Conditions for Observable Interference

The conditions for interference are the design rules of every interferometric instrument. Fourier-transform infrared spectrometers work only because a lamp's temporal coherence, though short, is enough near zero path difference — and the visibility-versus-delay curve they record is the spectrum. Gravitational-wave detectors like LIGO push the amplitude and coherence conditions to an extreme: kilometre arms fed from one ultra-stable laser, balanced to part-per-billion so the dark fringe is genuinely dark, because a residual $I_{min}$ would bury a passing wave's signal. Nomarski differential-interference-contrast microscopes turn the polarization condition into a feature, deliberately shearing light into orthogonal states and then recombining them through an analyzer so that phase gradients in a transparent cell become visible contrast. And every time an astronomer measures a star's diameter by watching fringes fade as a baseline grows, they are reading the spatial coherence condition directly off the sky.

Represent each wave as a spinning clock hand — its phasor — of length $\sqrt I$. The resultant is the vector sum of the two hands, and the brightness you see is the length of that sum squared. If the angle between the hands is held fixed as you move across the screen, the sum sweeps smoothly from fully aligned (a bright fringe, hands parallel) to fully opposed (a dark fringe, hands antiparallel): a stationary striped pattern. If instead the relative angle jitters randomly from one instant to the next — as it does for two independent sources — the sum's length takes every value in turn, and the exposure averages them into a flat, featureless grey. That contrast between a fixed relative phase and a scrambled one is the whole story, and the four conditions are four ways of keeping the relative phase both fixed and consequential.

Amplitude matching sets the contrast, not whether fringes exist. Two equal hands can point exactly opposite and cancel to zero — a perfectly black dark fringe. A long hand plus a short hand can never cancel, so the darkest point is still fairly bright and the pattern, though real, is faint. Polarization is a geometric gate on the cross term: only the projection of one field along the other, $\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2$, contributes, and orthogonal fields simply cannot cancel one another. Frequency matching is subtler — two slightly different frequencies keep a phase relationship that drifts at the beat rate $\Delta\omega$, so the pattern crawls across the screen and, given a long enough exposure, smears. The punchline figure isolates the amplitude condition, the one you can put a clean number on.

The curve is the amplitude condition made quantitative, and its shape carries the lesson: the penalty for imbalance is gentle near the top and brutal in the tails. A 4:1 intensity ratio still gives $V=0.80$; you have to reach $10:1$ before contrast halves, and $100:1$ before it collapses to $0.20$. This is why a lopsided beamsplitter is forgiving but a truly weak second beam is fatal — and why the coherence factor, which can drive $V$ to zero all by itself, is the more dangerous villain.

Everything follows from one master expression: the time-averaged intensity of two overlapping quasi-monochromatic beams. The first two terms are what you would get if you just added powers; the third — the interference term — is where all four conditions live, each as a separate factor that can switch it off.

where: $I_1,I_2$ — the two beams' intensities alone (time-averaged Poynting flux, recalled from aq-em-intensity-time-averaging); $\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2$ — polarization-overlap factor, between $0$ and $1$; $|\gamma_{12}(\tau)|$ — modulus of the complex degree of coherence at the time delay $\tau$ set by the path difference; $\delta$ — the deterministic phase difference from the optical path difference. Reading it left to right: kill any one factor of the third term and the intensity collapses to the plain sum $I_1+I_2$ — no fringes. The contrast of the resulting pattern is measured by the fringe visibility, the normalized swing between the brightest and darkest points. It packages the amplitude, polarization, and coherence factors into a single number you can read straight off a photograph.

where: $V$ — Michelson visibility, $0\le V\le1$; $I_{max},I_{min}$ — peak and trough intensities. When the beams are equal in power and co-polarized, the first two factors are one and $V=|\gamma_{12}|$ exactly — the fringe contrast is the degree of coherence, which is what makes the double slit a coherence meter. The coherence factor itself is a normalized correlation between the two fields, and its dependence on the delay $\tau$ encodes both the temporal and the spatial conditions at once.

where: $\langle\cdots\rangle$ — long-time average; $\tau$ — delay corresponding to the optical path difference. Its temporal reach is set by the source bandwidth through the coherence length, the maximum path difference over which fringes persist.

where: $\tau_c$ — coherence time; $\Delta\nu$ — spectral bandwidth; $\ell_c$ — coherence length; $\lambda,\Delta\lambda$ — central wavelength and spectral width. Interference disappears once the path difference exceeds $\ell_c$: a broadband source has a coherence length of about a wavelength, a stabilized laser one of hundreds of metres.

