Abstract

To understand the physical meaning of rational reflection coefficients in inverse-scattering theory for optical waveguide design [ J. Opt. Soc. Am. A 6, 1206 ( 1989)], we studied the relationship between the poles of the transverse reflection coefficient and the modes in inhomogeneous dielectrics. By using a stratified-medium formulation we showed that these poles of the spectral reflection coefficient satisfy the same equation as the guidance condition in inhomogeneous waveguides. Therefore, in terms of wave numbers, the poles are the same as the discrete modes in the waveguide. The radiation modes have continuous real values of transverse wave numbers and are represented by the branch cut on the complex plane. Based on these results, applications of the Gel’fand–Levitan–Marchenko theory to optical waveguide synthesis with the rational function representation of the transverse reflection coefficient are discussed.

© 1992 Optical Society of America

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References

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  1. A. K. Jordan, S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6, 1206–1212 (1989).
    [CrossRef]
  2. S. Yukon, B. Bendow, “Design of waveguides with prescribed propagation constants,”J. Opt. Soc. Am. 70, 172–179 (1980).
    [CrossRef]
  3. S. R. A. Dods, “Bragg reflection waveguide,” J. Opt. Soc. Am. A 6, 1465–1476 (1989).
    [CrossRef]
  4. I. Kay, “The inverse scattering problem,” (New York University, New York, 1955).
  5. A. K. Jordan, H. N. Kritikos, “An application of one-dimensional inverse scattering theory for inhomogeneous regions,”IEEE Trans. Antennas Propag. AP-22, 909–911 (1973).
    [CrossRef]
  6. D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,”IEEE Trans. Antennas Propag. AP-35, 934–946 (1987).
    [CrossRef]
  7. A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
    [CrossRef]
  8. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, New York, 1990), Chap. 3.
  9. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11; also see 2nd ed., 1990.
  10. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), Chaps. 2 and 6.
  11. R. L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 205.
  12. M. E. van Valkenburg, Introduction to Modern Network Synthesis (Wiley, New York, 1967).
  13. D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972).
  14. D. W. Mills, L. S. Tamil, “A new approach to the design of graded-index guided wave devices,”IEEE Microwave Guided Wave Lett. 1, 87–89 (1991).
    [CrossRef]

1991 (1)

D. W. Mills, L. S. Tamil, “A new approach to the design of graded-index guided wave devices,”IEEE Microwave Guided Wave Lett. 1, 87–89 (1991).
[CrossRef]

1989 (2)

1987 (1)

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,”IEEE Trans. Antennas Propag. AP-35, 934–946 (1987).
[CrossRef]

1980 (1)

1979 (1)

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

1973 (1)

A. K. Jordan, H. N. Kritikos, “An application of one-dimensional inverse scattering theory for inhomogeneous regions,”IEEE Trans. Antennas Propag. AP-22, 909–911 (1973).
[CrossRef]

Ahn, S.

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

Bendow, B.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), Chaps. 2 and 6.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11; also see 2nd ed., 1990.

Dods, S. R. A.

Frangos, P. V.

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,”IEEE Trans. Antennas Propag. AP-35, 934–946 (1987).
[CrossRef]

Jaggard, D. L.

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,”IEEE Trans. Antennas Propag. AP-35, 934–946 (1987).
[CrossRef]

Jordan, A. K.

A. K. Jordan, S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6, 1206–1212 (1989).
[CrossRef]

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

A. K. Jordan, H. N. Kritikos, “An application of one-dimensional inverse scattering theory for inhomogeneous regions,”IEEE Trans. Antennas Propag. AP-22, 909–911 (1973).
[CrossRef]

Kay, I.

I. Kay, “The inverse scattering problem,” (New York University, New York, 1955).

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, New York, 1990), Chap. 3.

Kritikos, H. N.

A. K. Jordan, H. N. Kritikos, “An application of one-dimensional inverse scattering theory for inhomogeneous regions,”IEEE Trans. Antennas Propag. AP-22, 909–911 (1973).
[CrossRef]

Lakshmanasamy, S.

Liboff, R. L.

R. L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 205.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972).

Mills, D. W.

D. W. Mills, L. S. Tamil, “A new approach to the design of graded-index guided wave devices,”IEEE Microwave Guided Wave Lett. 1, 87–89 (1991).
[CrossRef]

Tamil, L. S.

