I work with highly coherent laser light 1550 nm source traveling through singlemode fiber. EOM creates laser pulses ~100 ns length corresponding to $m\approx 10$ m length of pulse. I can tune laser temperature (so $\Delta k$) but this takes seconds of time, which is enough for fiber temperature to change, thus it is unusable for fiber structure research. Detector can give me direct intensity of rayleigh scattered back light with some discretization frequency corresponding to meters of fiber. As I understand, I can model this signal using bilinear forms. How can I extract that bilinear form?
Let's define the phase increment from the laser wavelength (or wavenumber) shift on unit length $\phi \equiv n\,(k_0+\Delta k) $.
Field vector with amplitudes $A_j$ and phase factors $\exp(\mathrm{i} j \phi)$ $$ E(k=k_0+\Delta k)=\begin{bmatrix} A_1 e^{\mathrm{i}\phi}\\ A_2 e^{\mathrm{i}2\phi}\\ \vdots\\ A_m e^{\mathrm{i}m\phi} \end{bmatrix}.$$
Total field as a coherent sum $$E_{\rm tot}(k_0+\Delta k)=\mathbf{1}^\top \mathbf{E}(k_0+\Delta k), \qquad \mathbf{1}= \begin{bmatrix} 1\\ \vdots\\ 1 \end{bmatrix}. $$
Intensity as a Hermitian quadratic (bilinear) form $$ I(k_0+\Delta k)=\left|E_{\rm tot}(k_0+\Delta k)\right|^2 =\mathbf{E}(k_0+\Delta k)^\dagger\,\mathbf{1}\mathbf{1}^\top\,\mathbf{E}(k_0+\Delta k), \qquad \mathbf{1}\mathbf{1}^\top= \begin{bmatrix} 1&1&\cdots&1\\ 1&1&\cdots&1\\ \vdots&\vdots&\ddots&\vdots\\ 1&1&\cdots&1 \end{bmatrix}. $$
Equivalent expanded bilinear sum $$ I(k_0+\Delta k)=\sum_{p=1}^m\sum_{q=1}^m A_p^*A_q\,e^{i(q-p)\phi}. $$
On the other hand, in reality bilinear $M_{pq}$ form does not consist of just ones, but represent additional phase differences due to irregularity if refraction index:
$$M_{pq} \sim \exp\left[\mathrm{i}k\int_p^q \delta n(l)\ dl\right]$$
Also, I can quickly change two different impulse forms $A_j$ and represent amlitude/wavelength changes as another diagonal matrix. How can I extract bilinear form matrix?
For clarity, I can modify impulse length and therefore it's shape changes:
