 − Δ(r/a)2] rather than in the form of Eq. (19). However, in the present development, the peak refractive index plays no role whereas the cladding index does.
  10. See, for example, P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  11. It has been shown that beam aberration and an erratic focusing pattern are general consequences of propagation in a nonquadratic lenslike medium. M. D. Feit, J. A. Fleck, J. R. Morris, J. Appl. Phys. 48, 3301 (1977).
    [CrossRef]
  12. S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
    [CrossRef]
  13. E. O. Brigham, Proc. IEEE 61, 141 (1973).
  14. We have expressed the modal z dependences in the form exp(−ikz + iβnz) in order to emphasize the similarity between the waveguide problem and the quantum mechanical problem of a particle in a potential well.
  15. J. M. Blatt, V. F. Weisskoff, Theoretical Nuclear Physics (Wiley, New York, 1952), p. 64.
  16. The short decay length for the leaky mode radiation may seem surprising in the light of previous discussion of the subject. See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1974); and A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [CrossRef]

1977 (1)

It has been shown that beam aberration and an erratic focusing pattern are general consequences of propagation in a nonquadratic lenslike medium. M. D. Feit, J. A. Fleck, J. R. Morris, J. Appl. Phys. 48, 3301 (1977).
[CrossRef]

1976 (1)

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

1975 (1)

For a comprehensive review of the application of the Fresnel approximation to problems in nonlinear optics see, for example, J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

1974 (1)

The short decay length for the leaky mode radiation may seem surprising in the light of previous discussion of the subject. See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1974); and A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

1973 (3)

Examples of modal theory can be found in D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973); D. Gloge, IEEE Trans. Microwave Theory Tech. MIT-23, 106 (1975); D. Marcuse, Theory of Dielectrical Optical Waveguides (Academic, New York, 1964); and R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

E. O. Brigham, Proc. IEEE 61, 141 (1973).

1965 (1)

See, for example, P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Arnaud, J. A.

See also J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Blatt, J. M.

J. M. Blatt, V. F. Weisskoff, Theoretical Nuclear Physics (Wiley, New York, 1952), p. 64.

Brigham, E. O.

E. O. Brigham, Proc. IEEE 61, 141 (1973).

This follows from the sampling theorem; see, for example, E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Engle-wood Cliffs, N.J., 1974), pp. 99–102.

Feit, M. D.

It has been shown that beam aberration and an erratic focusing pattern are general consequences of propagation in a nonquadratic lenslike medium. M. D. Feit, J. A. Fleck, J. R. Morris, J. Appl. Phys. 48, 3301 (1977).
[CrossRef]

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

Fleck, J. A.

It has been shown that beam aberration and an erratic focusing pattern are general consequences of propagation in a nonquadratic lenslike medium. M. D. Feit, J. A. Fleck, J. R. Morris, J. Appl. Phys. 48, 3301 (1977).
[CrossRef]

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

Gloge, D.

Examples of modal theory can be found in D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973); D. Gloge, IEEE Trans. Microwave Theory Tech. MIT-23, 106 (1975); D. Marcuse, Theory of Dielectrical Optical Waveguides (Academic, New York, 1964); and R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

Gordon, J. P.

See, for example, P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Li, T.

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Marburger, J. H.

For a comprehensive review of the application of the Fresnel approximation to problems in nonlinear optics see, for example, J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Marcatili, E. A. J.

Examples of modal theory can be found in D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973); D. Gloge, IEEE Trans. Microwave Theory Tech. MIT-23, 106 (1975); D. Marcuse, Theory of Dielectrical Optical Waveguides (Academic, New York, 1964); and R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Miller, S. E.

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Morris, J. R.

It has been shown that beam aberration and an erratic focusing pattern are general consequences of propagation in a nonquadratic lenslike medium. M. D. Feit, J. A. Fleck, J. R. Morris, J. Appl. Phys. 48, 3301 (1977).
[CrossRef]

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

Snyder, A. W.

The short decay length for the leaky mode radiation may seem surprising in the light of previous discussion of the subject. See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1974); and A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

Tien, P. K.

See, for example, P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Weisskoff, V. F.

J. M. Blatt, V. F. Weisskoff, Theoretical Nuclear Physics (Wiley, New York, 1952), p. 64.

Whinnery, J. R.