D. W. Mills, L. S. Tamil, “A new approach to the design of graded-index guided wave devices,”IEEE Microwave Guided Wave Lett. 1, 87–89 (1991).
[CrossRef]

van Valkenburg, M. E.

M. E. van Valkenburg, Introduction to Modern Network Synthesis (Wiley, New York, 1967).

Yukon, S.

IEEE Microwave Guided Wave Lett. (1)

D. W. Mills, L. S. Tamil, “A new approach to the design of graded-index guided wave devices,”IEEE Microwave Guided Wave Lett. 1, 87–89 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. K. Jordan, H. N. Kritikos, “An application of one-dimensional inverse scattering theory for inhomogeneous regions,”IEEE Trans. Antennas Propag. AP-22, 909–911 (1973).
[CrossRef]

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,”IEEE Trans. Antennas Propag. AP-35, 934–946 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Proc. Inst. Electr. Eng. (1)

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

Other (7)

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, New York, 1990), Chap. 3.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11; also see 2nd ed., 1990.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), Chaps. 2 and 6.

R. L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 205.

M. E. van Valkenburg, Introduction to Modern Network Synthesis (Wiley, New York, 1967).

D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972).

I. Kay, “The inverse scattering problem,” (New York University, New York, 1955).

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Figures (6)

Fig. 1
Fig. 1

One-dimensional inverse-scattering problem.

Fig. 2
Fig. 2

Mode diagram for discrete modes in slab waveguide.

Fig. 3
Fig. 3

Wave vectors in slab waveguide for (a) guided modes, (b) leaky modes, and (c) radiation modes.

Fig. 4
Fig. 4

Mode diagram for radiation modes in slab waveguide.

Fig. 5
Fig. 5

Discrete and continuous modes shown in complex (a) kx, (b) kz, and (c) k1x planes.

Fig. 6
Fig. 6

Mode diagram in inhomogeneous medium.

Tables (2)

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Table 1 Modes of Slab Waveguide at 12 GHz

Tables Icon

Table 2 Modes of Optical Slab Waveguide

Equations (47)

