Circle crosshairvalorant

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So, I'm given a certain wavelength $\lambda$ and the grating costant $d$ (distance between slits). I'm asked to find the maximum order of diffraction for this set of data. In general, when light falls upon the grating with angle $\theta_i$ and escapes (I don't know the right word in English, sorry) with an angle $\theta_s$, the total optical path difference is given by \begin{equation} \Delta=d(\sin \theta_i+\sin \theta_s) \end{equation} which combined with the maximum condition for bright zones $\Delta=m\lambda$ gives \begin{equation} d(\sin \theta_i+\sin \theta_s) =m\lambda \end{equation} where $m$ is the order of the maximum. So, the problem is very easy to solve when we have normal incidence (where $\theta_i=0$), but I can't manage to solve it in general where the two angles are involved. I have been told by colleagues that the maximum order is indeed $m=d/\lambda$ but I can't understand why. Every opinion is welcome. Thanks in advance for any response.

As a first observation, there is no maximum order. There is however a maximum propagating order, for which $\sin \theta_s = 1$. Higher orders will not propagate but exponentially decay in the propagation direction.

Circle crosshairValorant code

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The maximum propagating order propagates at 90$^{\circ}$ so parallel to the grating. It has $m=d/\lambda$ for perpendicular or $m=2d/\lambda$ for maximally oblique incidence.

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