Abstract

The modal wave propagation in a slab of transversally inhomogeneous complex medium, surrounded by a (generally complex) homogeneous medium, is studied both by an exact method of solution of the wave equation, and by several approximate approaches. Eigenvalues and eigenfunctions are determined for both the fundamental and the first higher-order even mode, in the case of an active parabolic medium, as a function of the “convergence” of the medium. Particular attention is devoted to the so-called no-boundary approximation, which is quite attractive in its simplicity but does not allow the determination of modal gain and losses.

© 1980 Optical Society of America

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References

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  1. R. Pratesi and L. Ronchi, “Thick-film liquid dye lasers,” Opt. Acta 23, 933–954 (1976).
    [Crossref]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  3. H. Kirchhoff, “The solution of Maxwell’s equations for inhomogeneous dielectric slabs,” Arch. Elektr. Uebertrag. 26, 537–541 (1972).
  4. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Ch. 19.
  5. J. Janta and J. Cturoky, “On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides,” Opt. Commun. 25, 49–52 (1978).
    [Crossref]
  6. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953).
  7. A. Consortini, P. Magi, and L. Ronchi, “Transmissione di informazione attraverso fibre ottiche,” Proc. 23rd Intern. Congr. Electron. “Sistemi di Comunicazione a Fibre Ottiche” (Roma, 1976), pp. 357–365.
  8. S. J. Maurer and L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
    [Crossref]
  9. S. Choudary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
    [Crossref]
  10. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [Crossref]
  11. R. E. Langer, “On the asymptotic solutions of ordinary differential equations with reference to the Stoke’s phenomenon about a singular point,” Trans. Am. Math. Soc. 37, 397–416 (1935).
  12. R. W. McKelvey, “Solution about a singular point of a linear differential equation involving a large parameter,” Trans. Am. Math. Soc. 91, 410–424 (1959).
    [Crossref]

1978 (1)

J. Janta and J. Cturoky, “On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides,” Opt. Commun. 25, 49–52 (1978).
[Crossref]

1976 (2)

R. Pratesi and L. Ronchi, “Thick-film liquid dye lasers,” Opt. Acta 23, 933–954 (1976).
[Crossref]

L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
[Crossref]

1974 (1)

S. Choudary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

1972 (1)

H. Kirchhoff, “The solution of Maxwell’s equations for inhomogeneous dielectric slabs,” Arch. Elektr. Uebertrag. 26, 537–541 (1972).

1967 (1)

S. J. Maurer and L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[Crossref]

1959 (1)

R. W. McKelvey, “Solution about a singular point of a linear differential equation involving a large parameter,” Trans. Am. Math. Soc. 91, 410–424 (1959).
[Crossref]

1935 (1)

R. E. Langer, “On the asymptotic solutions of ordinary differential equations with reference to the Stoke’s phenomenon about a singular point,” Trans. Am. Math. Soc. 37, 397–416 (1935).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Ch. 19.

Choudary, S.

S. Choudary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

Consortini, A.

A. Consortini, P. Magi, and L. Ronchi, “Transmissione di informazione attraverso fibre ottiche,” Proc. 23rd Intern. Congr. Electron. “Sistemi di Comunicazione a Fibre Ottiche” (Roma, 1976), pp. 357–365.

Cturoky, J.

J. Janta and J. Cturoky, “On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides,” Opt. Commun. 25, 49–52 (1978).
[Crossref]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953).

Felsen, L. B.

L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
[Crossref]

S. Choudary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

S. J. Maurer and L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[Crossref]

Janta, J.

J. Janta and J. Cturoky, “On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides,” Opt. Commun. 25, 49–52 (1978).
[Crossref]

Kirchhoff, H.

H. Kirchhoff, “The solution of Maxwell’s equations for inhomogeneous dielectric slabs,” Arch. Elektr. Uebertrag. 26, 537–541 (1972).

Langer, R. E.

R. E. Langer, “On the asymptotic solutions of ordinary differential equations with reference to the Stoke’s phenomenon about a singular point,” Trans. Am. Math. Soc. 37, 397–416 (1935).

Magi, P.

A. Consortini, P. Magi, and L. Ronchi, “Transmissione di informazione attraverso fibre ottiche,” Proc. 23rd Intern. Congr. Electron. “Sistemi di Comunicazione a Fibre Ottiche” (Roma, 1976), pp. 357–365.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953).

