Appendix

Appendix. Multimode interference formalism.

Definitions.
Index sizes
Cover size - equal to 1 for 1D collinear or equal to 2 for 1D conical or crossed;
Grating size - equal or greater than 1 and less or equal to total number of modes.
Complex vectors
INC, inc _i is incident from cover;
REF, ref _i is reflected in cover;
CG, cg _i is grating modes propagating from cover to substrate (down);
SG, sg _i is grating modes propagating from substrate to cove (up);
TR, tr _i is transmitted to substrate;
G, g _i is local grating modes;
Φ, φ _i is the vector of complex phase coefficients of the modes on the path from the cover to the substrate or vice versa;
φ _{j =} exp(i N_{eff j} k₀ Thi),
where k₀ is wavenumber in vacuum,
Thi isgrating layer thickness.
Scattering complex matrices
IR, ir _{i, j} is scattering matrix of INC to REF, Row size = Column size = Cover size;
IT, it _{i, j} is scattering matrix of INC to CG, Row size = Grating size, Column size = Cover size;
CR, cr _{i, j} is scattering matrix of G to CG at cover interface, Row size = Grating size, Column size = Grating size;
CT, ct _{i, j} is scattering matrix of G to REF at cover interface, Row size = Cover size, Column size = Grating size;
SR, sr _{i, j} is scattering matrix of G to SG at substrate interface, Row size = Grating size, Column size = Grating size;
ST, st _{i, j} is scattering matrix of G to TR at substrate interface, Row size = Cover size, Column size = Grating size;
E, δ _{i, j} and TMP, tmp _{i, j} is unity and temporary matrices, Row size = Column size = Grating size).
Summation is performed on repeated indices (exception – multiplication on φ _i - without summation).
INC depends on input conditions.

Suppose we know the vector CG. The mode vector g _{i =} cg _i * φ _i impinges the substrate interface. Thus, we can determine the transmission to the substrate and reflection from it
tr _i = st _{i, n} cg _n φ _n (1)
sg _i = sr _{i, n} cg _n φ _n. (2)
Reflected vector acquires a complex phase advance sg _i φ _i and impinges cover interface. Taking into account the INC vector, we obtain the expressions for vector CG:

cg _i = it _{i, n} inc _n + cr _{i, m} sg _m φ _m (3)
or
cg _i = it _{i, n} inc _n + cr _{i, m} sr _{m, n} cg _n φ _n φ _m and (4)
ref _i = ir _{i, n} inc _n + ct _{i, m} sr _{m, n} cg _n φ _n φ _m (5)
or
δ _{i, n} cg _n = it _{i, n} inc _n + cr _{i, m} sr _{m, n} cg _n φ _n φ _m (6)
or
(δ _{i, n} - cr _{i, m} φ _m sr _{m, n} φ _n) cg _n = it _{i, n} inc _n (7)
The multiplication on φ _n and φ _m does not involve summation, so in parentheses we have a square matrix. In the matrix form, the last equation will have the form:
(E – CR•TMP)•CG = IT•INC, where tmp _{i, j} = φ _i sr _{i, j} φ _j (8)
Finally
CG = (E – CR•TMP)^-1•IT•INC (9)
If we know the vector CG, then by the previous formulas it is possible to calculate the reflection, transmission and the resonant increase in the amplitudes of the grating modes.

A special functions that transform the conditional polarizations to the polarizations TE(s) and TM(p) were used.