How a Laser Achieves Coherence

Equivalent forms

N_2 > N_1 \quad (\text{population inversion})

\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}

G_{\text{round-trip}} = R_1 R_2\, e^{2g L} \ge 1 \;(\text{threshold})

Coherence is not built photon-by-photon by a designer — it is a runaway copying process. One lucky spontaneous photon seeds an avalanche of identical twins.

Symbol	Name	SI unit	Dimension	Range
$g$	Gain coefficientoutput Fractional intensity growth per unit length	m^{-1}	L^-1	—
$\sigma$	Stimulated-emission cross section	m^2	L^2	1e-24 – 1e-20
$N_2$	Upper-state population density	m^{-3}	L^-3	0 – 1e+24
$N_1$	Lower-state population density	m^{-3}	L^-3	0 – 1e+24

Where it holds

Two-level inversion is unstable; real lasers use 3- or 4-level schemes. Above threshold the gain saturates (clamps) so steady-state output is set by pump power, not runaway growth. Quantum noise sets the ultimate linewidth (Schawlow–Townes limit).

Discovery

Albert Einstein (theory) · Theodore Maiman (first laser) · 1960

Einstein predicted stimulated emission in 1917 while re-deriving Planck's law, introducing the A and B coefficients almost as an accounting trick. It sat dormant for decades. Charles Townes built the microwave MASER in 1954; then in May 1960 Theodore Maiman fired the first optical laser at Hughes Research Labs using a ruby rod and a photographic flashlamp — a device many had said was impossible. Within a year the helium–neon and semiconductor lasers followed.

Try this

How does a trillion atoms agree to emit light perfectly in step?

A laser turns disordered thermal energy into a beam where every photon shares the same phase, direction and color. What quantum condition makes that possible?

Real-world applications

Fiber-optic communication (semiconductor diode lasers)
LASIK eye surgery and surgical cutting
Barcode scanners, laser printers, optical drives
LIGO interferometry and laser cooling of atoms

Common misconceptions

Lasers don't 'create' energy — they reorganize pumped energy into one coherent mode, at modest efficiency
Population inversion is not a negative temperature gimmick you get for free; it requires continuous pumping
Stimulated photons copy direction and phase — that, not raw brightness, is what makes a laser a laser

Experimental verification

Threshold pump power, beam coherence length, and the e^{gL} gain scaling are measured directly in every laser lab. Photon-correlation (Hanbury Brown–Twiss) experiments confirm laser light is coherent (Poissonian) versus thermal light's bunching.

Derivation

Einstein balance for a two-level system: rate

up = B12 \rho N_{1}

; rate down

= A21\,\mathrm{N}_{2} + B21 \rho N_{2}

(spontaneous + stimulated).

Demanding consistency with the Planck blackbody spectrum at equilibrium forces

B12 = B21

and A

21/B21 = 8\pi h\nu ^{3}/c^{3}

.

Net stimulated rate

\propto (N_{2} - N_{1})

.

Light intensity along z obeys

dI/dz = \sigma (N_{2} - N_{1}) I

, giving exponential gain

I = I_{0} e^{gz}

with

g = \sigma (N_{2} - N_{1})

.

Gain is positive only when N2 > N1 — the inversion condition.

Lasing starts when round-trip gain beats mirror and absorption losses:

R_{1} R_{2} e^{2gL} \ge 1

.

What if…

What if you remove one mirror?

You get a single-pass amplifier (or superluminescent source) — high gain but no resonant feedback, so much poorer coherence.

What if

N_{2} = N_{1}

?

Gain is exactly zero (transparency): stimulated emission and absorption cancel. You're at the inversion threshold.

What if the medium is at thermal equilibrium?

Boltzmann guarantees N2 < N1, so g < 0 — the medium absorbs. No pumping, no laser.

1

Gain of a ruby-like medium

Given ·

sigma:: 1e-22 m^2
N2:: 6e23 m^-3
N1:: 2e23 m^-3

Find · gain coefficient g and single-pass amplification over

L = 0.1\,\mathrm{m}

Steps

$g = \sigma (N_{2} - N_{1}) = 1e-22 \times (6e23 - 2e23) = 40\,\mathrm{m}^-1$
Single pass: $I/I_{0} = e^{g\cdot L} = e^{40\times 0.1} = e^4 \approx 54.6$

Result ·

g = 1e-22\cdot (4e23) = 40\,\mathrm{m}^-1

;

I/I_{0} = e^{gL} = e^{4} \approx 54.6\times

2

Spontaneous-to-stimulated ratio at optical frequencies

Given ·

nu:: 4.74e14 Hz (633 nm HeNe)

Find · A21/B21 (energy-density form)

Steps

A $21/B21 = 8\pi h\nu ^{3}/c^{3}$
$= 8\pi (6.626e-34)(4.74e14)^{3} / (3e_{8})^{3}$
$\approx 2.8e-13 J\cdot s/m^{3}$

Result · A

21/B21 = 8\pi h\nu ^{3}/c^{3} \approx 2.8e-13 J\cdot s\cdot m^-3

, showing spontaneous emission dominates strongly at optical frequencies — hence the need for a resonant cavity to favor one mode.

Photon Energy (Planck Relation)→Bohr Hydrogen Energy Levels→Rabi Oscillations→Photoelectric Effect→