Abstract

Several types of waveguiding structures are known that can support the propagation of a finite number of bound electromagnetic modes. Two such structures are the dielectric slab and the optical fiber. In both structures the electromagnetic field associated with the bound modes extends beyond the central region; that part of the field that penetrates into the surrounding medium is termed evanescent. In this paper we use first-order perturbation theory to treat the effects caused by a surrounding medium with gain on the bound modes of the dielectric slab. A noteworthy effect is the amplification of these bound modes in accordance with formulas we present and which arises by evanescent-wave interaction with the surrounding medium.

© 1972 Optical Society of America

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References

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  1. See, for example, R. E. Collins, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).
  2. R. A. Kaplan, Proc. IEEE 51, 1144 (1963).
    [CrossRef]
  3. R. P. Flam, E. R. Schineller, Proc. IEEE 56, 195 (1968).
    [CrossRef]
  4. R. Shubert, J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).
  5. E. R. Schineller, R. P. Flam, Donald W. Wilmot, J. Opt. Soc. Am. 58, 1171 (1968).
    [CrossRef]
  6. S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).
  7. P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
    [CrossRef]
  8. M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
    [CrossRef]
  9. J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
    [CrossRef]
  10. P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  11. C. J. Koester, IEEE J. Quantum Electron. QE-2, 580 (1966).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), p. 10.
  13. The structure of Eq. (4) can be transformed into exactly that of the one-dimensional Schrödinger equation by allowing kx2 = ∊0μω2 − kz2. Equation (4) then becomes ∂2Ey/∂z2 + {kz2 − [∊0 − ∊(z)]μω2}Ey = 0, which is recognizable as being in the exact form of the one-dimensional Schrödinger equation.
  14. See, for example, J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  15. E. P. Ippen, C. V. Shank, A. Dienes, IEEE J. Quantum Electron. QE-7, 178 (1971).
    [CrossRef]
  16. J. E. Midwinter, IEEE J. Quantum Electron. QE-7, 339 (1971).
    [CrossRef]

1971 (3)

P. K. Tien, Appl. Opt. 10, 2395 (1971).
[CrossRef] [PubMed]

E. P. Ippen, C. V. Shank, A. Dienes, IEEE J. Quantum Electron. QE-7, 178 (1971).
[CrossRef]

J. E. Midwinter, IEEE J. Quantum Electron. QE-7, 339 (1971).
[CrossRef]

1970 (2)

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

1969 (2)

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

1968 (3)

R. P. Flam, E. R. Schineller, Proc. IEEE 56, 195 (1968).
[CrossRef]

R. Shubert, J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

E. R. Schineller, R. P. Flam, Donald W. Wilmot, J. Opt. Soc. Am. 58, 1171 (1968).
[CrossRef]

1966 (1)

C. J. Koester, IEEE J. Quantum Electron. QE-2, 580 (1966).
[CrossRef]

1963 (1)

R. A. Kaplan, Proc. IEEE 51, 1144 (1963).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), p. 10.

Collins, R. E.

See, for example, R. E. Collins, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Dienes, A.

E. P. Ippen, C. V. Shank, A. Dienes, IEEE J. Quantum Electron. QE-7, 178 (1971).
[CrossRef]

Flam, R. P.

Goell, J. E.

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

Harris, J. H.

R. Shubert, J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Ippen, E. P.

E. P. Ippen, C. V. Shank, A. Dienes, IEEE J. Quantum Electron. QE-7, 178 (1971).
[CrossRef]

Kaplan, R. A.

R. A. Kaplan, Proc. IEEE 51, 1144 (1963).
[CrossRef]

Koester, C. J.

C. J. Koester, IEEE J. Quantum Electron. QE-2, 580 (1966).
[CrossRef]

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Martin, R. J.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Midwinter, J. E.

J. E. Midwinter, IEEE J. Quantum Electron. QE-7, 339 (1971).
[CrossRef]

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

Schineller, E. R.

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Shank, C. V.

E. P. Ippen, C. V. Shank, A. Dienes, IEEE J. Quantum Electron. QE-7, 178 (1971).
[CrossRef]

Shubert, R.

