How does the diffraction formula nλ=dsinθ change when the diffraction grating isn't perpendicular to the light rays?

Question

If the diffraction grating is misaligned and therefore the incident angle towards the grating isn't=90º, would there be a formula relating the wavelength of the light emitted and the angle at which the bright fringe is found in the diffraction pattern, and the angle at which the diffraction grating is placed at (similar to nλ=dsinθ)?

Answer 1

Prologue

Just a few days ago I needed exactly this while in process of upgrading my spectroscopes from CD/DVD to BluRay gratings. The diffraction angles did not match my angular tweakables with new gratings so I needed to see where exactly the usable spectrum will be reflected. So I made an simulation to show and prepare new construction configuration for my devices...

The well known formula for reflective gratings is:

sin(a)+sin(b)=m*lambda/d

where:

a angle between grating normal and incoming light in [rad]
b angle between grating normal and diffracted or reflected light b in [rad]
m is the degree of diffraction (m=0 is simple reflection)
lambda is light wavelength [m]
d is distance between tracks of grating in [m].

here simple C++ computation

void grating(double lambda,double d,double a)   // [nm], [nm], [rad]
    {
    double m,b;
    int cnt=0;
    for (m=-3;m<=3;m++)
     if (m)
        {
        b=(m*lambda/d)-sin(a);
        if (fabs(b)>1.0) continue;
        b=asin(b);
        // here `b` is output light angle
        }
    }

here preview for d=1.6um (grating from CD) simulation based on this:

Done using simple 2D ray casting and RGB values of visible spectrum

The horizontal white line going from left to the grating is the light source. Diffracted rays length indicates the m as You can see some overlap. The higher the d the more this occurs.

Looks like formula and simulation is matching real gratings made from:

• CD d=1.60um
• DVD d=0.72um
• BR d=0.32um

that I use for my home made spectroscopes. Beware non linear tracks have major impact on the diffraction result. The angles does not change however the shape and clarity does ...

In case you have refractive gratings then I do not know if the formula stays the same or on top of it is snell's law applied. Sorry but I do not have experience with such gratings...