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Double-Slit Interference and Diffraction Gratings

Distinguish fine double-slit or grating directions from a finite-slit diffraction envelope.

AP Physics 2 Topic 14.8 · College Board CED printed p. 124, 125 / PDF p. 131, 132 · See this topic in the course outline

Two slits set the fine spacing. Each slit’s finite width sets a broader envelope. Many slits sharpen the bright directions.

Before you start
  • Single-slit diffraction
  • Superposition
  • Small-angle approximation

Fine fringes from two coherent sources

Young's double-slit pattern shows light's wave properties: a single source illuminates both slits, giving a stable phase relationship. Contributions from slits separated by d differ in path by Δ=dsin⁡θ\Delta=d\sin\theta. For initially in-phase coherent light, bright directions satisfy dsin⁡θ=mλd\sin\theta=m\lambda, and dark interference directions satisfy dsin⁡θ=(m+12)λd\sin\theta=(m+\tfrac12)\lambda. Equal path length gives the central bright fringe.

At |θ| < 10°, ymax⁡≈mλL/dy_{\max}\approx m\lambda L/d, so adjacent bright fringes are approximately uniformly spaced by λL/d. They are uniformly spaced in sin θ exactly, not generally in screen position at large angles. Increasing d packs the fringes closer; increasing λ or L spreads them.

Finite slit width a still produces single-slit diffraction. Fine interference peaks lie under the broad envelope with zeros at a sin θ = pλ. If an interference maximum coincides with an envelope zero, that bright order is missing. A photograph needs both scales: d sets fringe spacing, a sets envelope width. A perfectly narrow slit model suppresses this envelope and is an approximation.

Interference peaks inside a diffraction envelopeSmall-angle example: λ=600 nm, L=2.0 m, d=0.40 mm, a=0.10 mm. Fine interference-order reference positions are y=3m mm; the envelope zeros are at y=±12 mm, so m=±4 is missing. Relative intensity at order reference positions m=0,1,2,3,4 is 1,0.811,0.405,0.090,0. The fine peaks sit inside a broad envelope; the curve displays shape as well as positions. The slight envelope modulation can shift the exact local two-slit intensity maxima away from the ideal interference-only reference positions.-18-12-606121800.51Relative intensity (maximum = 1)Screen position y (mm)Dashed: single-slit envelopeBlue: two-slit intensity
Interference peaks inside a diffraction envelope. Small-angle example: λ=600 nm, L=2.0 m, d=0.40 mm, a=0.10 mm. Fine interference-order reference positions are y=3m mm; the envelope zeros are at y=±12 mm, so m=±4 is missing. Relative intensity at order reference positions m=0,1,2,3,4 is 1,0.811,0.405,0.090,0. The fine peaks sit inside a broad envelope; the curve displays shape as well as positions. The slight envelope modulation can shift the exact local two-slit intensity maxima away from the ideal interference-only reference positions.
Double-slit path-difference constructionTwo slits separated vertically by d emit in phase. The outgoing rays are parallel at angle θ to the horizontal normal. Project slit separation onto the ray direction: the extra lower-slit path is Δ=d sin θ. A reference line perpendicular to the rays completes a right triangle; the green projection segment is the extra path. Bright interference requires Δ=mλ; dark interference requires a half-integer wavelength.Far-field paths: Δ is the projection of slit separationnormalslit 1slit 2dΔ = d sin θθθParallel directions at large LOrange: perpendicular reference lineGreen: extra path from slit 2
Double-slit path-difference construction. Two slits separated vertically by d emit in phase. The outgoing rays are parallel at angle θ to the horizontal normal. Project slit separation onto the ray direction: the extra lower-slit path is Δ=d sin θ. A reference line perpendicular to the rays completes a right triangle; the green projection segment is the extra path. Bright interference requires Δ=mλ; dark interference requires a half-integer wavelength.

Grating orders and white-light dispersion

A diffraction grating contains many evenly spaced parallel openings. Neighboring openings have path difference d sin θ, so principal maxima again obey dsin⁡θ=mλd\sin\theta=m\lambda. Many coherent contributions reinforce strongly only near those directions, making narrower, more distinct peaks than two slits. If a grating has G lines per meter, d = 1/G.

An order exists only if |mλ/d| ≤ 1. For white light, m = 0 puts every wavelength at θ = 0, making a white central maximum. For the same nonzero order, longer wavelength needs a larger |sin θ|, so red appears farther from center than violet. Different orders can overlap; the grating does not guarantee a unique color at every location. Use the wavelength within the propagation medium; our numerical examples use air, approximated as vacuum.

