= Solution
At screen coordinate $X$, the propagation length from <aperture> coordinate $x$ is $[D^2+(X-x)^2]^{1/2}$. Its expansion has a constant term, a term linear in $x$ giving the observation angle, and a quadratic term of order $x^2/(2D)$. For $|x|\leq d/2$, the quadratic <wave phase> is at most of order $\pi d^2/(4\lambda D)$. Requiring it to be much smaller than one gives the sufficient condition \b[$D\gg d^2/\lambda$]; higher terms are then smaller in the paraxial geometry. In the <lens> arrangement, put the source at the first <lens>'s front focus, illuminate the grating with its collimated output, and put the screen at the second <lens>'s back <focal plane>. This obtains the angular <Fraunhofer diffraction> pattern without a physically very distant screen.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2011/paper-1-grating-layout.png]
{title=Two-lens arrangement for a grating diffraction pattern}
{description=Original schematic ray construction. Equal outgoing ray slopes meet at a common back-focal-plane position; distances and angles are illustrative.}
The slit pitch is $p=1/(200\,\mathrm{mm^{-1}})=5.00\,\mathrm{\mu m}$. Normal-incidence maxima satisfy $p\sin\theta=m\lambda$. Thus $\sin\theta_3=0.276$ and, using rectilinear focal-plane geometry,
$$
\boxed{x_3=\pm f\tan\theta_3=\pm86.1\,\mathrm{mm}.}
$$
The paraxial result $x_3\simeq fm\lambda/p=\pm82.8\,\mathrm{mm}$ is a roughly 4% lower approximation; an actual wide-angle <lens>'s mapping depends on its design. The angular result does not depend on that convention.
For $N$ uniformly illuminated equal slits of width $w$, the diffraction <intensity> has the form
$$
I(\theta)=Cw^2\operatorname{sinc}^2\!\left(\frac{\pi w\sin\theta}{\lambda}\right)
\left[\frac{\sin(N\pi p\sin\theta/\lambda)}{\sin(\pi p\sin\theta/\lambda)}\right]^2,\quad \operatorname{sinc}u=\frac{\sin u}u.
$$
Here $C$ is unchanged when the incoming <intensity> is fixed. The broad <single-slit diffraction> envelope multiplies the narrow grating orders. At a principal maximum,
$$
I_m=CN^2w^2\operatorname{sinc}^2(\pi mw/p).
$$
The <fixed-illumination grating peak change with slit width> gives the changes from $w=1.0$ to $1.5\,\mathrm{\mu m}$ are
$$
\boxed{I_{0,\rm new}/I_{0,\rm old}=2.25},\qquad
\boxed{I_{3,\rm new}/I_{3,\rm old}=\left[\frac{\sin(0.9\pi)}{\sin(0.6\pi)}\right]^2=0.1056.}
$$
The central peak increases by 125%, while the third-order peak decreases by about 89.4%. Separate normalizations to each central peak would hide this effect. The following width-specific solutions and profiles use a common incoming normalization.
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