The Exact Double Slit Equation

I am in favor of using dsinq = ml when possible, but with sound and microwaves it is sometimes inconvenient to have d very much smaller than the distance to the detector, so
dsinq = ml is not valid.

Let y = distance from the central maximum to a point of interest in the interference pattern. Let D = distance from the slits to the screen (on a line ^ to both-- if they are not parallel, forget it). We need the difference of length of the two rays, then set the difference equal to ml . Use Pythagorean theorem to get

[(y + d/2)² + D²]^1/2 - [(y - d/2)² + D²]^1/2 = ml

Move the 2^nd radical on the left over to the right (that sounds like a political change of heart), square both sides, simplify, and get

2yd = m² l ² + 2ml (y² - yd + d²/4 + D²)^1/2

which reduces to

yd/(ml ) -ml /2 = (y² -yd +d²/4 +D²)^1/2 .

Square both sides, do some algebra and show that

y = ml [D²/(d² - m² l ²) + 1/4]^1/2.

When m = 0, 1, 2, 3, … y is the distance from the center to the maxima. Now go to the main stuff on waves or back to the double slit.

Or go to one of the other main physics pages:
Mechanics
Fluids, heat, electricity and magnetism
Quantum
Index

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