Diffraction Gratings

A diffraction grating is an optical device containing a very large number of closely spaced, parallel slits or grooves. When light passes through or reflects from the grating, waves from the different openings interfere with one another.

At certain angles, waves from all the slits arrive in phase and produce strong constructive interference. These positions appear as bright diffraction maxima.

Diffraction gratings are especially useful because different wavelengths reach maximum intensity at different angles. This allows a grating to separate light into its component wavelengths and produce a spectrum.

The central equation is:

d sin θ = nλ

where:

• d = spacing between neighbouring slits
• θ = diffraction angle
• n = order of the maximum: 0, 1, 2, 3, ...
• λ = wavelength

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From Two Slits to Many Slits

A double slit uses two openings to produce interference.

A diffraction grating extends the same idea to hundreds, thousands, or even more regularly spaced openings or grooves.

Each opening produces waves that overlap with waves from all the others.

At most angles, the contributions partially cancel.

At certain specific angles, however, waves from all the openings reinforce one another.

This produces very narrow, intense maxima.

Therefore:

many equally spaced slits → interference → sharp bright maxima

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Diffraction and Interference

Two wave behaviours are involved.

Diffraction causes waves to spread after passing through each narrow opening.

Interference occurs when the diffracted waves overlap.

The grating therefore needs the light from neighbouring openings to overlap so that interference can occur.

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Coherent Waves

For a clear interference pattern, the waves emerging from the different slits must have a stable phase relationship.

Because all the slits are illuminated by the same incoming wave, the emerging waves can behave as coherent sources.

This allows stable maxima and minima to form.

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Constructive Interference

Consider light travelling from two neighbouring slits toward a distant screen.

The light from one slit travels slightly farther than the light from the next slit.

This difference is called the path difference.

If:

path difference = whole number of wavelengths

the waves arrive in phase.

Therefore:

constructive interference occurs

and a bright maximum is produced.

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The Diffraction Grating Equation

The path difference between light from neighbouring slits is:

path difference = d sin θ

For constructive interference:

path difference = nλ

Therefore:

d sin θ = nλ

This is the diffraction grating equation.

It predicts the angles at which bright maxima occur.

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Meaning of Diffraction Order

The integer n represents the order of the diffraction maximum.

For:

n = 0

we have the central or zero-order maximum.

For:

n = 1

we have the first-order maximum.

For:

n = 2

we have the second-order maximum.

Higher orders may also occur if the geometry allows them.

The pattern occurs symmetrically on both sides of the central maximum.

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The Zero-Order Maximum

For:

n = 0

the grating equation becomes:

d sin θ = 0

Therefore:

θ = 0°

All wavelengths travel straight ahead.

This produces the zero-order maximum.

For white light, the zero-order maximum generally contains all visible wavelengths overlapping, so it appears white.

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First-Order Maxima

For the first-order maxima:

n = 1

Therefore:

d sin θ = λ

or:

sin θ = λ/d

First-order spectra appear on both sides of the central maximum.

They are commonly used for observing the different wavelengths in a light source.

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Second-Order Maxima

For:

n = 2

the grating equation becomes:

d sin θ = 2λ

The second-order maximum occurs at a larger angle than the first-order maximum for the same wavelength.

Therefore:

higher order → larger diffraction angle

provided that the order is physically possible.

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Diffraction Pattern

For monochromatic light, the pattern contains:

• central maximum
• first-order maximum on each side
• second-order maximum on each side
• possibly higher-order maxima

The arrangement is approximately:

−3, −2, −1, 0, +1, +2, +3

where the numbers represent diffraction order.

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Why Grating Maxima Are Sharp

With only two slits, constructive interference produces relatively broad bright fringes.

With many slits, constructive interference must occur simultaneously among waves from many openings.

A small change in angle quickly causes the waves from different slits to become significantly out of phase.

Therefore, a grating produces:

• narrow maxima
• well-defined angles
• good separation between wavelengths

This makes diffraction gratings extremely useful for precise measurements.