See, for example, P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Appl. Phys. (2)

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

The short decay length for the leaky mode radiation may seem surprising in the light of previous discussion of the subject. See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1974); and A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

Bell Syst. Tech. J. (1)

Examples of modal theory can be found in D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973); D. Gloge, IEEE Trans. Microwave Theory Tech. MIT-23, 106 (1975); D. Marcuse, Theory of Dielectrical Optical Waveguides (Academic, New York, 1964); and R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

It has been shown that beam aberration and an erratic focusing pattern are general consequences of propagation in a nonquadratic lenslike medium. M. D. Feit, J. A. Fleck, J. R. Morris, J. Appl. Phys. 48, 3301 (1977).
[CrossRef]

Proc. IEEE (3)

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

E. O. Brigham, Proc. IEEE 61, 141 (1973).

See, for example, P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Prog. Quantum Electron. (1)

For a comprehensive review of the application of the Fresnel approximation to problems in nonlinear optics see, for example, J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Other (8)

This follows from the sampling theorem; see, for example, E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Engle-wood Cliffs, N.J., 1974), pp. 99–102.

It is more usual to write n as n = n0[1 − Δ(r/a)2] rather than in the form of Eq. (19). However, in the present development, the peak refractive index plays no role whereas the cladding index does.

Operator splitting results in a separation of the propagation and phase updating parts of the calculation. Since the operators inside the bracket in Eq. (7) do not commute, splitting must involve an approximation that holds for limited propagation distances Δz. By symmetrizing the splitting, the commutation error is further reduced. For a more complete discussion of this question see the Appendix of Ref. 5.

EVA Buffered Corguide Fibers Product Bulletin No. 2 (Telecommunication Products Dept., Corning Glass Works, Corning, N.Y. 14830, 1May1976).

For a comprehensive survey of research in this field see, for example, Optical Fiber Technology, D. Gloge, Ed. (IEEE Press, New York, 1976).

See also J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

We have expressed the modal z dependences in the form exp(−ikz + iβnz) in order to emphasize the similarity between the waveguide problem and the quantum mechanical problem of a particle in a potential well.

J. M. Blatt, V. F. Weisskoff, Theoretical Nuclear Physics (Wiley, New York, 1952), p. 64.

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Figures (15)

Fig. 1
Fig. 1

Algorithm for solving the Helmholtz equation replaces fiber by a system of lenses. In between lenses field satisfies the Helmholtz equation for a homogeneous medium. Algorithm treats large angle waves accurately and is accurate for small refractive index gradients.

Fig. 2
Fig. 2

Geometry of calculation showing Corning 1151 fiber. At the edge of the cladding, a strong absorber is placed to prevent waves from reentering cladding or reflections from computational mesh boundary.

Fig. 3
Fig. 3

On-axis intensity as a function of axial distance for the case of coherent illumination. Shown are intensity plots over two separate 1-cm path segments.

Fig. 4
Fig. 4

On-axis intensity as a function of axial distance for the coherent illumination case over first 15 cm of propagation.

Fig. 5
Fig. 5

Fractional power contours over two propagation ranges for coherent illumination case. Plotted as functions of axial distance are radii of circles containing 20%, 40%, 60%, and 80% of total beam power at the given z position.

Fig. 6
Fig. 6

Fractional power contours in transverse wavenumber space. Plotted are spectral radii containing specific fractions of spectral power as functions of propagation distance z. Fractions are the same as for Fig. 5. Note that the spectral radii are expressed both in terms of wavenumber and angle in degrees.

Fig. 7
Fig. 7

Percent core power change as a function of propagation distance for the case of coherent illumination.

Fig. 8
Fig. 8

Uncertainty product as a function of position for coherent illumination and power fractions 0.2, 0.4, 0.6, and 0.8. Curves with increasing radius indicate increasing power fraction. The uncertainty product for a Gaussian beam would be constant for a fixed power fraction.

Fig. 9
Fig. 9

Fractional power contours for the incoherent illumination case. Power fractions are 0.2, 0.4, 0.6, and 0.8.

Fig. 10
Fig. 10

Fractional power contours in transverse wavenumber space for the incoherent illumination case. The same information is plotted as in Fig. 6.

Fig. 11
Fig. 11

Uncertainty product ΔrΔκ over first cm of propagation. Power fractions are 0.2, 0.4, 0.6, and 0.8.

Fig. 12
Fig. 12

Percent change in core power as a function of propagation distance for incoherent illumination case.

Fig. 13
Fig. 13

Axial spectrum of coherently excited field over first 2.56 cm of propagation path.

Fig. 14
Fig. 14

Axial spectrum of coherently excited field over window extending from 16.13 cm to 18.58 cm.

Fig. 15
Fig. 15

Axial spectrum of incoherently excited field (a) over window extending from 0.45 cm to 1.06 cm, and (b) over window extending from 5.26 cm to 5.87 cm.

