Light Fundamentals

The physical nature of light — electromagnetic waves, photons, and the practical regimes in which each model applies. The foundational guide for all topics on this site.

1Introduction

1.1The Central Role of Light in Photonics

Light is the working medium of photonics. Every component on a photonics engineer's bench — lenses, mirrors, filters, detectors, fiber optics, lasers — exists to generate, shape, transport, or measure electromagnetic radiation. Understanding the physical nature of that radiation is not merely academic background; it is the foundation on which every design decision rests.

The term “light” in photonics extends well beyond the narrow visible band that the human eye can detect. Photonics systems routinely operate from the deep ultraviolet (below 200 nm) through the mid-infrared (beyond 10 μm), and some applications reach into the terahertz regime. The physical principles governing light — its wave nature, its particle nature, its speed, polarization, and coherence — apply uniformly across these spectral regions. What changes from one region to another is the available sources, the materials that transmit or absorb, and the detectors that respond. The physics is universal.

This guide provides that physical foundation. It covers the electromagnetic wave model, the photon model, and the practical regimes in which each applies. Many topics introduced here — polarization, interference, refraction, absorption — receive full treatment in their own dedicated guides elsewhere on this site.

1.2A Brief History

The debate over the nature of light is one of the longest-running in physics. Isaac Newton (1643–1727) favored a corpuscular model — light as a stream of particles — which explained rectilinear propagation and sharp shadows. His contemporary Christiaan Huygens (1629–1695) championed a wave model, showing that it could explain reflection, refraction, and the double refraction of calcite that the corpuscular model could not easily accommodate [1].

Newton's authority kept the particle model dominant for over a century. The tide turned in the early 1800s when Thomas Young demonstrated interference fringes (1801) and Augustin Fresnel developed a comprehensive wave theory of diffraction and polarization. By 1825 the wave model was ascendant. Fresnel's insight that light waves are transverse — oscillating perpendicular to the propagation direction — explained polarization phenomena that had puzzled earlier investigators [1, 5].

The decisive synthesis came from James Clerk Maxwell in 1865. By adding the displacement current term to Ampère's law, Maxwell showed that his equations of electromagnetism predicted transverse waves propagating at a speed determined entirely by the electric and magnetic constants of free space — a speed that matched the measured speed of light [6]. Heinrich Hertz confirmed this experimentally in 1888 by generating and detecting radio waves that exhibited reflection, refraction, and interference just as light does [1].

The electromagnetic wave picture seemed complete until the early twentieth century. Max Planck's quantization of blackbody radiation (1900), Albert Einstein's explanation of the photoelectric effect (1905), and Arthur Compton's scattering experiments (1923) collectively demonstrated that light also exhibits particle-like behavior. The energy of electromagnetic radiation is quantized into discrete packets — photons — each carrying energy proportional to the wave frequency. For nearly all optical engineering purposes, the practical question is: when does the wave model suffice, and when must photon effects be considered? Section 10 provides a decision framework.

1.3Three Models of Light

Ray optics (geometric optics) treats light as rays that travel in straight lines, refract at interfaces according to Snell's law, and reflect according to the law of reflection. This model ignores diffraction and interference entirely. It is valid when all optical elements and apertures are much larger than the wavelength — the typical condition for lens design, mirror systems, and most catalog component selection.

Wave optics (physical optics) treats light as an electromagnetic wave described by Maxwell's equations. This model accounts for diffraction, interference, polarization, and coherence — phenomena important when feature sizes approach the wavelength, when thin-film coatings must be designed, or when laser beams must be characterized. Wave optics is the primary model used throughout this site.

Quantum optics treats light as a stream of photons. This model is required when individual photons must be counted, when photon statistics matter (shot noise, squeezed light), or when light-matter interactions at the atomic scale must be calculated. Most laboratory photonics operates comfortably above the quantum limit, but understanding when photon effects intrude prevents design errors.

These three models are not competing theories; they are nested approximations. Ray optics is the short-wavelength limit of wave optics. Wave optics is the high-photon-number limit of quantum optics. Choosing the simplest adequate model for a given problem is itself an engineering skill.

2Electromagnetic Waves

2.1Maxwell's Equations in Free Space

All classical electromagnetic phenomena — including light — are governed by four equations formulated by James Clerk Maxwell. In free space (no charges, no currents) [6]:

Gauss's law for electricity: $\nabla \cdot \mathbf{E} = 0$ — the electric field has no sources or sinks in free space.

Gauss's law for magnetism: $\nabla \cdot \mathbf{B} = 0$ — magnetic field lines always close; there are no magnetic monopoles.

Faraday's law: $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$ — a time-varying magnetic field generates a circulating electric field.

Ampère-Maxwell law: $\nabla \times \mathbf{B} = \mu_0 \varepsilon_0\,\partial \mathbf{E}/\partial t$ — a time-varying electric field generates a circulating magnetic field.

The coupling is the key: a changing electric field creates a magnetic field, and a changing magnetic field creates an electric field. This mutual regeneration allows an electromagnetic disturbance to propagate through empty space indefinitely, with no medium required.

2.2The Wave Equation

Taking the curl of Faraday's law and substituting the Ampère-Maxwell law yields the electromagnetic wave equation [6]:

\nabla^2 \mathbf{E} = \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}

Comparing with the general wave equation identifies the propagation speed:

Speed of light from fundamental constants

c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

Worked Example: Speed of Light from Fundamental Constants

Problem: Calculate the speed of electromagnetic wave propagation from μ₀ and ε₀.

