Intensity, Amplitude Ratio & Coherence

PHYInterference & Young's Double Slit

When two coherent waves of amplitudes $a_1$ and $a_2$ superpose, the resultant intensity at any point depends on their phase difference $\phi$ through $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\,\cos\phi$ , since intensity scales as the square of amplitude ( $I \propto a^2$ ). This article covers converting amplitude ratios to intensity ratios, finding $I_{max} : I_{min}$ of the fringe pattern, and computing the intensity at a point from its phase or path difference.

It is a reliable numerical area in ISC wave optics because the algebra is short and the formulas are standard.

Intensity proportional to amplitude squared

\dfrac{I_1}{I_2} = \dfrac{a_1^{\,2}}{a_2^{\,2}}

I_1, I_2

are the intensities and

a_1, a_2

the amplitudes of the two waves; this follows from

I \propto a^2

Resultant intensity vs phase difference

I = I_1 + I_2 + 2\sqrt{I_1 I_2}\,\cos\phi

I

is the resultant intensity at a point and

\phi

the phase difference between the two superposing waves; it reduces to

I = 4I_0\cos^2(\phi/2)

when

I_1 = I_2 = I_0

Equal-source interference form

I = 4I_0\cos^2\!\left(\dfrac{\phi}{2}\right)

I_0

is the intensity of each (equal) source and

\phi

the phase difference;

I_{max} = 4I_0

\phi = 0

and

I = 0

\phi = \pi

Phase difference from path difference

\phi = \dfrac{2\pi}{\lambda}\,\Delta x

\phi

is the phase difference,

\Delta x

the path difference and

\lambda

the wavelength; a path difference of

\lambda

corresponds to

\phi = 2\pi

Max-to-min intensity ratio

\dfrac{I_{max}}{I_{min}} = \left(\dfrac{a_1 + a_2}{a_1 - a_2}\right)^{2} = \left(\dfrac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}}\right)^{2}

I_{max} \propto (a_1+a_2)^2

at constructive interference and

I_{min} \propto (a_1-a_2)^2

at destructive interference; the second form uses

a \propto \sqrt{I}

Coherence is the prerequisite: a steady (time-independent) phase difference $\phi$ between the sources is what makes the interference term $2\sqrt{I_1 I_2}\,\cos\phi$ stable. Without coherence the term averages to zero over time, leaving only $I = I_1 + I_2$ .
Amplitudes (with phase) combine to give the resultant, but intensities do NOT simply add — use $I \propto a^2$ to move between them, e.g. $a_1 : a_2 = 3:1$ gives $I_1 : I_2 = 9:1$ .
Constructive interference (maximum) occurs at $\phi = 2n\pi$ , i.e. path difference $\Delta x = n\lambda$ , giving $I_{max} \propto (a_1+a_2)^2$ . Destructive interference (minimum) occurs at $\phi = (2n+1)\pi$ , i.e. $\Delta x = (2n+1)\lambda/2$ , giving $I_{min} \propto (a_1-a_2)^2$ .
Always convert a given path difference to phase difference with $\phi = \dfrac{2\pi}{\lambda}\,\Delta x$ before using the cosine formula; e.g. $\Delta x = \lambda/3$ gives $\phi = 2\pi/3$ .
For two equal sources $(I_1 = I_2 = I_0)$ the resultant simplifies to $I = 4I_0\cos^2(\phi/2)$ , so the maximum is $4I_0$ (four times one source, not two) and the minimum is exactly zero.
The ratio $\dfrac{I_{max}}{I_{min}} = \left(\dfrac{a_1+a_2}{a_1-a_2}\right)^2$ ; plug in amplitudes directly, or use $a \propto \sqrt{I}$ when only intensities are given, e.g. $I_1 : I_2 = 1:4 \Rightarrow a_1 : a_2 = 1:2$ .
Energy is conserved: the average intensity over the whole pattern equals $I_1 + I_2$ . Interference does not create energy — the energy missing at the dark fringes appears at the bright fringes.
Keep ratios as pure numbers — $I_{max} : I_{min}$ and $I/I_{max}$ are dimensionless, so no units are involved; convert to actual values only if a numerical $I_0$ is supplied.

Where the marks go

Confusing amplitude ratio with intensity ratio by forgetting the square: $a_1 : a_2 = 3:1$ is $I_1 : I_2 = 9:1$ , NOT $3:1$ — and conversely $I_1 : I_2 = 1:4$ means $a_1 : a_2 = 1:2$ .
Halving the phase wrong in the cosine formula: it is $\cos^2(\phi/2)$ , not $\cos^2\phi$ . With $\phi = \pi/3$ the argument is $\phi/2 = \pi/6$ , so you need $\cos^2(\pi/6)$ .
Mixing up path difference and phase difference: a path difference of $\lambda$ is a phase difference of $2\pi$ (not $\pi$ ). Apply $\phi = (2\pi/\lambda)\,\Delta x$ every time rather than guessing.
Writing $I_{max} \propto a_1^2 + a_2^2$ or $I_{min} \propto a_1^2 - a_2^2$ . The correct forms square the SUM and DIFFERENCE of amplitudes: $I_{max} \propto (a_1+a_2)^2$ and $I_{min} \propto (a_1-a_2)^2$ .

How the board asks it

Numericalamplitude ratio to intensity ratio and fringe extremes
Two coherent sources have amplitudes in the ratio $a_1 : a_2 = 3 : 1$ . Calculate the ratio of (i) their intensities $I_1 : I_2$ and (ii) the maximum to minimum intensity $I_{max} : I_{min}$ in the resulting interference pattern.
Numericalresultant intensity from path difference
Two identical coherent sources, each of intensity $I_0$ , produce an interference pattern. Calculate the resultant intensity, in terms of $I_0$ , at a point where the path difference between the waves is $\lambda/3$ .
Give reasonscoherence is the prerequisite for sustained interference
Account for the fact that two independent sodium lamps placed side by side do not produce a sustained interference pattern, whereas the two slits in Young's double-slit experiment do. Explain in terms of coherence.
Give reasonsconservation of energy in interference
In a two-source interference pattern the average intensity over the whole pattern equals $I_1 + I_2$ . Explain how the existence of completely dark fringes, where no light arrives, is consistent with the law of conservation of energy.
Define / statecoherent sources
State what is meant by two coherent sources of light, and give the condition on their phase difference $\phi$ that allows a sustained interference pattern to be observed.
Assertion–Reasonintensity proportional to square of amplitude
Assertion: If the amplitudes of two interfering waves are in the ratio $1 : 2$ , their intensities are in the ratio $1 : 4$ . Reason: The intensity of a wave is directly proportional to the square of its amplitude. Choose the correct option regarding the truth of the assertion and the reason.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.