Tables (1)

Tables Icon

Table I Comparison of Modal Eigenvalues for an Infinite Square Law Refractive Index Medium and an Optical Fiber with a Parabolic Graded-Index Profile a

Equations (38)

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2 E x 2 + 2 E y 2 + 2 E z 2 + ω 2 c 2 n 2 ( ω , x , y ) E = 0 ,
E ( x , y , Δ z ) = exp [ ± i Δ z ( 2 + ω 2 c 2 n 2 ) 1 / 2 ] E ( x , y , 0 ) ,
( 2 + ω 2 c 2 n 2 ) 1 / 2 = 2 ( 2 + ω 2 c 2 n 2 ) 1 / 2 + ω c n + ( ω / c ) n .
( 2 + ω 2 c 2 n 2 ) 1 / 2 2 ( 2 + k 2 ) 1 / 2 + k + k + k [ ( n / n 0 ) 1 ] ,
k = ( n 0 ω ) / c .
E ( x , y , z ) = E ( x , y , z ) exp ( i k z ) .
E ( x , y , Δ z ) = exp { i Δ z [ 2 ( 2 + k 2 ) 1 / 2 + k + χ ( x , y ) ] } E ( x , y , 0 ) ,
χ = k [ ( n / n 0 ) 1 ] .
E ( x , y , Δ z ) = exp { i Δ z 2 [ 2 ( 2 + k 2 ) 1 / 2 + k ] } exp ( i Δ z χ ) × exp { i Δ z 2 [ 2 ( 2 + k 2 ) 1 / 2 + k ] } E ( x , y , 0 ) + 0 ( Δ z ) 3 ,
exp { i Δ z [ 2 ( 2 + k 2 ) 1 / 2 + k ] } E ( x , y , 0 )
( 2 x 2 + 2 y 2 + 2 z 2 ω 2 c 2 n 0 2 ) E = 0 ,
E ( x , y , z ) = m = N / 2 + 1 N / 2 n = N / 2 + 1 N / 2 E m n ( z ) exp [ 2 π i L ( m x + n y ) ] ,
E m n ( Δ z ) = E m n ( 0 ) exp { i Δ z [ κ x 2 + κ y 2 ( κ x 2 κ y 2 + k 2 ) 1 / 2 + k ] } ,
κ x = ( 2 π m ) / L , κ y = ( 2 π n ) / L .
E ( x , y ) = exp ( i Δ x χ ) E ( x , y , Δ z ) .
E ( j , l ) = E ( j Δ x , l Δ y ) ,
E m n D = j = 0 N 1 l = 0 N 1 E ( j , l ) exp { 2 π i ( m j + n l ) N } .
θ = sin 1 ( κ x 2 + κ y 2 ) 1 / 2 / k .
N π L = | κ x max | = | Δ κ y max | > k sin θ max .
n = n 0 1 Δ [ 1 Δ ( r a ) 2 ] , r a , n = n 0 , r a ,
E ( x , y , z ) = n E n exp ( i β n z ) u n ( x , y ) .
H ( z ) = 1 / 2 1 / 2 cos 2 π z Z , 0 z Z ,
β ( m , n ) = k Δ + ( n + m + 1 ) ( 2 Δ ) 1 / 2 a 1 ,
2 E z 2 = 2 E ω 2 c 2 n 2 ( x , y ) E ,
E ( x , y , z ) = exp ( ± i A z ) E ( x , y , 0 ) ,
exp ( ± i A z ) = 1 ± i A z 1 2 A 2 z 2 + .
A 2 = 2 + ω 2 c 2 n 2 = 2 + k 2 ( 1 + δ n n 0 ) 2 ,
A 1 = 2 2 k + k ( 1 + δ n n 0 ) ,
A 2 = ( 2 + k 2 ) 1 / 2 + k δ n / n 0 .
A 1 2 = ( 1 + δ n 2 n 0 ) 2 + 2 δ n 2 n 0 + 4 4 k 2 + k 2 ( 1 + δ n n 0 ) 2 .
E ( x , y , 0 ) = E 0 exp [ i ( κ · r ) ] ,
A 2 = κ 2 + k 2 ( 1 + δ n 0 n 0 ) 2 ,
A 1 2 = [ 1 + δ n 0 n 0 ( κ 2 k ) 2 ] κ 2 + k 2 ( 1 + δ n 0 n 0 ) 2 .
δ n 0 n 0 1 , κ k 1.
A 2 2 = 2 + k 2 + k ( 2 + k 2 ) 1 / 2 δ n n 0 + k δ n n 0 ( 2 + k 2 ) 1 / 2 + k 2 ( δ n n 0 ) 2 .
( 2 + k 2 ) 1 / 2 = k ( 1 + 2 2 k 2 1 8 4 k 4 + )
A 2 2 = ( 1 + δ n 2 n 0 ) 2 + 2 δ n 2 n 0 1 8 4 k 2 δ n n 0 1 8 δ n n 0 4 k 2 + k 2 ( 1 + δ n n 0 ) 2 + .
A 2 2 = [ 1 + δ n 0 n 0 + 1 4 δ n 0 n 0 ( κ k ) 2 ] κ 2 + k 2 ( 1 + δ n 0 n 0 ) 2 .

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