Marcuse, D.

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

Maurer, S. J.

S. J. Maurer and L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[Crossref]

McKelvey, R. W.

R. W. McKelvey, “Solution about a singular point of a linear differential equation involving a large parameter,” Trans. Am. Math. Soc. 91, 410–424 (1959).
[Crossref]

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953).

Pratesi, R.

R. Pratesi and L. Ronchi, “Thick-film liquid dye lasers,” Opt. Acta 23, 933–954 (1976).
[Crossref]

Ronchi, L.

R. Pratesi and L. Ronchi, “Thick-film liquid dye lasers,” Opt. Acta 23, 933–954 (1976).
[Crossref]

A. Consortini, P. Magi, and L. Ronchi, “Transmissione di informazione attraverso fibre ottiche,” Proc. 23rd Intern. Congr. Electron. “Sistemi di Comunicazione a Fibre Ottiche” (Roma, 1976), pp. 357–365.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Ch. 19.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953).

Arch. Elektr. Uebertrag. (1)

H. Kirchhoff, “The solution of Maxwell’s equations for inhomogeneous dielectric slabs,” Arch. Elektr. Uebertrag. 26, 537–541 (1972).

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. Pratesi and L. Ronchi, “Thick-film liquid dye lasers,” Opt. Acta 23, 933–954 (1976).
[Crossref]

Opt. Commun. (1)

J. Janta and J. Cturoky, “On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides,” Opt. Commun. 25, 49–52 (1978).
[Crossref]

Proc. IEEE (2)

S. J. Maurer and L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[Crossref]

S. Choudary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

Trans. Am. Math. Soc. (2)

R. E. Langer, “On the asymptotic solutions of ordinary differential equations with reference to the Stoke’s phenomenon about a singular point,” Trans. Am. Math. Soc. 37, 397–416 (1935).

R. W. McKelvey, “Solution about a singular point of a linear differential equation involving a large parameter,” Trans. Am. Math. Soc. 91, 410–424 (1959).
[Crossref]

Other (4)

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953).

A. Consortini, P. Magi, and L. Ronchi, “Transmissione di informazione attraverso fibre ottiche,” Proc. 23rd Intern. Congr. Electron. “Sistemi di Comunicazione a Fibre Ottiche” (Roma, 1976), pp. 357–365.

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Ch. 19.

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Figures (7)

FIG. 1
FIG. 1

Re (n0γ) vs η in the case n 0 r = 1.33, ne = 1.5, n 0 i = 10 5 , η ¯ = ± 0.05, and d = 40 λ. Solid lines: exact values. Dashed lines: no-boundary approximation. (a) Fundamental mode m = 0; (b) first higher-order even mode m = 2.

FIG. 2
FIG. 2

Im(n0γ) vs η in the same numerical case as Fig. 1. Solid lines: exact values. Dashed lines: no-boundary approximation for η ¯ = 0.05. Dot-dashed lines: no-boundary approximation for η ¯ = + 0.05.

FIG. 3
FIG. 3

Transverse field configurations of the first two even modes m = 0 and m = 2 for several values of η and η ¯ = ± 0.05. The abscissas represent the normalized transverse coordinate ρ = x/(d/2). Solid lines: exact solution. Dashed lines: no-boundary approximation.

FIG. 4
FIG. 4

Re[xc/(1/2)d] vs η for m = 0 and m = 2. Solid lines: exact values. Dashed lines and dot-dashed lines: WKB approximation Eq. (24). Little circles and black circles: from Eq. (25) for η ¯ = 0.05 and η ¯ = 0.05, respectively. Little crosses and little triangles: from Eq. (26), for η ¯ = 0.05 and η ¯ = + 0.05, respectively.

FIG. 5
FIG. 5

Re(n0γ) vs η for the same numerical values of the parameters as in Fig. 1. Solid lines: exact solutions. Other symbols as in Fig. 4.

FIG. 6
FIG. 6

Im(n0γ) vs η for the same numerical values of the parameters as in Fig. 1. Symbols as in Figs. 4 and 5.

FIG. 7
FIG. 7

Transverse field configurations of the first two even modes m = 0 and m = 2. Solid lines: exact solutions. Dashed lines: WKB approximation Eq. (17) with eigenvalue derived from Eq. (24).