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Double Slit vs Diffraction Grating

Both rely on the principle of superposition and constructive interference.

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Grating Spacing

The quantity d is the distance between neighbouring slits or grooves.

It is called the:

• grating spacing
• slit spacing
• grating constant

Diffraction gratings are often specified by the number of lines per unit length rather than directly giving d.

For example:

500 lines/mm

means that 500 lines fit into 1 mm.

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Converting Lines per Millimetre to Slit Spacing

If a grating contains N lines per metre:

d = 1/N

Suppose a grating contains:

500 lines/mm

Convert to lines per metre:

500 lines/mm × 1000 mm/m

= 500,000 lines/m

Therefore:

d = 1/500,000

d = 2.00 × 10⁻⁶ m

or:

d = 2.00 μm

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Worked Example: Finding the Diffraction Angle

A grating has:

500 lines/mm

Light has wavelength:

λ = 600 nm

Find the first-order diffraction angle.

First calculate d:

d = 1/(500 × 1000)

d = 2.00 × 10⁻⁶ m

For n = 1:

d sin θ = nλ

sin θ = λ/d

sin θ = (600 × 10⁻⁹)/(2.00 × 10⁻⁶)

sin θ = 0.300

Therefore:

θ ≈ 17.5°

The first-order maximum occurs approximately 17.5° from the central maximum.

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Worked Example: Second-Order Maximum

Using the same grating and wavelength:

d = 2.00 × 10⁻⁶ m

λ = 600 × 10⁻⁹ m

n = 2

Then:

sin θ = 2λ/d

sin θ = 2(600 × 10⁻⁹)/(2.00 × 10⁻⁶)

sin θ = 0.600

Therefore:

θ ≈ 36.9°

The second-order maximum occurs farther from the center.

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Finding Wavelength

The grating equation can be rearranged:

λ = d sin θ/n

Suppose:

d = 1.67 × 10⁻⁶ m

θ = 20.0°

n = 1

Then:

λ = (1.67 × 10⁻⁶)sin(20.0°)

λ ≈ 5.71 × 10⁻⁷ m

Therefore:

λ ≈ 571 nm

This wavelength lies in the visible region.

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Finding Grating Spacing

Rearranging:

d = nλ/sin θ

Suppose light has:

λ = 500 nm

and its second-order maximum occurs at:

θ = 30°

Then:

d = 2(500 × 10⁻⁹)/sin(30°)

d = 1.00 × 10⁻⁶/0.500

d = 2.00 × 10⁻⁶ m

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Finding Lines per Millimetre

If:

d = 2.00 × 10⁻⁶ m

then:

N = 1/d

N = 1/(2.00 × 10⁻⁶)

N = 500,000 lines/m

Convert to lines/mm:

500,000/1000

N = 500 lines/mm

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Effect of Wavelength

From:

d sin θ = nλ

for constant d and n:

larger λ → larger θ

Therefore, longer wavelengths are diffracted through larger angles.

For visible light:

red light → larger diffraction angle

blue/violet light → smaller diffraction angle

This is how a grating separates different wavelengths.

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Comparing Red and Blue Light

Suppose a grating has:

d = 2.00 × 10⁻⁶ m

For blue light:

λ = 450 nm

First order:

sin θ = (450 × 10⁻⁹)/(2.00 × 10⁻⁶)

sin θ = 0.225

θ ≈ 13.0°

For red light:

λ = 650 nm

sin θ = (650 × 10⁻⁹)/(2.00 × 10⁻⁶)

sin θ = 0.325

θ ≈ 19.0°

Red light therefore appears farther from the central maximum.

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White Light Through a Grating

White light contains many wavelengths.

At:

n = 0

all wavelengths overlap.

The central maximum therefore appears approximately white.

For:

n = 1

different wavelengths appear at different angles.

The light spreads into a spectrum.

The same occurs for:

n = 2, 3, ...

if those orders are possible.

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Producing a Spectrum

A spectrum is the distribution of light according to wavelength or frequency.

