6.3 Double Slit, Take II

Now that we understand the effects of a finite slit width, we can consider diffraction of light by two slits of width with their centers separated by distance .
Replacing Eq. 6.2 with Eq. 6.8 in Section 6.1, the reasoning through Eq. 6.5 produces

$\begin{displaymath} \bar{I}(\theta) = \bar{I}_o \left( \frac{r_o}{r(\theta)} \ri... ...n \theta \right)} {\left( \frac{ka}{2} \sin \theta \right)^2} \end{displaymath}$ (6.12)

and for small angles

$\begin{displaymath} \bar{I}(\theta) = \bar{I}_o \cos^2 \left( \frac{kd}{2} \sin... ...n \theta \right)} {\left( \frac{ka}{2} \sin \theta \right)^2} \end{displaymath}$ (6.13)
Dark fringes in this intensity distribution appear at locations described by both Eq. 6.6 and Eq. 6.11.
The bright fringes due to the slit separation appear at angular positions corresponding to the maxima of Eq. 6.5, described by

$\begin{displaymath} \frac{kd}{2} \sin \theta_m = m \pi \hspace{1cm} (m = 0, \pm1, \pm2,...) \end{displaymath}$

or

$\begin{displaymath} d \sin \theta_m = m \lambda \hspace{1cm} (m = 0, \pm1, \pm2,...) \end{displaymath}$ (6.14)
Bright fringes of the double slit diffraction pattern can fail to appear by coinciding with dark fringes due to the slit width. Comparing Eq.'s 6.11 and 6.14, this occurs if

$\begin{displaymath} \frac{a}{d} \frac{m}{m'} \hspace{1cm} (m,m' integers) \end{displaymath}$ (6.15)

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