Interference peaks inside a diffraction envelopeIllustration with eight equally spaced slits, width/separation a/d=1/4, and d/λ=10. The plotted |u|≤6 is within the physical |u|≤d/λ=10 domain. Principal interference directions have integer u. The finite slit width makes envelope zeros at u=±4, so those orders disappear; adding more slits sharpens principal peaks without moving their path-condition directions. The fine peaks sit inside a broad envelope; the curve displays shape as well as positions. Integer u marks principal maxima of the eight-slit interference factor. The finite-width diffraction envelope weights these orders and suppresses the u=±4 orders. Multiplying the interference factor by the envelope can slightly displace local maxima of the complete finite-grating intensity; the path-condition reference directions still satisfy d sin θ=mλ.-6-4-2024600.51Relative intensity (maximum = 1)u = d sin θ / λ (dimensionless)Dashed: single-slit envelopeBlue: eight-slit grating intensity
Interference peaks inside a diffraction envelope. Illustration with eight equally spaced slits, width/separation a/d=1/4, and d/λ=10. The plotted |u|≤6 is within the physical |u|≤d/λ=10 domain. Principal interference directions have integer u. The finite slit width makes envelope zeros at u=±4, so those orders disappear; adding more slits sharpens principal peaks without moving their path-condition directions. The fine peaks sit inside a broad envelope; the curve displays shape as well as positions. Integer u marks principal maxima of the eight-slit interference factor. The finite-width diffraction envelope weights these orders and suppresses the u=±4 orders. Multiplying the interference factor by the envelope can slightly displace local maxima of the complete finite-grating intensity; the path-condition reference directions still satisfy d sin θ=mλ.

Test whether an order can exist

A grating has 500 lines/mm. For λ = 600 nm find the first-order angle and the largest possible order.

  1. G = 5.00 × 10⁵ lines/m; d = 1/G = 2.00 × 10⁻⁶ m.

    Why this step? The equation uses slit spacing, not the number of lines in the original units.

  2. sin θ₁ = λ/d = 0.300, so θ₁ = 17.5°.

    Why this step? Use the exact angular relation because the small-angle boundary is exceeded.

  3. |m| ≤ d/λ = 3.33; the largest integer order is 3.

    Why this step? A sine cannot exceed one, so the fourth order is physically impossible.

Answer and check

θ₁ = 17.5°; maximum order m = 3 on each side.

Why a grating’s white center becomes colored orders

Each evenly spaced grating opening diffracts; the many coherent contributions superpose. In a principal direction, the path difference between neighboring openings is mλ, so all corresponding phases reinforce. Adding openings at the same spacing makes this phase agreement more selective and sharpens the principal peaks without changing d sin θ = mλ.

For m = 0, every wavelength has zero path difference at θ = 0, so white light’s colors overlap in a white central maximum. For a particular nonzero order, sin θ = mλ/d depends on wavelength. Red, with longer λ, appears farther from center than violet. The diagram places six sample colors on the same measured angular axis; a real white spectrum fills the intervening colors. It complements a monochromatic intensity graph, which cannot show white-light dispersion.

A grating-normal angular axis runs from −20 to +20 degrees. All white-light wavelengths share the m = 0 central white direction. In each first order, six illustrative wavelengths have angle magnitude arcsin(lambda/2500 nm): violet 400 nm at 9.21 degrees, blue 470 nm at 10.84, green 530 nm at 12.24, yellow 580 nm at 13.41, orange 610 nm at 14.12, and red 650 nm at 15.07 degrees. Violet is closest and red farthest in each nonzero order. Negative and positive orders are mirror images. Bars sample a continuous white spectrum; no exact category cutoff or intensity scale is implied.
Exact first-order color directions for a 400 lines/mm grating in air.
Inspect diagram labels

The diagram keeps its full size here. Focus the view and use the arrow keys, or scroll across it, to inspect every label.

A grating-normal angular axis runs from −20 to +20 degrees. All white-light wavelengths share the m = 0 central white direction. In each first order, six illustrative wavelengths have angle magnitude arcsin(lambda/2500 nm): violet 400 nm at 9.21 degrees, blue 470 nm at 10.84, green 530 nm at 12.24, yellow 580 nm at 13.41, orange 610 nm at 14.12, and red 650 nm at 15.07 degrees. Violet is closest and red farthest in each nonzero order. Negative and positive orders are mirror images. Bars sample a continuous white spectrum; no exact category cutoff or intensity scale is implied.