R. Shubert, J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

Standley, R. D.

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

Stratton, J. A.

See, for example, J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tien, P. K.

P. K. Tien, Appl. Opt. 10, 2395 (1971).
[CrossRef] [PubMed]

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Wilmot, Donald W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), p. 10.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Bell Syst. Tech. J. (1)

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

IEEE J. Quantum Electron. (3)

C. J. Koester, IEEE J. Quantum Electron. QE-2, 580 (1966).
[CrossRef]

E. P. Ippen, C. V. Shank, A. Dienes, IEEE J. Quantum Electron. QE-7, 178 (1971).
[CrossRef]

J. E. Midwinter, IEEE J. Quantum Electron. QE-7, 339 (1971).
[CrossRef]

IEEE Trans. (1)

R. Shubert, J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

J. Opt. Soc. Am. (1)

Proc. IEEE (3)

R. A. Kaplan, Proc. IEEE 51, 1144 (1963).
[CrossRef]

R. P. Flam, E. R. Schineller, Proc. IEEE 56, 195 (1968).
[CrossRef]

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

Other (4)

See, for example, R. E. Collins, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), p. 10.

The structure of Eq. (4) can be transformed into exactly that of the one-dimensional Schrödinger equation by allowing kx2 = ∊0μω2 − kz2. Equation (4) then becomes ∂2Ey/∂z2 + {kz2 − [∊0 − ∊(z)]μω2}Ey = 0, which is recognizable as being in the exact form of the one-dimensional Schrödinger equation.

See, for example, J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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Figures (8)

Fig. 1
Fig. 1

The spatial variation of the dielectric constant (z) for a dielectric slab whose normal is in the z direction. 1, n1, 2, and n2 are the dielectric constants and refractive indices of the core and cladding layers, respectively. The thickness of the core layer is 2l.

Fig. 2
Fig. 2

Graphical solution for the eigenvalues of the TE modes. The intersections of a circle with the solid lines give the solutions for the even modes and with the dashed lines for the odd modes. The smaller circle is drawn for k1/k2 = 1.01 and 2l = 10−3 cm and the large circle for k1/k2 = 1.01 and 2l = 3 × 10−3 cm; k1 = 2π × 104 cm−1 in both cases. The circle for k1/k2 = 1.1 and 2l = 10−3 cm is of almost the same radius as that of the larger circle.

Fig. 3
Fig. 3

The z variation of the imaginary part of the dielectric constant of the active slab.

Fig. 4
Fig. 4

The ratio of the gain coefficient of a slab mode to the gain coefficient of a plane wave propagating in the active mediums plotted against the mode designation for the even TE modes for four slab geometrics. k1 = 2π × 104 cm−1 in both cases.

Fig. 5
Fig. 5

The gain coefficient ratio of the low-order TE modes of a slab as a function of the product k1l for the index ratio n1/n2 = 1.01.

Fig. 6
Fig. 6

The gain coefficient ratio of the low-order TE modes of a slab as a function of the index ratio n1/n2 for k1l = 2π.

Fig. 7
Fig. 7

The strength of the evanescent-waye component at the boundary relative to the peak of the homogeneous-wave component as a function of the angle θ. K1 = 2π × 104 cm−1.

Fig. 8
Fig. 8

The geometric factor Gn for the low-order TE modes as a function of kzl [see Eq. (B3)].

Equations (31)

