Let us define:

• $d(\lambda_x, \lambda_m)$ = detection efficiency, proportional to solid angle geometry, detector quantum efficiency and optical path trasmission efficiency (comprising both excitation and emission)
• $f(\lambda)$ = distribution density function of emitted fluorescence (spectrum)
• $\phi$ = Quantum yield of the fluorophore
• $I(\lambda_x,\cdot)$ = Intensity of the excitation light

Emitted fluorescence ($F$) from a solution of N fluorescent species can be expressed as a function of excitation (x) and emission (m) wavelengths as follows.

$$ F(\lambda_x,\lambda_m) = \sum_{i=1}^N 2.303 \epsilon_i(\lambda_x) c_i l I(\lambda_x) \phi_i f_i(\lambda_m) d(\lambda_x,\lambda_m) $$

When FRET occurs among (at least 2) species and after defining $\kappa = 2.303 l$ and $i = D, A$. The length of the optical path l in a microscope can be taken as a constant.

$$ F(\lambda_x,\lambda_m) = \kappa I(\lambda_x) d(\lambda_x,\lambda_m) [\epsilon_D(\lambda_x) c_D \phi_D f_D(\lambda_m) (1-E) + \epsilon_A(\lambda_x) c_A \phi_A f_A(\lambda_m) + E \epsilon_D(\lambda_x) c_D \phi_A f_A(\lambda_m)] $$

$$ \int_X \int_M I(\lambda_x) d(\lambda_x, \lambda_m) f(\lambda_m) d\lambda_x d\lambda_m = \int_X \int_M I(\lambda_x) d(\lambda_x) d(\lambda_m) f(\lambda_m) d\lambda_x d\lambda_m = \int_X I(\lambda_x) d(\lambda_x) d\lambda_x \int_M d(\lambda_m) f(\lambda_m) d\lambda_m \equiv I_X F_i^M $$

$$ F(X, M) \equiv \int_X \int_M F(\lambda_x, \lambda_m) d\lambda_x d\lambda_m = \kappa I_X $$