See, e.g., D. L. Jaggard, K. E. Olson, “Numerical reconstruction of dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).

[CrossRef]

H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985).

[CrossRef]

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).

[CrossRef]

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.

[CrossRef]

I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).

I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).

[CrossRef]

B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)

J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)

[CrossRef]

S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).

J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).

[CrossRef]

J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).

[CrossRef]

M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).

[CrossRef]

D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.

[CrossRef]

See, e.g., D. L. Jaggard, K. E. Olson, “Numerical reconstruction of dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).

[CrossRef]

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.

[CrossRef]

D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).

See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).

I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.

[CrossRef]

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).

[CrossRef]

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)

[CrossRef]

S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

S. A. Schelkunoff, “Remarks concerning wave propagation in stratified media,” in The Theory of Electromagnetic Waves, M. Kline, ed. (Interscience, New York, 1951), pp. 181–192.

J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).

[CrossRef]

J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).

[CrossRef]

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).

[CrossRef]

J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)

[CrossRef]

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.

[CrossRef]

S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).

[CrossRef]

See, e.g., D. L. Jaggard, K. E. Olson, “Numerical reconstruction of dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).

[CrossRef]

H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985).

[CrossRef]

B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)

J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).

[CrossRef]

I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).

I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).

[CrossRef]

M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).

Note that r(p) must satisfy energy conservation and causality. However, r̂(p) is a mathematical transformation that does not necessarily satisfy the restriction |r̂(p)|<1.

See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).

The techniques include invariant embedding, use of the impedance concept and a direct manipulation of Maxwell’s equations or the Helmholtz equation. See Refs. 5–8 for examples.

S. A. Schelkunoff, “Remarks concerning wave propagation in stratified media,” in The Theory of Electromagnetic Waves, M. Kline, ed. (Interscience, New York, 1951), pp. 181–192.

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).