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2 E - ( r ) μ E ¨ - grad ( div E ) = 0 ,
2 H - ( r ) μ H ¨ + { grad [ log ( r ) ] } × curl H = 0 ,
( r ) ( z ) .
( 2 E y / z 2 ) + [ ( z ) μ ω 2 - k x 2 ] E y = 0.
z [ 1 ( z ) H y z ] + [ ω 2 - k x 2 ( z ) ] H y = 0.
( E y ) I , III = [ A exp ( K z l ) cos ( k z l ) ] exp ( - K z z ) for z l ( E y ) II = A cos ( k z z ) for z l , where A = [ k z tan ( k z l ) 1 + k z l tan ( k z l ) ] 1 2 = [ K z 1 + K z l ] 1 2 } even modes ( E y ) I , III = ± [ A exp ( K z l ) sin ( k z l ) ] exp ( - K z z ) for z l ( E y ) II = A sin ( k z z ) for z l , where A = [ - k z cot ( k z l ) 1 - k z l cot ( k z l ) ] 1 2 = [ K z 1 + K z l ] 1 2 . } odd modes
and             K z = ( k x 2 - 2 μ ω 2 ) 1 2 k z = ( 1 μ ω 2 - k x 2 ) 1 2 .
- E y * E y d z = 1.
K z l = k z l tan ( k z l ) even modes , K z l = - k z l cot ( k z l ) odd modes ,
( K z l ) 2 + ( k z l ) 2 = ( k 1 2 - k 2 2 ) l 2 ,
( z ) = ( z ) + j ( z ) .
k x ; n 2 = k x ; n 2 + δ n .
δ n = E y ; n j ( z ) μ ω 2 E y ; n ;
δ n = j μ ω 2 [ cos 2 ( k z ; n l ) 1 + k z ; n l tan ( k z ; n l ) ] even modes , δ n = j μ ω 2 [ sin 2 ( k z ; n l ) 1 + k z ; n l cot ( k z ; n l ) ] odd modes .
exp [ - j ( δ n / k x ; n ) x ] ,
α n = [ - j ( δ n / k x ; n ) ] .
α 0 = ( k 0 / ) ,
θ n = sin - 1 [ k z ; n / k 1 ] = sin - 1 [ k z ; n / ( k z ; n 2 + k x ; n 2 ) 1 2 ] .
θ c = ( π / 2 ) - sin - 1 ( n 2 / n 1 ) .
( H y ) I , III = [ A exp ( K z l ) cos ( k z l ) ] exp ( - K z z ) for z l ( H y ) II = A cos ( k z z ) for z l , where A = [ ( 2 / 1 ) k z tan ( k z l ) cos 2 ( k z l ) + ( 2 / 1 ) sin 2 ( k z l ) + ( 2 / 1 ) k z 2 l tan ( k z l ) ] 1 2 } even modes , ( H y ) I , III = [ A exp ( K z l ) sin ( k z l ) ] exp ( - K z z ) for z l , ( H y ) II = A sin ( k z z ) for z l , A = [ - ( 2 / 1 ) k z cos ( k z l ) sin 2 ( k z l ) + ( 2 / 1 ) cos 2 ( k z l ) - ( 2 / 1 ) k z l cot ( k z l ) ] 1 2 } odd modes .
K z l = ( 2 / 1 ) k z l tan ( k z l ) even modes , K z l = ( 2 / 1 ) k z l cot ( k z l ) odd modes .
δ n = j μ ω 2 × [ cos 2 ( k z ; n l ) cos 2 ( k z ; n l ) + ( 2 / 1 ) sin 2 ( k z ; n l ) + ( 2 / 1 ) k z ; n l tan ( k z ; n l ) ] ( even modes )
δ n = j μ ω 2 × [ sin 2 ( k z ; n l ) cos 2 ( k z ; n l ) + ( 2 / 1 ) sin 2 ( k z ; n l ) - ( 2 / 1 ) k z ; n l cot ( k z ; n l ) ] ( odd modes ) .
δ n k x ; n 2 - k x ; n + 1 2 = δ n k z ; n + 1 2 - k z ; n 2 < 1.
k 2 2 < 2 k z ; n Δ k z ; n + ( Δ k z ; n ) 2 k 2 G n ,
G n = cos 2 ( k z ; n l ) 1 + k z ; n l tan ( k z ; n l ) , even TE modes , G n = sin 2 ( k z ; n l ) 1 - k z ; n l cot ( k z ; n l ) , odd TE modes .
G n < 1.
k z ; n ( n - 1 2 ) ( π / 2 l )
( k z ) max = ( k 1 2 - k 2 2 ) 1 2 ,
( n - 1 2 ) max ( 2 l / π ) ( k 1 2 - k 2 2 ) 1 2 ,
( / 2 ) k 2 < 1350 cm - 1 ,

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