L01 · Laser foundations

Laser vs Lamp vs LED

Can a very bright light still make interference fringes with poor contrast? Yes. Turning up the power scales the bright and dark fringes alike, so their contrast stays the same. Contrast comes from coherence, which depends on how narrow the spectrum is and on how well points across the beam stay in step. A bright, tightly collimated beam is no proof of a laser.

Move a detector along the beam, then examine the same source with a spectrometer, a double slit, a polarizer and a fast photodiode. A controlled counterexample shows that power, directionality and coherence are separate properties.

Detector beam radius $\text{—}$

Far-field half-angle $\text{—}$

Peak irradiance $\text{—}$

Fringe visibility $\text{—}$

Temporal correlation $\text{—}$

Slit spatial correlation $\text{—}$

Finite-screen collected power $\text{—}$

Power after analyzer $\text{—}$

Actual path difference $\text{—}$

Counterexample challenge $\text{—}$

Configure the source

Tell the beam properties apart

Beam expanderExpand the waist tenfold; reduce angular spread consistently.High spatial coherenceChange the transverse coherence width; the divergence changes with it.Narrow spectral linesSwitch between 5 MHz and 2 THz line widths; temporal coherence follows.Five longitudinal modesEqual weights separated by 500 MHz; inspect coherence revivals.Linear polarizationWhen off, the light is an unpolarized ensemble.Technical intensity variation20 MHz field-envelope modulation adds spectral sidebands.Equal optical pathsSet temporal delay to zero without changing spatial coherence.Mean source power

P

10\,\mathrm{mW}

Detector distance

z

250\,\mathrm{cm}

Path difference, logarithmic scale

\log_{10}(\Delta L/\mathrm{mm})

2

Analyzer rotation

\theta

0\,\mathrm{^\circ}

Challenge: expand the beam, raise the mean power to at least 80 mW, and keep the fringe visibility below 0.1. Can a bright, well-directed beam still have poor coherence?

The source presets are teaching examples. Changing the settings builds a synthetic source; it does not describe every real lamp, LED or laser.

All sources are idealized, conditioned beams at 633 nm, not complete lamp or LED spectra. The beams propagate as Gaussian-Schell beams, and the longitudinal modes have equal power and Lorentzian lines. Beam expansion changes both the beam width and the coherence width. The screen shows relative brightness with a square-root display gamma. The diagnostics are ideal and add no measurement noise. The photodiode passes frequencies below 100 MHz and shows technical amplitude modulation, not photon statistics.

Start with the essentials

Focus question: Does raising the power improve the normalized fringe visibility?
One-sentence intuition: Power scales the signal, and coherence sets the contrast of the fringes. Temporal coherence must match the spectrum; spatial coherence describes how well points across the beam are correlated.

Core mathematical model

Power and irradiance

I(r,z)=\frac{2P}{\pi w^2(z)}e^{-2r^2/w^2(z)}

Integrating the irradiance across the beam gives the mean power. The screen is 40 mm across and can miss the faint tails. The peak irradiance readout is still a linear physical quantity.

Beam spread in the Gaussian-Schell model

w^2(z)=w_0^2+\theta_d^2z^2,\quad\theta_d=\frac{\lambda}{\pi}\sqrt{w_0^{-2}+\xi^{-2}}

In the Gaussian-Schell model, a narrower coherence width means a larger angular spread. The beam expander makes both the waist and the coherence width ten times larger and keeps the power the same. The drawn beam outline is schematic; no lens train is simulated.

Spectrum determines temporal coherence

g^{(1)}(\tau)=e^{-\pi\Delta\nu|\tau|}\sum_j p_j e^{2\pi i\nu_j\tau},\quad\sum_jp_j=1

Every spectral component has the same Lorentzian full width at half maximum. Five longitudinal modes can make the correlation revive. Technical modulation of the field envelope adds sidebands to the same list of components, so the time trace always stays tied to the spectrum.

Fringe visibility from two equal slits

V=e^{-d^2/(2\xi^2)}|g^{(1)}(\Delta L/c)|,\qquad d=0.5\,\mathrm{mm}

The two slits sample the beam at the source exit plane, after any beam expansion. The fringe position is counted in periods, and both arms carry equal intensity. Equal optical paths remove the time delay, but they cannot make up for low spatial coherence.

Common difficulties

A source name does not fix its properties

Typical misconceptionEvery laser is perfectly monochromatic and every LED is completely incoherent.

Better mental modelThe presets are idealized examples for comparison. Real sources have finite coherence and may run on several modes, and their light can be filtered or spatially conditioned.

Display brightness is not a measurement

Typical misconceptionIf two detector spots look alike on screen, they have the same irradiance.

Better mental modelThe screen shows relative brightness through a display gamma. Compare the calibrated irradiance readouts and profiles instead of judging by screen brightness alone.

Run the experiment

01
More power does not mean more contrast
Choose Lamp, turn on Beam expander, and raise the power to 80 mW or more. Open the Double slit diagnostic and press Check counterexample.
What to observe: The irradiance goes up, but the normalized visibility stays low. Collimating the beam changes the spatial coherence that the fixed slits sample, but it does not narrow the spectrum.
02
Separate spatial and temporal coherence
Choose Single mode. Turn off High spatial coherence and turn on Equal optical paths. Then turn High spatial coherence back on, switch off Equal optical paths and Narrow spectral lines, and keep a nonzero path difference.
What to observe: Losing either spatial or temporal correlation is enough to wash out the fringes. The readouts show which one limits this measurement.
03
A spectrum is more than one width
Choose Multiple modes and open the Spectrum diagnostic. Sweep the path difference and look for revivals of the correlation. Toggle Technical intensity variation and compare the photodiode trace with the spectral sidebands.
What to observe: The five modes are 500 MHz apart. Beats between the modes lie above the 100 MHz cutoff of the ideal photodiode, so it does not show them. It does show the technical 20 MHz modulation and its intensity harmonic.
04
Check polarization on its own
Open the Polarizer diagnostic and rotate the analyzer. Repeat with Linear polarization turned off, then drag the detector in the apparatus.
What to observe: Unpolarized light passes half its mean power at every analyzer angle. Moving the detector changes the beam size and the peak irradiance, but the total source power stays the same. A wide beam can spill past the edge of the screen, so less power is collected.