The Bragg Condition for plane, parallel grating with fringes normal to the grating surface is given by

$$ m \lambda = \Lambda 2 n_2 \sin(\alpha) $$

\eqref{eq:bragg-condition} \cite{Barden1999-vq}

Where:

$$

\lambda: \text{Wavelength of light in free space}\\ \Lambda: \text{Fringe spacing}\\ \alpha: \text{Incident angle in air, equal and opposite to diffraction angle in air $\beta$ with respect}\\ \text{to the surface normal}\\ \\ m: \text{Order of diffraction}\\ n_2: \text{Refractive index of grism}

$$

For orthogonal fringe (unslanted) plane gratings, the efficiency $\eta$ at the 1st order Bragg condition is given by

$$

\eta = \sin^2\left(\frac{\pi \Delta{n_2}d}{2 \cos(\alpha))}\right)

$$

Where:

$$ \eta: \text{Diffraction efficiency}\\ \alpha: \text{Incident angle in air}\\ d: \text{Thickness of grating volume}\\ \Delta n_2: \text{Semiamplitude (half of peak-to-peak amplitude) of variation of $n_2$ inside }\\\text{grating volume.}

$$

With the Bragg condition satisfied, the Volume-Phase Holographic (VPH) Grism Diffraction Efficiency can thus be given as a function of fringe spacing by

$$ \eta = \sin^2\left(\frac{m \pi \Delta{n_2}d}{2 \Lambda n_2 \sin(2 \sin^{-1}(\frac{n_0}{n_2}\sin(\alpha))}\right) $$

Where:

$$ \eta: \text{Diffraction efficiency} \\ m: \text{Order of diffraction.}\\ \Delta n_2: \text{Semiamplitude (half of peak-to-peak amplitude) of variation of $n_2$ inside }\\ \text{grating volume.}\\ d: \text{Thickness of grating volume}\\ \Lambda: \text{Fringe spacing}\\ n_0: \text{Refractive index of air ($\approx 1$)}\\ n_2: \text{Refractive index of VPH}\\ \alpha: \text{Incident angle in air}

$$

The fringe spacing is the inverse of the fringe density or frequency