A diffraction grating separates wavelengths because:

d sin θ = nλ

Different values of λ require different values of θ.

Therefore:

different wavelength → different diffraction angle → spatial separation

This allows individual wavelengths to be studied.

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Diffraction Grating vs Prism

Both gratings and prisms can separate white light into a spectrum, but they use different mechanisms.

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Emission Spectra

Hot gases can produce light at particular wavelengths.

When this light passes through a diffraction grating, it produces a set of bright spectral lines.

This is an emission spectrum.

Different elements produce characteristic patterns.

For example, the positions of spectral lines can reveal which elements are present in a light source.

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Absorption Spectra

When continuous light passes through a cooler gas, atoms can absorb particular wavelengths.

A diffraction grating can separate the remaining light into a spectrum containing dark lines.

This is an absorption spectrum.

The positions of the lines can help identify the substances responsible for the absorption.

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Spectroscopy

Spectroscopy is the study of spectra.

Diffraction gratings are important tools in spectroscopy because they can separate wavelengths precisely.

Spectroscopy can provide information about:

• chemical composition
• temperature
• motion
• density
• physical conditions of a source

It is used throughout physics, chemistry, astronomy, environmental science, and engineering.

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Spectrometers

A spectrometer is an instrument used to measure properties of a spectrum.

A basic optical spectrometer may include:

• entrance slit
• collimating optics
• diffraction grating
• focusing optics
• detector

The grating separates the wavelengths.

The detector measures the intensity at different wavelengths.

Modern instruments often record the entire spectrum electronically.

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Measuring Unknown Wavelengths

Suppose an unknown spectral line produces a maximum at a measured angle.

If d and n are known:

λ = d sin θ/n

Therefore, diffraction gratings provide a direct way to measure wavelengths.

This can be far more precise than estimating colour by eye.

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Astronomy

Astronomers collect light from:

• stars
• galaxies
• nebulae
• planets
• other astronomical objects

A diffraction grating can separate this light into a spectrum.

The spectrum can reveal information that cannot be obtained from an ordinary image alone.

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Determining Chemical Composition of Stars

Atoms produce characteristic spectral lines.

If astronomers observe these lines in starlight, they can identify elements present in the star's atmosphere.

For example, spectra can provide evidence for elements such as:

• hydrogen
• helium
• sodium
• calcium
• iron

Astronomers can therefore investigate the composition of objects that are far too distant to sample directly.

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Measuring Stellar Motion

Spectral lines can also shift because of the Doppler effect.

If spectral lines move toward longer wavelengths:

redshift

If they move toward shorter wavelengths:

blueshift

A diffraction grating can separate the wavelengths accurately enough for these shifts to be measured.

Therefore, diffraction gratings and the Doppler effect work together in astronomical spectroscopy.

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Studying Galaxies

Spectra from galaxies can provide information about:

• chemical composition
• motion
• star formation
• gas
• temperature
• redshift

The ability to separate and measure wavelengths makes diffraction gratings essential components of many astronomical instruments.

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Spectral Resolution

Two wavelengths may be extremely close together.

A good spectroscopic instrument should be able to distinguish them as separate spectral lines.

This ability is called resolution.

Using many illuminated grating lines can produce narrower maxima, allowing nearby wavelengths to be distinguished more easily.

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Resolving Power

The resolving power of an ideal diffraction grating can be expressed as:

R = λ/Δλ = nN

where:

• R = resolving power
• λ = wavelength
• Δλ = smallest wavelength difference that can be resolved
• n = diffraction order
• N = number of illuminated grating lines

Therefore, resolution can improve when:

• more grating lines are illuminated
• a higher diffraction order is used

This is one major advantage of diffraction gratings.

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Worked Example: Resolving Power

Suppose:

n = 2

and:

N = 5000 illuminated lines

Then:

R = nN

R = 2(5000)

R = 10,000

For light near:

λ = 600 nm

the approximate minimum wavelength difference is:

Δλ = λ/R

Δλ = 600/10,000

Δλ = 0.060 nm

This illustrates how gratings can distinguish very closely spaced wavelengths.

