# Diffraction Grating Has A

All waves interfere. In regions where two light waves overlap, their electric field vectors add. Light waves with the same polarization can interfere constructively or destructively. Waves that interfere constructively are in phase, waves that interfere destructively are 180^{o} out of phase. For the interference to not change with time, the waves have to maintain their phase relationship, they have to be **coherent**.

How do we make sure two interfering waves have the same polarization? We split one wave into two waves. A way to split one wave onto two waves is called **division of wave front.** We pass the same wave front through two closely spaced slits.

### The double slit

If light is incident onto an obstacle which contains two very small slits a distance d apart, then the wavelets emanating from each slit will interfere behind the obstacle. Waves passing through each slit are diffracted and spread out. At angles where the single slit diffraction pattern produces nonzero intensity, the waves from the two slits can now constructively or destructively interfere.

If we let the light fall onto a screen behind the obstacle, we will observe a pattern of bright and dark stripes on the screen, in the region where with a single slit we only observe a diffraction maximum. This pattern of bright and dark lines is known as an interference **fringe pattern**. The bright lines indicate constructive interference and the dark lines indicate destructive interference.

The bright fringe in the middle of the diagram on the right is caused by constructive interference of the light from the two slits traveling the same distance to the screen. It is known as the **zero-order fringe**. Crest meets crest and trough meets trough. The dark fringes on either side of the zero-order fringe are caused by destructive interference. Light from one slit travels a distance that is ½ wavelength longer than the distance traveled by light from the other slit. Crests meet troughs at these locations. The dark fringes are followed by the **first-order fringes**, one on each side of the zero-order fringe. Light from one slit travels a distance that is one wavelength longer than the distance traveled by light from the other slit to reach these positions. Crest again meets crest.

Note: We need single-slit diffraction to observe double-slit interference. Without the spreading, waves light waves passing through different slits would not meet and therefore could not interfere.

The diagram on the right shows the geometry for the fringe pattern. If light with wavelength λ passes through two slits separated by a distance d, we will observe constructively interference at certain angles. These angles are found by applying the condition for constructive interference, which is

A plane diffraction grating has lines marked on a plane surface; a concave grating has lines marked on a concave, usually spherical, surface. Diffraction gratings may also be classified as reflective of transmission. The lines of reflective gratings are marked on a mirror surface (usually metal), and observations are made in reflected light. A commercial diffraction grating has 600 lines per mm. When a student shines a 540 nm laser through this grating, how many bright spots could be seen on a screen behind the grating?

d sinθ = mλ, m = 0, 1, 2, ... .

The distances from the two slits to the screen differ by an integer number of wavelengths. Crest meets crest.

The angles at which dark fringes occur can be found be applying the condition for destructive interference, which is

d sinθ = (m+½)λ, m = 0, 1, 2, ... .

The distances from the two slits to the screen differ by an integer number of wavelengths + ½ wavelength. Crest meets trough.

If the interference pattern is viewed on a screen a distance L from the slits, then the wavelength can be found from the spacing of the fringes.

We have sinθ = z/(L^{2} + z^{2})^{½} and λ = zd/(m(L^{2} + z^{2})^{½}), where z is the distance from the center of the interference pattern to the mth bright line in the pattern.

If L >> z then (L^{2} + z^{2})^{½} ~ L and we can write

λ = zd/(mL).

Diffraction, and interference are phenomena are observed with all waves.

Link: Observe single and double slit diffraction with water waves

### Diffraction gratings

We have seen that diffraction patterns can be produced by a single slit or by two slits. When light encounters an entire array of identical, equally-spaced slits, called a diffraction grating, the bright fringes, which come from constructive interference of the light waves from different slits, are found at the same angles they are found if there are only two slits. But the pattern is much sharper.

The figure below shows the interference pattern for various numbers of slits. The width of all slits is 50 micrometers and the spacing between all slits is 150 micrometers. The location of the maxima for two slits is also the location of the maxima for multiple slits. The single slit diffraction pattern acts as an envelope for the multiple slit interference patterns.

Diffraction gratings contain a large number of parallel, closely spaced slits or grooves. They produce interference maxima at angles θ given by d sinθ = mλ. Because the spacing between the slits is generally very small, the angles θ are generally quite large. We cannot use the small angle approximation for relating wavelength and the position of the maxima on a screen for gratings, but have to use

sinθ = z/(L^{2} + z^{2})^{½}.

## Diffraction Grating Definition

Diffraction gratings disperse white light into its component colors because different wavelengths produce bright fringes at different angles.

d sinθ = mλ,

for a given m, bigger wavelength <> bigger angle

The spectral pattern is repeated on either side of the main pattern. These repetitions are called 'higher order spectra'. There are often many of them, each one fainter than the previous one. If the distance between slits is d, and the angle to a bright fringe of a particular color is θ, the wavelength of the light can be calculated.

