The diffraction grating
| English | Chinese | Pinyin |
|---|---|---|
| diffraction grating | 衍射光栅 | yǎn shè guāng shān |
| slit | 狭缝 | xiá fèng |
| coherent | 相干 | xiāng gān |
| maxima | 极大 | jí dà |
| order | 级次 | jí cì |
| monochromatic | 单色 | dān sè |
The rainbow on a CD
- Tilt a CD in the light and you see bright rainbow colours.
- The disc's fine tracks act as a diffraction grating 衍射光栅, splitting the light by wavelength.
- Gratings are how we measure wavelengths precisely.
What a grating is
- A grating has many equally spaced slits 狭缝 — often hundreds per millimetre.
- Each slit is a coherent 相干 source, and they all interfere together.
Fundamental mode on a stretched string — one loop, with L equal to half a wavelength
Why the grating gives sharp maxima
Two waves add when in phase and cancel when out of phase — change the phase and watch the resultant. A grating's many slits make the bright fringes razor-sharp.
A diffraction grating is made of:
Hundreds or thousands of equally spaced slits, each acting as a coherent source.
The grating equation
- Bright maxima 极大 appear at angles given by $d\sin\theta = n\lambda$.
- $d$ = slit spacing, $n = 0, 1, 2, \ldots$ is the order 级次, $\theta$ measured from the normal.
Light hits a grating of slit spacing $2.0\ \mu\text{m}$. The first-order ($n=1$) maximum is where $\sin\theta = 0.30$. Find the wavelength, in nm.
$\lambda = \dfrac{d\sin\theta}{n} = \dfrac{2.0 \times 10^{-6} \times 0.30}{1} = 6.0 \times 10^{-7}\ \text{m} = 600\ \text{nm}$.
Sharper than two slits
- With many slits, every "wrong" direction is cancelled by lots of slits.
- So a grating gives much sharper maxima than a double slit.
Young's double-slit experiment — the single slit makes the two slits coherent sources
A diffraction grating gives sharper maxima than a double slit.
Yes — many slits cancel every "wrong" direction, leaving narrow, bright maxima.
Slit spacing and highest order
- $N$ lines per mm → $d = \dfrac{1}{N}\ \text{mm}$.
- Since $\sin\theta \le 1$, the highest order is $n_{\text{max}} = \left\lfloor \dfrac{d}{\lambda} \right\rfloor$.
A diffraction grating splits monochromatic 单色 light into sharp maxima on a screen
A grating has $500$ lines per mm. What is the slit spacing $d$, in µm?
$d = \dfrac{1}{500}\ \text{mm} = 0.002\ \text{mm} = 2.0\ \mu\text{m}$.
For a grating and wavelength with $\dfrac{d}{\lambda} = 3.27$, the highest order seen is:
$n_{\text{max}} = \left\lfloor \dfrac{d}{\lambda} \right\rfloor = \lfloor 3.27 \rfloor = 3$. Order 4 would need $\sin\theta > 1$.
Finding a wavelength
- Shine the light straight at the grating and measure the angle of a maximum.
- Then $\lambda = d\sin\theta$ (use $n = 1$ for the first order); average over orders to reduce error.
To find a wavelength with a grating, you measure the ____ of a maximum.
Measure $\theta$ for a known order, then $\lambda = \dfrac{d\sin\theta}{n}$.
You've got it
- a grating is many equally spaced slits → sharp maxima
- grating equation $d\sin\theta = n\lambda$; $d = \dfrac{1}{N}$ from "$N$ lines per mm"
- highest order $n_{\text{max}} = \left\lfloor \dfrac{d}{\lambda} \right\rfloor$