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Effect of Grating Spacing

From:

d sin θ = nλ

smaller d produces a larger diffraction angle for a given wavelength and order.

Therefore, a grating with more lines per millimetre generally spreads wavelengths more strongly.

Since:

d = 1/N

where N here represents line density in lines per metre:

more lines per unit length → smaller d → greater angular separation

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Maximum Possible Order

Not every value of n is physically possible.

Since:

sin θ ≤ 1

the grating equation requires:

nλ ≤ d

Therefore:

n ≤ d/λ

The largest possible integer satisfying this condition is the highest possible diffraction order.

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Worked Example: Maximum Order

Suppose:

d = 2.0 × 10⁻⁶ m

and:

λ = 600 nm = 6.0 × 10⁻⁷ m

Then:

n ≤ d/λ

n ≤ (2.0 × 10⁻⁶)/(6.0 × 10⁻⁷)

n ≤ 3.33

Therefore, the largest possible integer order is:

n_max = 3

The fourth order cannot occur because it would require:

sin θ > 1

which is impossible.

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Different Wavelengths Can Overlap

Different combinations of wavelength and order can sometimes produce the same diffraction angle.

For example:

n₁λ₁ = n₂λ₂

would give the same value of:

d sin θ

This means that higher-order spectra can sometimes overlap.

For example, second-order light of one wavelength may appear at the same angle as first-order light of twice that wavelength.

This can complicate spectral measurements.

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Transmission Gratings

A transmission grating contains many transparent openings separated by opaque regions.

Light passes through the openings.

The transmitted waves interfere and produce diffraction maxima.

These gratings are useful for:

• demonstrations
• spectrometers
• wavelength measurements
• optical experiments

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Reflection Gratings

A reflection grating uses many closely spaced grooves on a reflective surface.

Light reflects from the grooves rather than passing through them.

The reflected waves interfere.

Reflection gratings are widely used in professional spectroscopic instruments.

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CDs and DVDs as Gratings

The closely spaced tracks on optical discs can behave somewhat like a reflection diffraction grating.

White light reflecting from the surface can separate into different colours.

This produces the familiar rainbow appearance seen on CDs and DVDs.

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Why a CD Produces Colours

The surface contains regularly spaced structures.

Reflected light from neighbouring structures travels slightly different distances.

At some angles, a particular wavelength undergoes constructive interference.

At other angles, different wavelengths do.

As a result, different colours are observed in different directions.

This is an everyday example of diffraction and interference.

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Optical Communication

Diffraction gratings can be used in optical communication systems to separate or combine different wavelengths.

Modern fibre-optic networks can transmit multiple wavelengths through the same fibre.

Optical components can separate these wavelengths at the receiving end.

This allows large amounts of information to be transmitted efficiently.

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Lasers

Diffraction gratings are also used in some laser systems.

They can help:

• select particular wavelengths
• tune laser output
• analyze laser spectra
• separate unwanted wavelengths

Because gratings can distinguish small wavelength differences, they are valuable wherever precise wavelength control is required.

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Chemical Analysis

Spectroscopy using diffraction gratings can help identify substances.

A sample may:

• emit light
• absorb light
• reflect light

The resulting spectrum contains information about the material.

Applications include:

• laboratory chemistry
• environmental monitoring
• materials science
• industrial quality control

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Environmental Monitoring

Spectroscopic instruments can identify substances by how they interact with light.

This can help measure:

• atmospheric gases
• pollutants
• water composition
• industrial emissions

Diffraction gratings allow the relevant wavelengths to be separated and measured.

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Remote Sensing

Satellites and aircraft can carry spectrometers.

These instruments analyze light reflected or emitted by Earth's surface and atmosphere.

Spectral measurements can provide information about:

• vegetation
• water
• minerals
• atmospheric gases
• environmental change

The grating allows the incoming light to be separated into wavelength bands.