View diagram in a separate tab ↗

14.8.A.414.8.A.5

Place red and violet on an angular scale

White light strikes a grating with 400 lines/mm normally. Compare red λ = 650 nm and violet λ = 400 nm in air. Use exact angular trigonometry.

A grating-normal angular axis runs from −20 to +20 degrees. All white-light wavelengths share the m = 0 central white direction. In each first order, six illustrative wavelengths have angle magnitude arcsin(lambda/2500 nm): violet 400 nm at 9.21 degrees, blue 470 nm at 10.84, green 530 nm at 12.24, yellow 580 nm at 13.41, orange 610 nm at 14.12, and red 650 nm at 15.07 degrees. Violet is closest and red farthest in each nonzero order. Negative and positive orders are mirror images. Bars sample a continuous white spectrum; no exact category cutoff or intensity scale is implied.
Exact first-order color directions for a 400 lines/mm grating in air.
Inspect diagram labels

The diagram keeps its full size here. Focus the view and use the arrow keys, or scroll across it, to inspect every label.

A grating-normal angular axis runs from −20 to +20 degrees. All white-light wavelengths share the m = 0 central white direction. In each first order, six illustrative wavelengths have angle magnitude arcsin(lambda/2500 nm): violet 400 nm at 9.21 degrees, blue 470 nm at 10.84, green 530 nm at 12.24, yellow 580 nm at 13.41, orange 610 nm at 14.12, and red 650 nm at 15.07 degrees. Violet is closest and red farthest in each nonzero order. Negative and positive orders are mirror images. Bars sample a continuous white spectrum; no exact category cutoff or intensity scale is implied.

View diagram in a separate tab ↗

  1. 400 lines/mm = 4.00 × 10⁵ lines/m, so d = 1/G = 2.50 × 10⁻⁶ m.
  2. For m = 1, sin θred = 0.260 and sin θviolet = 0.160. Thus θred = 15.07° and θviolet = 9.21°; the negative order is symmetric.
  3. At m = 0, θ = 0 for both and all intervening wavelengths: their light overlaps as white.
  4. More coherent openings at the same d sharpen the principal peaks without moving these reference directions. A finite slit-width envelope can still weight or suppress orders.

Answer and check

The white center stays at zero; each first order spreads violet closer to center and red farther out. Red’s angle exceeds 10°, so the exact arcsine must be used.

14.8.A.414.8.A.5

Make the model respond

Focused Physics 2 model

Interference orders and standing waves

Predict first. Choose a model. Predict how wavelength changes bright orders or how harmonic number changes frequency.

Bright orders: d sin θₘ = mλ; exact screen y = L tan θₘ; small-angle estimate y ≈ Lmλ/d

Interference orders and standing waves: graphOnly integer orders are physical maxima. Lines connect the sampled values to aid comparison. The display clips |y| > 4 m; the table retains those exact positions and labels forbidden or grazing orders. Both apparatus choices share the same ideal maximum-angle condition; the model does not compute peak width or a single-slit envelope. Exact values and their units appear in the open data table below.-5-2.502.55-4-2024Selected orderScreen coordinate y (m)Integer bright order m (sampled −5…+5)
  • Exact finite screen positions
  • Small-angle estimates
  • Selected or reference point
Only integer orders are physical maxima. Lines connect the sampled values to aid comparison. The display clips |y| > 4 m; the table retains those exact positions and labels forbidden or grazing orders. Both apparatus choices share the same ideal maximum-angle condition; the model does not compute peak width or a single-slit envelope.

On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.

Selected angular order14.48 °
Exact screen position0.5164 m
Maximum angular |m|4

Order m = 1: θ = 14.48°, exact y = 0.5164 m; small-angle y ≈ 0.5 m.

Read the model data and comparisons
Selected-state values. SI units are shown explicitly; display controls may use named SI prefixes.
QuantityValueUnitMeaning or limit
Wavelength5.000 × 10^−7m
Slit spacing2.000 × 10^−6m
Screen distance2m
Selected integer order1
Required path difference mλ5.000 × 10^−7m
Selected angle14.48°
Exact screen coordinate0.5164m
Small-angle estimate0.5mOnly accurate when |mλ/d| ≪ 1
Maximum allowed angular |m|4Equality gives grazing order, with no finite screen intersection
Order -5 exact yforbiddenm|mλ/d| > 1
Order -4 exact ygrazingmθ = -90°
Order -3 exact y-2.268mθ = -48.59°
Order -2 exact y-1.155mθ = -30°
Order -1 exact y-0.5164mθ = -14.48°
Order 0 exact y0mθ = 0°
Order 1 exact y0.5164mθ = 14.48°
Order 2 exact y1.155mθ = 30°
Order 3 exact y2.268mθ = 48.59°
Order 4 exact ygrazingmθ = 90°
Order 5 exact yforbiddenm|mλ/d| > 1