Equations (37)

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2 u ( x , z ) + k 2 n 2 ( x ) u ( x , z ) = 0 ,
n 2 ( x ) = n 0 2 n 2 x 2 ,
n ( x ) = n r ( x ) + i n i ( x )
n r ( x ) = n 0 r [ 1 1 2 η ( x / 1 2 d ) 2 ] n i ( x ) = n 0 i [ 1 η ¯ ( x / 1 2 d ) 2 ]
u ( x , z ) = υ ( x ) exp ( i k n 0 γ z ) ,
υ ( x ) + k 2 [ n 2 ( x ) γ 2 ] υ ( x ) = 0.
u ( x ) = D ν ( X ) + D ν ( X ) ,
ν = 1 / 2 k 2 n 0 2 ( γ 2 1 ) / α 2 , X = α x
α 4 = 4 k 2 n 2 .
υ ( d / 2 ) υ ( d / 2 ) = i k ( n e 2 n 0 2 γ 2 ) 1 / 2 ,
Re ( n e 2 n 0 2 γ 2 ) 1 / 2 0 ,
u ( x , z ) = u 0 exp [ i k n e ( α e x + γ e z ) ]
α D ν ( X ¯ ) D ν ( X ¯ ) D ν ( X ¯ ) + D ν ( X ¯ ) = i k ( n e 2 n 0 2 γ 2 ) 1 / 2 ,
n e = [ n 0 2 γ 2 1 k 2 υ 2 ( d / 2 ) υ 2 ( d / 2 ) ] 1 / 2 .
υ ( x ) = H m ( 2 x / w ) exp ( x 2 / w 2 )
1 / w 2 = ( 1 / 2 ) k n 2 γ 2 = 1 2 m + 1 k n 0 2 n 2 .
n 2 ( x c ) n 0 2 γ 2 = 0 ,
Re x c = ( 1 / 2 ) d .
υ ( x ) = [ S ( x ) ] 1 / 2 [ a e i k S ( x ) + b e i k S ( x ) ] ,
S ( x ) = 0 x [ n 2 ( x ) n 0 2 γ 2 ] 1 / 2 d x
Re x c < ( 1 / 2 ) d ,
υ ( x ) = [ S ( x ) ] 1 / 2 [ a e i k S ( x ) + b e i k S ( x ) ] .
n 2 ( x ) n 0 2 γ 2 ( x x c ) f
f = d d x n 2 ( x c ) .
d 2 υ d x 2 ( k 2 f ) ( x x c ) υ = 0.
υ ( x ) = a ¯ Ai ( Y ) + b ¯ Bi ( Y ) ,
Y = ( k 2 f ) 1 / 3 ( x c x ) .
a = 2 e i ( k ϕ ¯ π / 4 ) cos ( k ϕ ¯ + π / 4 ) , b = e i ( k ϕ ¯ + π / 4 ) cos ( k ϕ ¯ π / 4 ) , a ¯ = 2 ( k / f ) 1 / 6 π cos ( k ϕ ¯ π / 4 ) , b ¯ = 2 ( k / f ) 1 / 6 π cos ( k ϕ ¯ + π / 4 ) ,
ϕ ¯ = 0 x c [ n 2 ( x ) n 0 2 γ 2 ] 1 / 2 d x .
S d tan k S d + 1 2 k S d S d = i [ n e 2 n 0 2 γ 2 ] 1 / 2 ,
a ¯ Ai ( Y d ) + b ¯ Bi ( Y d ) a ¯ Ai ( Y d ) + b ¯ Bi ( Y d ) ( k 2 f ) 1 / 3 = i k [ n e 2 n 0 2 γ 2 ] 1 / 2
S d tan ( k S d + ϕ 0 ) + 1 2 k S d S d = i [ n e 2 n 0 2 γ 2 ] 1 / 2
e 2 i ϕ 0 = a / b = 2 e 2 i k ϕ ¯ tan ( k ϕ ¯ π / 4 ) .
S ( x ) = ( 1 / 2 ) n 0 2 1 γ 2 n 2 ( θ + sin θ cos θ )
sin θ = n 2 x / n 0 ( 1 γ 2 ) 1 / 2 .
x c = n 0 ( 1 γ 2 ) 1 / 2 / n 2 , θ ( x c ) = ( 1 / 2 ) π
ϕ ¯ = ( π / 4 ) n 0 2 ( 1 γ 2 ) / n 2 .