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Gratings in Modern Optical Technology

Diffraction gratings are found in or contribute to technologies including:

• spectrometers
• astronomical spectrographs
• optical communication systems
• laser equipment
• scientific instruments
• environmental sensors
• remote-sensing instruments

Their importance comes from their ability to separate wavelengths accurately.

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Experimental Measurement of Wavelength

A classroom experiment can use:

• monochromatic laser
• diffraction grating
• screen
• metre ruler

The grating-to-screen distance L is measured.

The distance y from the central maximum to a diffraction maximum is measured.

The angle can then be found using:

tan θ = y/L

Then:

λ = d sin θ/n

This allows the laser wavelength to be determined experimentally.

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Worked Experimental Example

A grating contains:

600 lines/mm

The screen is:

L = 1.50 m

from the grating.

The first-order maximum occurs:

y = 0.60 m

from the center.

First calculate the grating spacing.

600 lines/mm = 600,000 lines/m

d = 1/600,000

d = 1.67 × 10⁻⁶ m

Now find θ:

tan θ = y/L

tan θ = 0.60/1.50

tan θ = 0.400

Therefore:

θ ≈ 21.8°

Now:

λ = d sin θ

λ = (1.67 × 10⁻⁶)sin(21.8°)

λ ≈ 6.20 × 10⁻⁷ m

Therefore:

λ ≈ 620 nm

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Small-Angle Approximations

For very small diffraction angles:

sin θ ≈ tan θ ≈ θ

when θ is measured in radians.

This can simplify calculations.

However, diffraction gratings often produce relatively large angles.

Therefore, it is usually safer to calculate θ using the actual trigonometric functions unless the small-angle approximation is clearly appropriate.

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A Complete Grating Problem

A grating has:

400 lines/mm

Light of wavelength:

500 nm

falls on the grating.

Find the first- and second-order angles.

First:

400 lines/mm = 400,000 lines/m

d = 1/400,000

d = 2.50 × 10⁻⁶ m

For first order:

sin θ₁ = λ/d

sin θ₁ = (500 × 10⁻⁹)/(2.50 × 10⁻⁶)

sin θ₁ = 0.200

θ₁ ≈ 11.5°

For second order:

sin θ₂ = 2λ/d

sin θ₂ = 0.400

θ₂ ≈ 23.6°

Therefore:

first order ≈ 11.5°

second order ≈ 23.6°

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Choosing the Correct Equation

For a diffraction grating, the main equation is:

d sin θ = nλ

If line density is given:

d = 1/N

If screen measurements are given:

tan θ = y/L

These relationships can be combined.

A reliable sequence is:

line density → slit spacing → angle → wavelength

or:

line density → slit spacing → wavelength → angle

depending on what is unknown.

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Units

Unit conversion is especially important in diffraction-grating calculations.

Useful conversions include:

1 mm = 10⁻³ m

1 μm = 10⁻⁶ m

1 nm = 10⁻⁹ m

For line density:

1 line/mm = 1000 lines/m

Therefore:

500 lines/mm = 500,000 lines/m

A common mistake is forgetting this conversion before calculating d.

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Common Mistakes

Confusing slit spacing with slit width

The diffraction grating equation uses the distance between neighbouring slits or grooves.

Using lines/mm directly as d

Lines per millimetre is a line density. It must first be converted into a spacing.

Using n = 0 to calculate wavelength

The zero-order maximum gives θ = 0° for all wavelengths and does not separate them.

Thinking shorter wavelengths appear at larger angles

For the same order, longer wavelengths generally appear at larger angles.

Forgetting that spectra appear on both sides

The diffraction pattern is symmetrical around the central maximum.

Assuming every diffraction order must exist

A proposed order is impossible if it requires sin θ > 1.

Assuming higher orders are always better

Higher orders can provide greater separation but may be weaker and can overlap with other spectral orders.

Using the small-angle approximation at large angles

Use the exact trigonometric relationships when the angle is not small.

Confusing a diffraction grating with a prism

A grating separates wavelengths mainly through interference. A prism separates them through wavelength-dependent refraction.