Two coherent slits produce constructive interference when the path difference is an integer wavelength. A bright-order number is an integer, not a continuous position. Convert the allowed angle to a flat-screen coordinate with tan θ. Using sin θ ≈ tan θ ≈ θ is a separate approximation and can noticeably underestimate positions at larger angles.

Assumptions and limits. Monochromatic coherent illumination, identical narrow slits, uniform center spacing, far-field interference, screen perpendicular to the central beam. The maximum-order condition uses exact trigonometry within this ideal far-field geometry. Finite slit width, missing orders from the diffraction envelope, brightness, peak width, and thin-film interference are not modeled.

Try it from memory

Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.

Question 1

A double-slit pattern has fine maxima under a broad envelope. Which length determines each scale?

Worked answer and reason

Separation d sets fine fringe spacing; width a of each slit sets the broad diffraction envelope.

Both slit interference and each slit’s diffraction contribute.

Question 2

Why is the central grating maximum white, but the first order separates colors?

Worked answer and reason

At m = 0 every λ has zero path difference and the same direction. At m = 1, sin θ = λ/d differs by color, with red farther out.

The grating selects equal integer path differences for each wavelength.

JavaScript enables saved drafts and self-assessment. You can always work on paper.

Catch the tempting mistake

Every bright fringe in a real double-slit pattern has equal intensity.
Its finite-width slits supply an envelope that modulates brightness and can remove orders entirely.

Put it to work

Give the physical reason as well as the calculation. The worked solutions are on a separate page.

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For the teacher
  • Ask for both a and d on every sketch. Require sin θ ≤ 1 before accepting a grating order.

Unit planning and inquiry guidance →

Source objectives and scope
  • 14.8.A — Describe the behavior of a wave and the diffraction pattern resulting from the wave passing through multiple openings.

Suggested skills are teaching guidance. The course applies all science practices across its objectives.

Science references for this teaching
Glossary and deeper reading · 8 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Diffraction

    Diffraction is the spreading and interference of waves near openings or obstacles. The spreading is especially pronounced when the opening or obstacle size is comparable to wavelength. For example, sound spreads through a doorway much more noticeably than visible light because sound wavelengths are far larger.

    Read more: Diffraction in detail →

  • Wave

    A wave is a propagating disturbance that can transfer energy without transporting the medium as a whole. Mechanical waves require a material medium, while electromagnetic waves can propagate through vacuum. For example, a pulse travels along a rope while individual rope segments move locally around their equilibrium positions.

    Read more: Wave in detail →

  • Oscillation phase

    Phase specifies the position within a repeating oscillation cycle. A phase difference tells how much one oscillation leads or lags another at a chosen place and time. For example, two sinusoidal waves half a cycle out of phase can cancel when their amplitudes are equal.

  • Wave interference

    Wave interference is the combination of overlapping wave disturbances. In a linear medium, the signed disturbances add at each location and time without permanently destroying the passing waves. For example, two overlapping upward rope pulses temporarily make a larger upward disturbance.

    Read more: Wave interference in detail →

  • Diffraction grating

    A diffraction grating is an array of many regularly spaced openings or scattering regions. Their coherent wave contributions produce narrow reinforced directions determined by spacing and wavelength. For example, a grating spreads white light into colored orders because different wavelengths reinforce at different angles.

  • Coherence

    Coherence describes a sufficiently stable phase relationship between waves for a sustained interference pattern. It depends on the sources and on the time or distance over which their phase relationship remains predictable. For example, illuminating two slits with one suitable monochromatic source produces a stable relative phase at the slits.

  • Wavelength

    Wavelength is the spatial distance between neighboring points at the same phase of a periodic wave. It connects propagation speed and frequency through speed equal to frequency times wavelength. For example, neighboring crests of a sinusoidal water wave are separated by one wavelength.

    Read more: Wavelength in detail →

  • Wave intensity

    Wave intensity is the time-averaged energy-transfer rate per area perpendicular to propagation. It is measured in watts per square meter and differs from total power because area matters. For example, spreading the same sound power over a larger area reduces the intensity there.

    Read more: Wave intensity in detail →

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