Please watch: Passing Light Through Multiple Apertures - Exploring Wave Motion (YouTube)

#### Problem:

The first order bright line appears 0.25 cm from the center bright line when a double slit grating is used. The distance between the slits is 0.5 mm and the screen is 2.7 m from the grating. Find the wavelength.

Solution:

- Reasoning:

When light with wavelength λ passes through equally spaced slits separated by a distance d, constructively interference occurs at angles θ given by d sinθ = mλ, m = 0, 1, 2, ... . Here m = 1 and we are asked to solve this equation for λ. - Details of the calculation:

z = 0.25 cm = 2.5*10^{-3 }m.

L = 2.7 m.

d = 0.5 mm = 5*10^{-4 }m.

L >> z, therefore sinθ ~ z/L and λ = zd/(mL).

λ = (2.5*10^{-3 }m)(5*10^{-4 }m)/(2.7 m).

λ = 4.63*10^{-7 }m = 463 nm.

#### Problem:

A diffraction grating has 420 lines per mm. The grating is used to observe normally incident light with a wavelength of 440 nm. The grating is placed 1.3 m from a screen. Where on the screen will the first order bright line appear?

Solution:

- Reasoning:

When light with wavelength λ passes through equally spaced slits separated by a distance d, constructively interference occurs at angles θ given by d sinθ = mλ, m = 0, 1, 2, ... . Here m = 1 and we are asked to solve this equation for sinθ and then for z. - Details of the calculation:

d sinθ = λ.

d = (1/420) mm = 2.38*10^{-3 }mm = 2.38*10^{-6 }m.

λ = 440 nm = 4.40*10^{-7 }m.

L = 1.3 m = distance to the screen.

sinθ = λ/d = 10.65^{o}.

z = L tanθ = 24.5 cm = distance from the central maximum.

#### Problem:

A monochromatic wave of wavelength λ illuminates an opaque mask with two slits as shown in the figure. The diffraction pattern is recorded on a screen a distance L from the mask. You may assume that λ << d. D << L.

(a) What is the distance Λ between adjacent interference fringes observed on the screen?

(b) What is the width Δx of the central lobe of the interference pattern on the screen?

Solution:

- Reasoning:

The distance Λ between adjacent interference fringes is the distance between adjacent maxima of the double slit interference pattern.

The width Δx of the central lobe of the interference pattern equals twice the distance from the central maximum to the first minimum of the single slit interference pattern. - Details of the calculation:

(a) For the distance between adjacent maxima of the double slit pattern we have Dsinθ = λ.

Here sinθ ≈ θ ≈ Λ/L. So Λ = λL/D.

(b) For the distance from the central maximum to the first minimum of the single slit interference pattern we have dsinθ = λ.

Here sinθ ≈ θ ≈ Δx /(2L). So Δx = λ2L/d.

#### Module 9, Question 1

- What is the difference between diffraction and interference?
- Compare the formulas for the bright maxima and dark minima in the diffraction and interference patterns.

When you solve problems using these formulas, what do you have to be careful about?

**Discuss these questions with your fellow students in the discussion forum!**

Diffraction and interference patterns can be observed when light passes through a set of regularly spaced slits. For diffraction to produce an observable pattern, the spacing of the slits must be comparable to the wavelength of the radiation. Visible light has a wavelength range from ~400 nm to ~700 nm. A typical diffraction grating for visible light with 300 grooves per mm has a slit spacing of (1/300)mm = 3 mm = 3000 nm. This spacing is 4 to 8 times larger than the wavelengths of visible light and produces an easily observable pattern.

The wavelengths of x-rays lie in the 1 nm to 1 pm range. A typical diffraction grating will not produce an observable pattern. But the wavelengths of x-rays are comparable to the spacing of atoms in common crystals, and material with a regularly spaced grid of atoms can diffract x-rays and produce diffraction patterns that can be captured on photographic film.

Interference patterns are only observed if the interfering light from the various sources is **coherent**, i.e. if the phase difference between the sources is constant. Splitting the light from a single source into various beams is one way to produce coherent sources. Light from two different light bulbs is incoherent and will not produce an interference pattern. Lasers are sources of monochromatic, (single wavelength), coherent light. Two lasers can maintain a constant phase difference between each other for relatively long time intervals.

If you miss having regular lectures, consider these video lectures

Lecture 33: Double-Slit Interference and Interferometers

Lecture 34: Gratings and Resolving Power

“The diffraction grating is a useful device for analyzing light sources. It consists of a large number of equally spaced parallel slits.” Its working principle is based on the phenomenon of diffraction. The space between lines acts as slits and these slits diffract the light waves thereby producing a large number of beams that interfere in such a way to produce spectra.