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Key Terms

Diffraction grating — optical device containing many regularly spaced slits or grooves that produce interference.

Grating spacing — distance d between neighbouring slits or grooves.

Line density — number of grating lines per unit length.

Diffraction maximum — direction of strong constructive interference.

Diffraction order — integer n identifying a particular set of maxima.

Zero-order maximum — central maximum where θ = 0°.

First-order spectrum — wavelength-separated pattern corresponding to n = 1.

Constructive interference — reinforcement of waves that arrive in phase.

Path difference — difference in distance travelled by waves from neighbouring openings.

Spectrum — distribution of electromagnetic radiation according to wavelength or frequency.

Spectroscopy — study and measurement of spectra.

Spectrometer — instrument used to measure a spectrum.

Emission spectrum — pattern of wavelengths emitted by a source.

Absorption spectrum — spectrum containing wavelengths removed through absorption.

Spectral line — narrow wavelength feature in a spectrum.

Resolving power — ability of an optical system to distinguish closely spaced wavelengths.

Transmission grating — grating in which light passes through many openings.

Reflection grating — grating in which light reflects from regularly spaced grooves.

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Key Takeaways

• A diffraction grating contains many closely spaced, regularly arranged slits or grooves.
• Waves from the different openings interfere according to the principle of superposition.
• Constructive interference produces sharp bright maxima at specific angles.
• The grating equation is d sin θ = nλ.
• The central maximum corresponds to n = 0.
• First-order maxima correspond to n = 1.
• Higher diffraction orders occur at larger angles for a given wavelength.
• Grating spacing is the distance between neighbouring lines, not the number of lines per unit length.
• If line density is known, spacing can be calculated using d = 1/N.
• Longer wavelengths are diffracted through larger angles than shorter wavelengths in the same order.
• White light is separated into its component wavelengths because each wavelength satisfies the grating equation at a different angle.
• The zero-order maximum does not separate wavelengths.
• Diffraction gratings generally produce sharper maxima than double slits because many waves contribute to the interference.
• Increasing the number of illuminated grating lines can improve spectral resolution.
• Higher diffraction orders can provide greater wavelength separation but may overlap.
• Diffraction gratings can be used to measure unknown wavelengths precisely.
• Spectroscopy uses gratings to investigate the composition and properties of light sources.
• Astronomers use grating-based spectrographs to investigate stars, galaxies, and other astronomical objects.
• Gratings are also important in lasers, optical communications, environmental monitoring, remote sensing, and scientific instruments.
• The diffraction grating demonstrates how interference can become a powerful measurement tool rather than simply a wave phenomenon.

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Check Your Understanding

1. Explain how a diffraction grating differs from a double slit.
2. Why do many closely spaced slits produce sharper interference maxima than two slits?
3. State the diffraction grating equation and explain each quantity.
4. Explain what is meant by the zero-order, first-order, and second-order maxima.
5. A grating has 500 lines/mm. Calculate its grating spacing in metres.
6. Light of wavelength 600 nm passes through the grating in Question 5. Calculate the angle of the first-order maximum.
7. Calculate the angle of the second-order maximum for the same wavelength and grating.
8. A first-order spectral line occurs at 25° using a grating with 600 lines/mm. Calculate the wavelength.
9. Explain why red light appears at a greater diffraction angle than blue light in the same diffraction order.
10. A grating has spacing 2.0 × 10⁻⁶ m and is illuminated with 500 nm light. Determine the highest possible diffraction order.
11. Explain why the zero-order maximum of white light is normally white while the first-order pattern is separated into colours.
12. Compare how a prism and a diffraction grating produce a spectrum.
13. Explain how astronomers can use a diffraction grating to determine the chemical composition of a star.
14. Explain why increasing the number of illuminated grating lines can help distinguish two wavelengths that are very close together.
15. A scientist observes a spectral line at a known laboratory wavelength, but an astronomical spectrograph shows the line at a slightly longer wavelength. Explain how the diffraction grating allows the wavelength to be measured and how the observed shift could then provide information about the motion of the astronomical object.