1 · The amplitude penalty of an unbalanced interferometer (confidence builder)

A perfectly coherent, monochromatic beam is split so one arm carries four times the intensity of the other ($I_2=4I_1$), with parallel polarization. Find the fringe visibility. Then the two arms are made to cross their polarizations at $30^\circ$ — what is the visibility now?

Step 1. The intensity ratio is $r=I_2/I_1=4$, so the amplitude factor is $V_{amp}=2\sqrt r/(1+r)=2\sqrt4/(1+4)=4/5=\mathbf{0.80}$. Even a 4:1 power imbalance — which looks severe — only drops the contrast to $0.80$, because the visibility depends on the amplitude ratio $\sqrt r=2$, not the intensity ratio.

Step 2. Now tilt one polarization by $30^\circ$. The polarization factor is $\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2=\cos30^\circ=0.866$, and it multiplies the amplitude factor: $V=0.80\times0.866=\mathbf{0.69}$.

Step 3. Sanity: had the polarizations been fully crossed ($90^\circ$), $\cos90^\circ=0$ and the fringes would vanish outright regardless of the coherence or amplitude. The $30^\circ$ tilt is a mild gate; the amplitude imbalance is the larger loss here. V = 0.80 (balanced-pol), 0.69 (30° crossed).

2 · Two lasers, one detector: when do the fringes survive? (exam level)

Two lasers at $\lambda\approx633$ nm are beat together. A photodetector integrates for $T=1$ ms. First their frequencies differ by $\Delta\nu=1$ MHz; then by $\Delta\nu=100$ Hz. In which case does a steady interference signal survive the average?

Step 1. The cross term ticks at the beat frequency, and the detector applies the factor $\mathrm{sinc}(\Delta\omega\,T/2)=\mathrm{sinc}(\pi\,\Delta\nu\,T)$. Form the dimensionless product $\Delta\nu\,T$ in each case.

Step 2. For $\Delta\nu=1$ MHz: $\Delta\nu\,T=(10^6)(10^{-3})=1000$, so $\mathrm{sinc}(\pi\cdot1000)$ has magnitude below $1/(\pi\cdot1000)\approx3\times10^{-4}$ — the beat oscillates a thousand times during one exposure and averages to essentially zero. No steady fringes.

Step 3. For $\Delta\nu=100$ Hz: $\Delta\nu\,T=(100)(10^{-3})=0.1$, so $\mathrm{sinc}(\pi\cdot0.1)=\sin(0.314)/0.314=\mathbf{0.984}$ — the beat barely advances during the exposure and the contrast is essentially full. Fringes survive (and slow-scanning this beat is exactly heterodyne detection).

Step 4. The crossover sits where $\Delta\nu\,T\sim1$, i.e. $\Delta\nu\sim1/T=1$ kHz here: sources must match in frequency to better than the inverse exposure time. Survives at 100 Hz (V ≈ 0.98), washes out at 1 MHz (V < 3×10⁻⁴).

The four conditions are all facets of one idea — that light can be correlated with itself, at two times or at two places — and the person's-eye view of that idea is the concept of coherence itself, which grew from a qualitative "sameness of light" in Young's day into the precise, measurable quantity $\gamma_{12}$ at the heart of statistical optics.

Texts. Hecht, Optics, 5th ed., §12.1 (Conditions for Interference) and §7.1 · Born & Wolf, Principles of Optics, 7th ed., §10.3 (partial coherence) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 11 (Statistical Optics) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 3 · Foot, Atomic Physics, appendix on coherence.

Provenance. The master intensity and its four-factor decomposition follow Hecht §12.1 and Born & Wolf §10; the amplitude visibility $V=2\sqrt{I_1I_2}/(I_1+I_2)$ and the identity $V=|\gamma_{12}|$ for equal co-polarized beams are from Saleh & Teich ch. 11 and verify by extremizing $I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta$. The intensity-as-time-averaged-Poynting relation is recalled from aq-em-intensity-time-averaging; the transverse, two-orthogonal-component nature of the field from aq-em-polarization-em-waves, with the resulting non-interference of orthogonal polarizations being the Fresnel–Arago law of optics. The coherence length $\ell_c=\lambda^2/\Delta\lambda$ verifies by differentiating $\nu=c/\lambda$. All worked numbers ($V=0.80,0.69$; $\mathrm{sinc}$ washout at $\Delta\nu T=0.1$ vs $1000$; $\ell_c$ for white light and the sodium doublet) are recomputed in verify_conditions-for-interference.py.

Prerequisites. Superposition Principle of Electromagnetic Waves · Two-Wave Interference and the Interference Term · Optical Path Difference and Phase Retardation. Leads to. Fringe Visibility and Contrast · Temporal Coherence and Coherence Length · Spatial Coherence and the Van Cittert–Zernike Theorem · Mutual Coherence Function and Degree of Coherence.