For time‐independent wave propagation in focusing media or wave guides, backscattering and coupling between propagation modes are caused by deterministic or random variations of the refractive index in the distinguished (x) direction of propagation. Various splittings of the wave field into forward and backward traveling components, which lead to coupled equations involving abstract operator coefficients, are presented. Choosing a natural explicit representation for these operators immediately yields a coupled mode form of these equations. The splitting procedure also leads naturally to abstract transmission and reflection operators for slabs of finite thickness (a≤x≤b), and abstract invariant imbedding equations satisfied by these. The coupled mode form of these equations, together with such features as reciprocity (associated with an underlying symplectic structure) are also discussed. The example of a square law medium is used to illustrate some of these concepts.

1.
H.
Breimner
,
Commun. Pure Appl. Math.
4
,
105
(
1951
).
Google Scholar
 
2.
F. W.
Sluijter
,
J. Math. Anal. Appl.
27
,
282
(
1969
);
Google Scholar
Crossref
Search ADS
 
F. W.
Sluijter
,
J. Opt. Soc. Am.
60
,
8
(
1970
).
Google Scholar
Crossref
Search ADS
 
3.
J. A.
Arnaud
,
Bell Syst. Tech. J.
49
,
2311
(
1970
).
Google Scholar
Crossref
Search ADS
 
4.
J.
Corones
,
J. Math. Anal. Appl.
50
,
361
(
1975
).
Google Scholar
Crossref
Search ADS
 
5.
M. E. Davison, “A general approach to splitting and invariant imbedding techniques for linear wave equations,” Ames Laboratory preprint, 1982.
6.
J. P.
Corones
 and
R. J.
Krueger
,
J. Math. Phys.
24
,
2301
(
1983
).
Google Scholar
Crossref
Search ADS
 
7.
J. W.
Evans
,
J. Math. Phys.
26
,
2196
(
1985
).
Google Scholar
Crossref
Search ADS
 
8.
V. A. Fock, Electromagnetic Diffraction and Propagation Problems (McMillan, New York, 1960).
9.
L.
Fishman
 and
J. J.
McCoy
,
Proc. Soc. Photo Opt. Instrum. Eng.
358
,
168
(
1982
);
Google Scholar
 
L.
Fishman
 and
J. J.
McCoy
,
J. Math. Phys.
25
,
285
(
1984
).
Google Scholar
Crossref
Search ADS
 
10.
D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972);
Theory of Optical Dielectric Waveguides (Academic, New York, 1974).
11.
J. M. Arnold, in Hybrid Formulation of Wave Propagation and Scattering, edited by L. B. Felsen (Nijhoff, Dordrecht, 1984).
12.
M. Reed and B. Simon, Methods of Modern Mathematical Physics: IV Analysis of Operators (Academic, New York, 1978), p. 100.
13.
E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961).
14.
G. M. Wing, An Introduction to Transport Theory (Wiley, New York, 1962);
R. Bellman and R. Vasudevan, Wave Propagation: An Invariant Imbedding Approach (Reidel, Dordrecht, 1986).
15.
For any differentiable Φ(x), the following are equivalent:
(i) Φ(y)C̄(x,y) = C(x,y)Φ(x),
 
(ii) Φ(x)Ā(x)−A(x)Φ(x)+Φx(x) = 0.
16.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part 1 (McGraw‐Hill, New York, 1953), (a) pp. 810 and 822; (b) p. 828.
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© 1988 American Institute of Physics.
1988
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