How minimum intensity in interference is found: Imin = (√I1 − √I2)² explained

Short answer first: the minimum intensity in an interference setup with two coherent light sources is I_min = (√I1 − √I2)². That’s the result you get when the two waves cancel each other out as much as they can, i.e., during destructive interference. Now let me walk you through why this is the right formula and how it fits with what you already know.

Two waves, one story: how intensity adds up in interference

Imagine two light waves coming together. Each has its own intensity, I1 and I2. Intuitively, you might think the total light is just I1 + I2, but interference isn’t that simple. Light waves aren’t just blobs of energy moving independently; they’re oscillating electric fields that can push and pull at the same time. When two waves meet, their electric fields add. If the peaks line up with peaks (in phase), you get a bigger crest—constructive interference. If a peak meets a trough (out of phase by π), they partially cancel—destructive interference.

Mathematically, the rule for two coherent waves looks neat:

I = I1 + I2 + 2√(I1 I2) cos φ

Here φ is the phase difference between the two waves. The term 2√(I1 I2) cos φ is the interference term. If cos φ = 1 (φ = 0, constructive interference), you get the largest possible intensity:

I_max = (√I1 + √I2)²

If cos φ = −1 (φ = π, destructive interference), you get the smallest possible intensity:

I_min = (√I1 − √I2)²

That last expression is the star of the show. It’s saying: the minimum isn’t simply I1 − I2 or any other simple difference. It’s driven by the square roots of the individual intensities, then their difference, and finally squaring that difference. It’s a compact way to capture how whole-wave cancellation can be in practice.

Why the square roots matter

Think of √I1 and √I2 as the amplitudes of the two waves. Intensity is proportional to the square of amplitude. When you subtract the amplitudes before squaring, you’re encoding the idea that the waves partly cancel. If the amplitudes are equal, the subtraction wipes them out completely and I_min becomes zero:

If I1 = I2, then √I1 − √I2 = 0, so I_min = 0.

That’s the dream scenario for a perfect destructive interference: a dark fringe where no light would be detected, at that particular phase difference.

A quick numeric intuition

Let’s play with a couple of numbers to see it in action.

• Suppose I1 = 9 (units), I2 = 4. Then √I1 = 3 and √I2 = 2.

I_min = (3 − 2)² = 1.

So even at its darkest, you still have a little light, but it’s much reduced.

• If I1 = I2 = 16, then √I1 = √I2 = 4.

I_min = (4 − 4)² = 0.

A perfect dark fringe is possible when the phases line up just right.

Now, contrast that with the maximum case:

I_max = (√I1 + √I2)²

When the amplitudes add, you get the brightest fringe you can observe with those two sources.

A broader view: what this looks like in a lab

In a classic double-slit setup, light from two slits acts as two coherent sources. As you adjust the path difference between the two beams, you slide through a whole sequence of bright and dark fringes. The heights of those fringes—the fringe contrast—depends on the relative strengths I1 and I2. If one beam is much brighter than the other, the brightest fringes pop more dramatically, but the darkest fringes still obey the I_min formula. The math behind these patterns is exactly the I = I1 + I2 + 2√(I1 I2) cos φ rule.

A practical tangent: what if you’re comparing two light sources?

The same idea applies whether you’re looking at laser beams, filtered LEDs, or filtered sunlight in a clever optical experiment. The “square-root trick” is a handy mental heuristic: treat each intensity as an amplitude, then see how the two amplitudes combine under constructive or destructive interference. It’s a neat bridge between the wave picture and the energy picture.

Common pitfalls worth spotting

• Don’t assume I_min equals I1 − I2. That’s a tempting but incorrect simplification if you’re thinking only in terms of intensities and not the phase relationship.

• Remember that I_min and I_max depend on φ. If you hold the phase at π, you’re in the destructive zone and you hit I_min. If you hold it at 0, you’re in the constructive zone and you reach I_max.

• If one source is much dimmer than the other (say I2 is tiny), the minimum gets close to I1 − 2√(I1 I2) + I2, which is near I1 for small I2, but you won’t get all the way to zero unless the amplitudes match exactly.

A mini-tangent: beyond light

The same formula surfaces in other wave phenomena. Sound waves, water waves, and even radio waves obey the same core principle: the total energy depends on how the individual wave amplitudes combine. Understanding the two-wave case makes you fluent in more complex interference patterns later on, like adding a third slit or introducing a phase-changing element.

Putting it all together for a quick check

If someone hands you a multiple-choice setup and asks for the minimum intensity in an interference situation, here’s a quick checklist:

• Identify I1 and I2—the two source intensities.

• Remember the general interference form: I = I1 + I2 + 2√(I1 I2) cos φ.

• For a minimum, set cos φ = −1 (destructive interference).

• Compute I_min = (√I1 − √I2)².

That tiny checklist is a compact tool you can keep in your pocket for discussions or quick problem-solving sessions.

A few light-hearted lessons you can carry forward

• Interference highlights a broader truth: waves don’t just add energy; they carry phase information. That phase is the reason some fringes glow bright while others retreat into darkness.

• The square-root perspective isn’t just algebra; it’s a window into the wave nature of light. It helps you connect what you measure (intensity) with what the waves actually do (amplitude alignment and cancellation).

• The math is elegant but practical. It gives you a precise target for estimating how dark a fringe can get, based on how strong the two beams are.

A final, friendly nudge

If you’re curious how this all ties into real devices, look at interference coatings on lenses or the fringe patterns in holography. The same math underpins how engineers tune brightness, contrast, and even color fidelity in optical systems. The feeling of watching a pattern emerge—bright spots, then a sudden dip to near nothing—is not just pretty; it’s a direct whisper from the physics inside the glass and air.

In case you’re checking your understanding with a quick mental test: take any two sources with different intensities, predict the darkest point using I_min = (√I1 − √I2)², and then imagine how tweaking the phase shifts that darkest point. The two ideas—the amplitude viewpoint and the phase dance—are the heart of interference, and they’re surprisingly approachable once you see the pattern.

If you want to explore more, you can look at how adding a third slit changes the picture, or how coherence length limits the interference when the light isn’t perfectly coherent. The journey from a simple two-wave sketch to a richer interference story is short, and the payoff is a sharper intuition for how light behaves when it’s thick with waves and possibility.