Intuition The ONE core ideaTwo identical light-waves start out marching in step, but one has to walk a little farther to reach a spot on the screen — and that tiny extra walk is measured in wavelengths . When the extra walk is a whole number of wavelengths the waves add up to a bright band; a half-wavelength off and they cancel to darkness.
This page builds every letter and symbol the parent derivation quietly assumes, from absolute zero — one wave-picture at a time. By the last section you will meet the final formula for fringe spacing and understand exactly what each mark on it is a picture of . We start with nothing but a ripple.
Before any letters, look at what light is here: a repeating up-and-down ripple travelling forward.
Figure 1 below shows one such wave.
λ λ distance from one crest of the wave to the next crest . It is the wave's own built-in ruler.
Picture: in Figure 1, λ Why the topic needs it: the whole bright/dark game is about comparing a travel-distance to how long one wave is . Without a length to compare against, "path difference" would be a naked number with no meaning.
Definition Crest and troughA crest is the highest point of the ripple (wave pushed up); a trough is the lowest (wave pushed down).
Picture: in Figure 1, the lavender dot marks a crest (top of a red hump), the mint dot marks a trough (bottom of a dip).
Why: "crest meets crest" (bright) and "crest meets trough" (dark) is the entire mechanism — see Interference of light .
Definition Coherent sourcesTwo sources are coherent when they keep a fixed rhythm relative to each other — every crest leaves source 1 at a predictable moment relative to a crest leaving source 2.
Picture: two people tapping a pond at the exact same beat forever, never drifting.
Why the topic needs it: if the two slits drifted out of step randomly, the bright and dark spots would flicker and smear into grey. A steady pattern requires coherence — that is why we use one lamp behind two slits rather than two separate lamps. Full story in Coherence and coherent sources .
Figure 2 below sets up the whole apparatus — refer to it for the next four definitions.
S 1 S 2 d S 1 S 2 two narrow openings that light passes through; they behave like two tiny coherent sources. d distance between them .
Picture: in Figure 2, the two mint dots on the left barrier; d Why: the two waves are born here. Because S 1 + 2 d S 2 − 2 d d
d Why it feels right: both are small distances at the slits.
Why it's wrong: d gap between the two slits' centres , not how fat a single slit is. Slit width controls diffraction envelope (see Diffraction grating ); slit separation d Fix: d S 1 S 2
Still on Figure 2 :
Definition Screen distanceD D straight-across distance from the slit barrier to the screen .
Picture: in Figure 2, the long horizontal lavender arrow from the slits to the wall.
Why: the farther the screen, the more the two paths spread apart on the screen, so D
O O directly opposite the midpoint of the two slits .
Picture: in Figure 2, where the dashed central axis hits the wall.
Why: it is our "zero" — all screen heights are measured from here.
y P P any chosen point on the screen ; y O Picture: in Figure 2, the coral dot P y O Why: we want a formula for where bright fringes land. That "where" is a value of y
This is the heart, so it gets its own figure. Figure 3 below zooms into the two rays reaching P
Definition The fringe angleθ θ angle the two (nearly parallel) rays make with the central axis as they head from the slits to P Picture: in Figure 3, the small butter-coloured wedge between the dashed central axis and a ray. When P θ P = O θ = 0 Why we introduce it now: distances on the screen (y θ
Definition Path differenceΔ Δ how much farther the wave from the far slit travels to reach P Δ = S 2 P − S 1 P Picture: in Figure 3, drop a perpendicular from S 1 S 2 P coral stub between S 2 Δ Why: everything — bright, dark, spacing — is decided by comparing Δ λ Path difference and phase difference .
Δ λ Δ ratio Δ/ λ Δ λ
Definition Pythagoras' theoremFor a right triangle with legs a b c c = a 2 + b 2 Picture: the large right triangle in Figure 3 formed by the horizontal D P S 1 P S 2 P Why this tool and not another? Each ray goes across (D up (a height). That is literally a right triangle, and Pythagoras is the one rule that turns two perpendicular distances into the straight-line distance. That is why the parent writes S 1 P = D 2 + ( y − 2 d ) 2
≫ D ≫ d D much greater than d θ Picture: in Figure 3 the two rays look nearly parallel — squint and they merge; the wedge θ Why we need it: it lets us swap sin θ tan θ θ S 1 P + S 2 P 2 D Δ = D y d Small angle approximation .