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G ( r , r ) = - d k s [ A ( k s ) + r ( k s ) B ( k s ) ] × exp [ i k s · ( r s - r s ) ] ,
r ( k 1 x , ω ) = R - 10 + R 01 exp ( i 2 k x d ) 1 + R - 10 R 01 exp ( i 2 k x d ) ,
R - 10 = - R 01 ,
R 01 2 exp ( i 2 k x d ) = 1
R 01 = k x - k 1 x k x + k 1 x .
k 1 x 2 + k 1 z 2 = k 1 2 ,
k x 2 + k z 2 = k 2
r ( k 1 x = 0 ) = - 1 ,
E ˜ - 1 = E - 1 exp ( i k z z ) exp ( - i k 1 x x )             for region ( - 1 ) , x < 0 ,
E ˜ 0 = exp ( i k z z ) [ A exp ( i k x x ) + B exp ( - i k x x ) ]             for region ( 0 ) , 0 < x < d ,
E ˜ 1 = E 1 exp ( i k z z ) exp ( i k 1 x x )             for region ( 1 ) , x > d .
A = B R 01 .
B exp ( - i k x d ) = A exp ( i k x d ) R 01 ,
exp ( 2 i k x d ) R 01 2 = 1.
k 1 x 2 = k x 2 - ( k 2 - k 1 2 ) < k x 2 < 0 ,
R 01 = | ( k x - k 1 x ) 2 + ( k x - k 1 x ) 2 ( k x + k 1 x ) 2 + ( k x + k 1 x ) 2 | 1 / 2
k c 0 = m π d 1 ( r - 1 r ) 1 / 2 .
f c m = c m 2 d 1 ( r - 1 r ) 1 / 2 .
E ˜ - 1 = E - 1 exp ( i k z z ) exp ( i k 1 x x ) + r ( k 1 x ) E - 1 exp ( i k z z ) × exp ( - i k 1 x x )             for region ( - 1 ) , x < 0 ,
E ˜ 0 = exp ( i k z z ) [ A exp ( i k x x ) + B exp ( - i k x x ) ]             for region ( 0 ) , 0 < x < d ,
E ˜ 1 = E 1 exp ( i k z z ) exp ( i k 1 x x )             for region ( 1 ) , x > d ,
r ( k 1 x , ω ) = B - 1 A - 1 ,
r ( k 1 x , ω ) = 1 R - 10 + ( 1 - 1 / R - 10 2 ) 1 / R - 10 + B 0 / A 0 .
B 0 A 0 = - 1 R - 10 .
B 0 A 0 = exp ( i 2 k x d 1 ) R 01 + ( 1 - 1 / R 01 2 ) exp [ i 2 ( k 1 x + k x ) d 1 ] 1 / R 01 exp ( i 2 k 1 x d 1 ) + B 1 / A 1 .
B l A l = exp ( i 2 k l x d l + 1 ) R l ( l + 1 ) + [ 1 - 1 / R l ( l + 1 ) 2 ] exp { i 2 [ k ( l + 1 ) x + k l x ] d l + 1 } 1 / R l ( l + 1 ) exp [ i 2 k ( l + 1 ) x d l + 1 ] + B l + 1 / A l + 1 ,
A l B l = exp ( - i 2 k l x d l ) R l ( l - 1 ) + [ 1 - 1 / R l ( l - 1 ) 2 ] exp { - i 2 [ k ( l - 1 ) x + k l x ] d l } 1 / R l ( l - 1 ) exp [ - i 2 k ( l - 1 ) x d l ] + A l - 1 / B l - 1 .
B n A n = R n t exp ( i 2 k n x d n ) .
r ( k 1 x = 0 ) = - 1.
E - 1 = B - 1 exp ( i k z z ) exp ( - i k - 1 x x )             for region ( - 1 ) , x < 0 ,
E 0 = exp ( i k z z ) [ A 0 exp ( i k x x ) + B 0 exp ( - i k x x )             for region ( 0 ) , 0 < x < d 1 ,
E 1 = exp ( i k z z ) [ A 1 exp ( i k 1 x x ) + B 1 exp ( - i k 1 x x )             for region ( 1 ) , d 1 < x < d 2 ,
E n = exp ( i k z z ) [ A n exp ( i k n x x ) + B n exp ( - i k n x x )             for region ( n ) , d n < x < d t ,
E t = A t exp ( i k z z ) exp ( i k t x x )             for region ( t ) , x > d t .
R + l R - l = 1 ,
R + 0 R - 0 = 1.
R + 0 = B 0 A 0 = exp ( i 2 k x d 1 ) R 01 + ( 1 - 1 / R 01 2 ) exp [ i 2 ( k 1 x + k x ) d 1 ] 1 / R 01 exp ( i 2 k 1 x d 1 ) + B 1 / A 1 ,
R - 0 = A 0 B 0 = - R - 10 .
B l A l ( exp ( - i 2 k l x d l ) R l ( l - 1 ) + [ 1 - 1 / R l ( l - 1 ) 2 ] exp { - i 2 [ k ( l - 1 ) x + k l x ] d l } 1 / R l ( l - 1 ) exp [ - i 2 k ( l - 1 ) x d l ] + A l - 1 / B l - 1 ) = 1 ,
( exp [ i 2 k ( l - 1 ) x d 1 R ( l - 1 ) l + [ 1 - 1 / R ( l - 1 ) l 2 ] exp { i 2 [ k l x + k ( l - 1 ) x ] d l } 1 / R ( l - 1 ) l exp ( i 2 k l x d l ) + B l / A l ) A l - 1 B l - 1 = 1.
B l A l ( [ 1 - 1 / R ( l - 1 ) l 2 ] exp { i 2 [ k l x + k ( l - 1 ) x ] d l } - exp [ i 2 k ( l - 1 ) x d l ] / R ( l - 1 ) l + B l - 1 / A l - 1 - 1 R ( l - 1 ) l exp ( i 2 k l x d l ) ) - 1 = 1 ,
B l A l ( ( B l - 1 / A l - 1 ) R l ( l - 1 ) exp { - i 2 [ k l x + k ( l - 1 ) x ] d l } + exp ( - i 2 k l x d l ) B l - 1 / A l - 1 exp [ - i 2 k ( l - 1 ) x d l + R l ( l - 1 ) ) = 1.
2 E ( k 1 x , x ) x 2 + k x 2 E ( k 1 x , x ) = 0             for x 0 ,
2 E ( k 1 x , x ) x 2 + k 1 x 2 E ( k 1 x , x ) = k 0 2 [ 1 - r ( x ) ] E ( k 1 x , x )             for x 0.
R - 10 + R 01 exp ( i 2 k x d ) = 0.
B 0 A 0 = - R - 10 = R 01 .
r ( k 1 x ) = r 0 i N ( k 1 x - k i z ) i M ( k 1 x - k i p ) ,

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