Light Quanta · Introduction

Continuous waves explain
purely optical phenomena.

What does a continuous wave description of light explain well, and what exactly does its intensity measure?

Read the Introduction of Einstein’s 1905 paper →

LQ-01 · An executable model

Wave description and energy spreading laboratory

Wave description and energy spreading, host calculation

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In the introduction of his 1905 light paper, Einstein affirms that the wave theory has proven excellent for purely optical phenomena and will presumably never be replaced. Explore two-source wave interference, time-averaged versus instantaneous readouts, and spherical inverse-square energy spreading.

Discovery Mode: Predict Before Calculating

Two equal waves meet at the center of the screen. What happens to the intensity there when the phase difference goes from 0 to π?

It stays the same

Interference does not change the total brightness at the center⟨I⟩(π) = ⟨I⟩(0)

It halves

Shifting one wave cancels half the incoming radiant power⟨I⟩(π) = 0.5 * ⟨I⟩(0)

It drops to zero

Equal waves out of phase by half a cycle cancel completely⟨I⟩(π) = 0

Display Mode:Two-Source InterferenceSpherical Energy Spreading (Inverse-Square)

Screen Intensity Profile ⟨I(y)⟩

Time-Averaged

Classical linear superposition of two coherent point sources. Central intensity: 4.000, Fringe visibility: 1.000, Fringe spacing: 33.3333 m.

2D Wavefield Crest Superposition

λ = 1000000000 nm | d = 3.0 λ | δ = 0.00 π

Concentric circular wavefront crests radiate from coherent sources S₁ and S₂.

Accepted Laboratory Telemetry Snapshot

Quantity ID	Status	Value / Result	Unit	Owner ID
centerIntensity	value	4.0000	1	radiation.twoSourceIntensity
instantaneousCenterIntensity	value	4.0000	1	radiation.twoSourceIntensity
fringeVisibility	value	1.0000	1	radiation.fringeVisibility
fringeSpacing	value	33.3333	lambda	radiation.fringeSpacingSmallAngle
pathDifference	value	0.0000	lambda	radiation.twoSourceIntensity
selectedPositionIntensity	value	4.0000	1	radiation.twoSourceIntensity
screenIntensity	value	[object Object]	1	radiation.twoSourceIntensity
pointSourceIntensity	value	0.0796	W/m2	radiation.inverseSquareIntensity
shellPower	value	1.0000	W	radiation.shellPowerIdentity
smallAperturePower	value	7.9577e-6	W	radiation.aperturePower
exactDiskPower	value	7.9576e-6	W	radiation.aperturePower
relativeDifference	value	-2.3873e-5	1	radiation.aperturePower

Limits of this Classical Wave Model (Not Modeled)

This reference owner implements continuous classical wave optics and geometric energy spreading. The following physical regimes require vector electrodynamics, microscopic matter coupling, or quantum optics and are explicitly not modeled:

Polarization and vector electromagnetic field components.
Photon statistics, photon anti-bunching, or quantum optics.
Absorption, emission, or quantum detection by matter.
Non-monochromatic or finite-coherence-length light sources.
Diffraction effects beyond the coherent two-source idealization.
An absolute physical intensity scale unless a radiant power and detector geometry are declared.

Show the code

The Physical Context

The Successes of Continuous Wave Optics

In the opening paragraph of his 1905 paper, Einstein contrasts the continuous spatial functions of Maxwellian electrodynamics with the atomistic discrete description of ponderable matter. He emphasizes that the wave theory of light:

“...has proved to be excellently suited for the description of purely optical phenomena and will probably never be replaced by any other theory.”

Diffraction, reflection, refraction, and interference are macroscopic triumphs of the continuous wave picture. When two coherent monochromatic waves superpose at a point, their scalar fields add linearly:

\psi(\mathbf{r}, t) = \frac{A_1}{r_1}\cos(k r_1 - \omega t + \phi_1) + \frac{A_2}{r_2}\cos(k r_2 - \omega t + \phi_2)

Time Averages Versus Instantaneous Values

Optical detectors (the eye, photographic plates, chemical actinometers) cannot resolve oscillations at optical frequencies (

\nu \sim 10^{14}\text{--}10^{15}\text{ Hz}

). They record exclusively the time-averaged intensity over millions of optical periods:

\langle I \rangle = \frac{\kappa}{2}\left[ a_1^2 + a_2^2 + 2 a_1 a_2 \cos\delta \right]

For two equal-amplitude coherent waves in phase (

\delta = 0

), the time-averaged intensity at constructive interference is 4 times that of a single wave. When shifted by half a wave (

\delta = \pi

), the intensity drops to identically zero.

Geometric Energy Spreading

On the wave theory, energy emitted by an isotropic point source of power

P

spreads continuously over expanding spherical wavefronts of surface area

4\pi r^2

. The radiant intensity at distance

r

is:

I(r) = \frac{P}{4\pi r^2}

The total energy flux integrated over any enclosing spherical surface is strictly conserved:

\oint_{\text{sphere}} I(r)\, dA = P

It is precisely this continuous dilution of energy throughout an expanding volume that Einstein challenges when analyzing the generation and transformation of light in fluorescence, cathode-ray excitation, and photoionization.

Return to the Light Quanta Paper →Explore Journey I →