# 10.4.2 Diffraction Grating Interference Pattern

With today’s technology, we can etch extremely thin, closely packed and equally spaced slits on a transparent material. Such an optical component is called a diffraction[1] grating. Typically, a grating has hundreds, if not thousands of equally spaced slits (aka lines) per millimetre. When we shine a beam of laser into the grating, each slit acts as a wave source! As these hundreds, if not thousands of light waves, propagate out of the slits towards a screen, they superpose and form the ultimate interference pattern.

Compared to the double-slit, the fringes produced by the grating are a lot brighter and narrower but fewer. Why?

Bright Fringes

Just like a double-slit, a grating produces (bright) fringes at angles where

$\displaystyle d\sin \theta =n\lambda$ , $n=0,1,2...$

However, because gratings usually have much smaller d compared to double-slits (~um vs ~0.1 mm), they produce fringes at much larger values of θ. In fact, usually just a handful of fringes are formed before we hit $\theta =90{}^\circ$.

$\displaystyle d\sin {{\theta }_{n}}=n\lambda \quad \Rightarrow \quad {{\theta }_{n}}={{\sin }^{{-1}}}(\frac{{n\lambda }}{d})$

Since the inverse sine function is non-linear at large values of θ, the grating typically produces irregularly spaced fringes. This is why the $\displaystyle \Delta y=\frac{{L\lambda }}{d}$ formula is not applicable to gratings, since $\displaystyle \sin {{\theta }_{n}}=\frac{{n\lambda }}{d}\text{ }\Rightarrow \text{ }\frac{{{{y}_{n}}}}{L}=\frac{{n\lambda }}{d}\text{ only if }\theta \text{ is small}$.

Compared to the double-slit, the grating produces much brighter (and narrower) fringes. This is because we are talking about the constructive interference of hundreds of waves from hundreds of slits (compared to two for the double-slit). Even with just 100 slits, we are talking about a bright fringe with amplitude 100A and intensity 10,000I (compared to 2A and 4I  for the double-slit[2]).

Dark Fringes: Actually, we don’t talk about dark fringes for gratings. When there are hundreds of waves, the slightest phase difference between adjacent rays results in mostly destructive interference among them, such that the amplitude of the resultant wave is always negligibly small if not zero. In this sense, the grating is a lot more finely tuned than the double-slit. All the energy is concentrated very precisely at angles where $\displaystyle d\sin \theta =n\lambda$ , $n=0,1,2...$, and little is left elsewhere. This also explains why gratings produce such narrow bright fringes.

Excellent contrast and resolution

Since the grating produces very narrow and bright fringes, interference pattern produced by a grating offers very good contrast and spatial resolution. This explains why the grating is widely used for spectral analysis.

Example

A parallel beam of light of wavelength 600 nm is incident normally on a diffraction grating. The grating has 500 lines per millimeter.

a) Determine the total number of bright fringes formed.

b) Determine the angles at which the fringes are formed.

Solution

a)

$\displaystyle d=\frac{{1.0\times {{{10}}^{{-3}}}}}{{500}}=2.0\times {{10}^{{-6}}}\text{ m}$

\begin{aligned}(d\sin \theta &=n\lambda )\\(2.0\times {{10}^{{-6}}})sin90{}^\circ &=n(700\times {{10}^{{-9}}})\\n&=2.857\end{aligned}

The highest order fringe formed is the $n=2$ order. Therefore, 5 fringes are formed.

b)

0th order fringe: \begin{aligned}(d\sin \theta &=n\lambda )\\(2.0\times {{10}^{{-6}}})sin{{\theta }_{1}}&=(0)(700\times {{10}^{{-9}}})\\{{\theta }_{1}}&=0{}^\circ \end{aligned}

1st order fringe:      \begin{aligned}(d\sin \theta &=n\lambda )\\(2.0\times {{10}^{{-6}}})sin{{\theta }_{1}}&=(1)(700\times {{10}^{{-9}}})\\{{\theta }_{1}}&=20.5{}^\circ \end{aligned}

2nd order fringe:     \begin{aligned}(d\sin \theta &=n\lambda )\\(2.0\times {{10}^{{-6}}})sin{{\theta }_{2}}&=(2)(700\times {{10}^{{-9}}})\\{{\theta }_{2}}&=44.4{}^\circ \end{aligned}

Demonstration

Laser Beam through a Grating

Applet

Multiple Sources (ngsir)

[1] We will study diffraction in detail in later sections. For the time being, just take it to mean spreading.

[2] I should mention that in practice, a grating usually has narrower slits compared to a double-slit. So the amplitude of the light wave through one slit of a grating is probably lower than that through one slit of a double-slit. However, overall, the much greater number of slits gives the grating a comfortable win in this contest.