Calculating Highest Order Possible

The equation $d\text{sin}\theta =\mathrm{m\lambda }\text{(for}m=\mathrm{0,}\mathrm{1,}-\mathrm{1,}\mathrm{2,}-\mathrm{2,}\dots )$ describes constructive interference. For fixed values of $d$ and $\lambda$, the larger $m$ is, the larger $\text{sin}\theta$ is. However, the maximum value that $\text{sin}\theta$ can have is 1, for an angle of $\text{90º}$. (Larger angles imply that light goes backward and does not reach the screen at all.) Let us find which $m$ corresponds to this maximum diffraction angle.

Solving the equation  for $m$ gives

Taking $\text{sin}\theta =1$ and substituting the values of $d$ and $\lambda$ from the preceding example gives

Therefore, the largest integer $m$ can be is 15, or

The number of fringes depends on the wavelength and slit separation. The number of fringes will be very large for large slit separations. However, if the slit separation becomes much greater than the wavelength, the intensity of the interference pattern changes so that the screen has two bright lines cast by the slits, as expected when light behaves like a ray. We also note that the fringes get fainter further away from the center. Consequently, not all 15 fringes may be observable.