Problem 3

Monochromatic light of wavelength

\lambda

is incident normally on a transmission grating of spacing

d=3\lambda

with many narrow parallel slits per unit length. Neglect absorption and consider principal maxima; angle

\theta

is measured from the straight-through direction. (a) State the maximum condition and find the first-order angle. (b) List all observable orders and the largest order. (c) For a screen a distance

L

from the grating, find the transverse position

y_1

of the first-order maximum using exact geometry.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{\theta_1=\arcsin(1/3),\quad m=0,\pm1,\pm2,\pm3,\quad y_1=\frac{L}{2\sqrt2}}

problems.proof.analysis

Maxima on the two sides are symmetric about the direct beam. The screen coordinate uses exact geometry without a small-angle approximation.

problems.proof.pitfall. An observable order must be an integer satisfying

|m|\lambda\le d

; do not round an impossible order into existence.