A transmission grating can be made by cutting parallel lines on a glass plate with a precision ruling machine. The space between the lines is transparent to the light and hence acts as separate slits. A reflection grating can be made by cutting parallel lines on the surface of refractive material. Gratings that have many lines very close to each other can have very small slit spacing. For example, a grating ruled with 5000 lines/cm has a slit spacing d=1/5000 cm=2.00×10^{-4 }cm.

A section of a diffraction grating is illustrated in the figure. A plane wave is an incident from the left, normal to the plane of the grating. A converging lens brings the rays together at point P. The pattern observed on the screen is the result of the combined effects of interference and diffraction. Each slit is produced diffraction, and the diffracted beams interfere with one another to produce the final pattern.

The waves from all slits are in phase as they leave the slits. However, for some arbitrary direction θ measured from the horizontal, the waves must travel different path lengths before reaching point p. From the figure, we note that the path difference’ δ ‘ between rays from any two adjacent slits is equal to *d* sin θ. If this path difference is equal to one wavelength or some integral multiple of a wavelength, then waves from all slits are in phase at point P and a bright fringe is observed. Therefore, the condition for maxima in the interference pattern at the angle θ is.

## Diffraction grating formula

*d* sin θ =mλ

Where m=0,1,2,3,4…

We can use this expression to calculate the wavelength if we know the grating spacing and the angle 0. If the incident radiation contains several wavelengths, the mth-order maximum for each wavelength occurs at a specific angle. All wavelengths are seen at θ =0, corresponding to m=0, the zeroth-order maximum (m=1) is observed at the angle that satisfies the relationship sin θ =λ/d: the second-order maximum (m=2) is observed at a larger angle θ, and so on.

The intensity distribution for a diffraction grating obtained with the use of a monochromatic source. Note the sharpness of the principal maxima and the broadness of the dark areas. This is in contrast to the broad bright fringes characteristic of the double-slit interference pattern. Because the principal maxima are so sharp, they are very much brighter than double-slit interference maxima.

### Grating element definition

Distance between two consecutive slits (lines) of the grating is called a grating element. Grating element ‘d’ is calculated as:

Grating element =Length of grating/Number of lines

### Dispersion and resolving power

### Dispersion

## A Diffraction Grating Has 2000 Lines Per Centimeter

The ability of a grating to produce spectra that permit precise measurement of wavelengths is determined by two intrinsic properties of grating.

- The separation Δθ between the spectral lines that differ in wavelength by small amount Δλ.
- The width or sharpness of the lines.

The dispersion D of the grating is defined as:

“The angular separation Δθ per unit wavelength Δλ is called the dispersion D of the grating.”

D = Δθ/Δλ

For lines of nearly equal wavelengths to appear as widely as possible,we would like our grating to have the largest possible dispersion.

Since the grating equation is:

d Sinθ =mλ

Differentiating the above equation we have:

d cosθdθ = mλ

Now in terms of small differences the above equation we have:

d cosθ dθ =m dλ

Now in terms of small differences, the above relation can be written as:

d cosθ Δθ =mΔλ

Δθ/Δλ =m/d cosθ

D = m/d cosθ

From the above relation, we see that the dispersion D increases as the spacing between the slits ‘d’ decreases. We can also increase the dispersion by working at higher-order ( large m). Note that the dispersion does not depend on the number of rulings N.

### Resolving power definition

“The resolving power of an instrument is its ability to reveal minor details of the object under examination.”

#### Resolving Power of grating

“The resolving power of grating is a measure of how effectively it can separate or resolve two wavelengths in a given order of their spectrum”.

The diffraction grating is most useful for measuring accurately. Like the prism, the diffraction grating can be used to disperse a spectrum into its wavelength components. The grating is the more precise device if we want to distinguish two closely spaced wavelengths.

According to Rayleigh’s Criterion ” For two nearly equal wavelengths λ_{1} and λ_{2} between which a diffraction grating can just barely distinguish, the resolving power R of the grating is defined as:

R = λ/Δλ

Thus,a grating that has a high resolving power can distinguish small differences in wavelength.

Where λ = λ_{1} + λ_{2 }_{/}2 and Δλ = λ_{1} – λ_{2 }

Thus, resolving power increases with the increasing order number and with an increasing number of illuminated slits. If the lines are to be narrow, the angular separation δθ is small, then corresponding wavelengths interval Δλ must be small, and by equation(1) the resolving power must be large. To find the physical property of the grating that determines to resolve power R, we write the spacing between nearby lines as:

⇒R = Nm

Thus resolving power increases with the order number m and number of lines N.Resolving power is independent of the separation d of the slits.

Watch also video

External source

- https://en.wikipedia.org/wiki/Diffraction_grating
- http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html