Δ = d sin θ Δ = y d / D Why it feels right: they look different.
Why it's wrong: they are the same statement . Since sin θ ≈ tan θ = y / D d sin θ ≈ y d / D Fix: identical under the small-angle approximation — one is the angle form, the other the screen-position form.
The extra walk Δ everything . There are exactly two special cases, and we build both.
Definition Bright-fringe conditionCrests aligned means the extra walk is a whole number of wavelengths:
Δ = nλ , n = 0 , ± 1 , ± 2 , … n which bright band.
Definition Dark-fringe conditionCrest-on-trough means the extra walk is a whole number of wavelengths plus half a wavelength :
Δ = ( n + 2 1 ) λ , n = 0 , ± 1 , ± 2 , … Δ = ( 2 n + 1 ) 2 λ odd multiple of a half-wavelength.
Common mistake "Dark meansΔ = nλ /2 Why it feels right: "half a wavelength = cancel" sounds clean, and at n = 1 λ /2 Why it's wrong: at n = 2 Δ = λ whole wave → bright , not dark!
Fix: dark = ( n + 2 1 ) λ λ /2
Now feed each condition into the master formula Δ = D y d y
n n integer label (n = 0 , ± 1 , ± 2 , … n = 0 n = 1 Picture: number the bright stripes outward from the centre: 0 , 1 , 2 , … Why: to locate a specific fringe we need to name it; n
β β gap on the screen between two neighbouring bright bands (equivalently, between two neighbouring dark bands): β = y n + 1 − y n Picture: the constant spacing between adjacent stripes.
Why: it is the final target of the whole derivation — the one number that says how spread-out the pattern is. Subtracting consecutive bright positions, the n β = d ( n + 1 ) λ D − d nλ D = d λ D . Read it back with everything you now know: λ D d
The chain below shows how each foundation on this page feeds the next, ending at the fringe-width result. Figure 5 draws the same chain as a picture in case the diagram does not render for you.
Wave and wavelength lambda
Interference bright and dark
Screen distance D and height y
Pythagoras right triangle
Small angle approx D much bigger
Fringe order n and positions y_n
Fringe width beta equals lambda D over d
Test yourself — say each answer aloud before revealing.
What does λ The distance from one wave crest to the next — the wave's own ruler.
What is d The gap between the two slit centres; NOT the width of a single slit.
What is D The straight-across distance from the slit barrier to the screen.
Where is the point O On the screen, directly opposite the midpoint of the two slits — the "zero" of y
What does y The height of the chosen point P O
What is the fringe angle θ The angle the rays to P
Define path difference Δ How much farther one slit's wave travels to reach P Δ = S 2 P − S 1 P
What are the two equal forms of Δ Δ = d sin θ ≈ D y d
State the bright-fringe condition. Δ = nλ
State the dark-fringe condition. Δ = ( n + 2 1 ) λ
Where do dark fringes sit relative to bright ones? Exactly halfway between neighbouring bright fringes.
Why is sin θ ≈ tan θ ≈ θ Corrections start at θ 3
Why must the sources be coherent? So the pattern holds still instead of flickering into grey.
Which theorem turns "across D y Why is S 1 P + S 2 P ≈ 2 D Each ray is ≈ D D ≫ y , d ( d /2 ) 2 y 2
Why does tan θ = y / D Tangent is opposite over adjacent; opposite side is y D
What does the integer n y n n y n = nλ D / d y n = ( n + 2 1 ) λ D / d
What is β The spacing between two neighbouring bright (or dark) fringes, y n + 1 − y n = λ D / d
Young's double slit — fringe width derivation — the parent this page prepares you for.Interference of light — the crest-meets-crest mechanism.Coherence and coherent sources — why one lamp, two slits.Path difference and phase difference — the meaning of Δ Small angle approximation — the tool that cleans up the geometry.Refractive index — needed for the "in water" example next.Diffraction grating — where slit width vs. separation matters. 
