Slit And Grating Angle Ledgers

grating-order-can-be-accepted-or-rejected

A grating order is accepted by exact spacing-times-sine arithmetic and rejected when the order is impossible. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A grating order starts as spacing times sine

The accepted row multiplies grating spacing by a rational sine marker and compares that product with an integer wavelength count.

d=5 msin⁡θ=45d=5\ \text{m}\qquad \sin\theta=\frac{4}{5}
Accepted grating rowThe accepted order is a checked spacing-times-sine row.sinegratingSpacing=5 msineFactor=4/5wavelength=2 morder=2 countacceptedBit=1 bit

Orders stop when the required sine is impossible

The first two orders fit inside the sine range. The third row would require the endpoint and still fails the exact product test, so it is rejected instead of drawn as valid.

ssin⁡θmaccepted5 m25115 m45215 m130\begin{array}{c|c|c|c}s&\sin\theta&m&\text{accepted}\\5\ \text{m}&\frac{2}{5}&1&1\\5\ \text{m}&\frac{4}{5}&2&1\\5\ \text{m}&1&3&0\\\end{array}

A grating order can be accepted or rejected exactly

The accepted order reaches the wavelength ledger, while the impossible order is kept as a rejected check instead of drawing an invalid angle.

dsin⁡θ=5 m⋅45=2λ;impossible accepted=0d\sin\theta=5\ \text{m}\cdot\frac{4}{5}=2\lambda;\quad \text{impossible accepted}=0
Grating accepted and rejected order ledgersThe impossible order remains a finite rejected ledger.sinesinegratingSpacing=5 msineFactor=4/5wavelength=2 morder=2 countacceptedBit=1 bitgratingSpacing=5 msineFactor=1wavelength=2 morder=3 countacceptedBit=0 bit