3.2.5

# Diffraction Gratings

Test yourself

## Diffraction Grating Equation

The pattern produced by a diffraction grating can be described by the diffraction grating equation.

### Equation

• Diffraction gratings produce a pattern which is described by:
• $d\sin\theta = n\lambda$
• Where d is the distance between slits in the grating, θ is the angle between the maximum and the zero order line, λ is the wavelength of incident light, and n is the order of the maximum.

### Slit spacing

• A diffraction grating is made of many slits.
• If there are 1000 slits per metre, then slit spacing is 1/1000 metres.
• In general, if there are x slits per metre, then the slit spacing is 1/x metres.
• This gives the value of "d".

### Orders

• The variable "n" stands for the order of the maximum.
• Knowing d and λ lets us predict the angle of the central maximum.
• The central maximum is the zero order. "n" is zero.
• This implies that θ is zero. This is what we expect.
• The first order maximum will have n = 1, the second order maximum will have n = 2, and so on.

### Not all orders exist

• Remember that sin θ cannot be greater than 1.
• If you are using the equation and find that sin θ is larger than 1, the order you are looking at must not exist.

### Conclusions

• As we increase λ, sin θ increases so θ increases. This means the pattern becomes more spread out.
• If we increase the distance between slits, d, sin θ decreases so θ decreased. This means the pattern becomes less spread out.

## Derivation of Grating Equation

You need to know how to derive the diffraction grating equation.

### Step 1 - producing coherent sources

• Light enters the grating as parallel rays.
• It diffracts through each slit.
• The slits then act as coherent (in phase) and monochromatic (same wavelength) sources of light.
• These diffracted waves then interfere with each other to produce the pattern.

### Step 2 - angle to 1st order

• The 1st order maximum happens at an angle such that the path difference between two sources is one wavelength, λ (for constructive interference).
• Let's call this angle θ.

### Step 3 - create triangle

• We create a triangle as shown above.
• We know the distance between the two slits is d.
• We know the path difference is λ.
• We know the angle θ by geometry.

### Step 4 - use trigonometry

• Using trigonometry, we can see that:
• path difference $=\lambda = d \sin\theta$

### Step 5 - generalise for all n

• We know that maxima always occur for when the path difference is a whole number multiple of λ (constructive interference).
• Therefore, we can generalise the equation so that path difference = nλ
• $n\lambda = d \sin \theta$

## Applications of Diffraction Gratings

Diffraction gratings are very useful for physicists.

### Analysing light

• Diffraction gratings can be used to separate wavelengths in light from different substances.
• The wavelengths present can help us learn new things about the substance being tested.

### Atomic spacing in crystals

• Atoms in a crystal can act like a diffraction grating, with little gaps for light to pass through.
• We can find the spacing between the atoms in the crystal by looking at how X-rays diffract through the crystal.