μ₀ = 4π × 10⁻⁷ T·m/A
ε₀ = 8.854 × 10⁻¹² F/m

Step 1: Compute the product μ₀ε₀.

μ₀ε₀ = (1.2566 × 10⁻⁶)(8.8542 × 10⁻¹²) = 1.1127 × 10⁻¹⁷ s²/m²

Step 2: Take the inverse square root.

c = 1/√(1.1127 × 10⁻¹⁷) = 1/(3.3356 × 10⁻⁹)
c = 2.998 × 10⁸ m/s

This matches the defined value. The agreement between a speed calculated from electrostatic and magnetostatic measurements and the independently measured speed of light was Maxwell's most stunning prediction.

2.3Plane Wave Solutions

The simplest solution to the wave equation is a linearly polarized plane wave propagating along the z-axis [1, 2]:

\mathbf{E}(z,t) = E_0 \cos(kz - \omega t)\,\hat{x}

\mathbf{B}(z,t) = B_0 \cos(kz - \omega t)\,\hat{y}

Maxwell's equations impose three constraints: transversality (E and B perpendicular to propagation), orthogonality (E ⊥ B), and an amplitude relation:

B_0 = \frac{E_0}{c}

Because c is large (~3 × 10⁸ m/s), B₀ is numerically much smaller than E₀ in SI units. For a visible light wave with E₀ = 1000 V/m, B₀ ≈ 3.3 × 10⁻⁶ T. Despite this numerical disparity, both fields carry equal energy in the wave.

Figure 2.1 — Plane electromagnetic wave. The electric field E (copper) oscillates perpendicular to the propagation direction z, with wavelength λ marked. The magnetic field B oscillates in the orthogonal plane.

2.4Phase Velocity, Wavelength & Frequency

Fundamental wave relation

c = \lambda f

Related quantities used throughout photonics:

\omega = 2\pi f \quad\text{(angular frequency, rad/s)}

k = \frac{2\pi}{\lambda} \quad\text{(wavenumber, rad/m)}

The dispersion relation in vacuum is $\omega = ck$ . For visible light, wavelengths span roughly 380–750 nm and frequencies span roughly 400–790 THz. A green photon at 532 nm oscillates at 5.6 × 10¹⁴ Hz — no electronic detector can follow individual oscillation cycles; all optical measurements are inherently time-averaged.

3The Electromagnetic Spectrum

3.1Spectrum Overview

Electromagnetic radiation spans more than 20 orders of magnitude in wavelength. The classification into named spectral regions is a practical convention reflecting different sources, detectors, and interaction mechanisms [1].

Region	Wavelength	Frequency	Photon Energy	Photonics Relevance
Gamma rays	< 10 pm	> 30 EHz	> 124 keV	Radiation detection
X-rays	10 pm – 10 nm	30 PHz – 30 EHz	124 eV – 124 keV	X-ray optics, synchrotron
Extreme UV	10 – 121 nm	2.5 – 30 PHz	10.2 – 124 eV	EUV lithography
Ultraviolet	200 – 400 nm	750 THz – 1.5 PHz	3.1 – 6.2 eV	UV curing, fluorescence
Visible	380 – 750 nm	400 – 790 THz	1.65 – 3.26 eV	Imaging, display
Near-IR	750 nm – 1.4 μm	214 – 400 THz	0.89 – 1.65 eV	Telecom, Nd:YAG
SWIR	1.4 – 3 μm	100 – 214 THz	0.41 – 0.89 eV	Moisture sensing
MWIR	3 – 8 μm	37 – 100 THz	0.16 – 0.41 eV	Thermal imaging
LWIR	8 – 15 μm	20 – 37 THz	0.08 – 0.16 eV	Thermal imaging
Far-IR	15 μm – 1 mm	0.3 – 20 THz	1.2 meV – 83 meV	THz spectroscopy
Microwave	1 mm – 1 m	300 MHz – 300 GHz	1.2 μeV – 1.2 meV	Radar
Radio	> 1 m	< 300 MHz	< 1.2 μeV	Broadcasting

Table 3.1 — Electromagnetic spectrum regions with photonics relevance.

Figure 3.1 — The electromagnetic spectrum. The optical spectrum — UV through visible to infrared — is the domain of photonics. Key laser wavelengths are marked.

3.2The Optical Spectrum

In photonics, the “optical spectrum” extends far beyond visible light. It encompasses roughly 10 nm to 1 mm — UV through visible to infrared — where radiation can be manipulated using optical components such as lenses, mirrors, gratings, fibers, and semiconductor detectors [2]. This wide range is unified by the applicability of geometric and wave optics.

3.3Visible Light

The visible spectrum spans approximately 380 to 750 nm, defined by the spectral sensitivity of the human eye, which peaks near 555 nm (green-yellow) under photopic conditions [8].

Color	Wavelength (nm)	Frequency (THz)
Violet	380 – 450	670 – 790
Blue	450 – 495	605 – 670
Green	495 – 570	525 – 605
Yellow	570 – 590	510 – 525
Orange	590 – 620	485 – 510
Red	620 – 750	400 – 485

Table 3.2 — Visible light color ranges.

These boundaries are approximate. The spectrum is continuous; color perception is a psychophysical phenomenon. Many perceived colors (magenta, brown, white) cannot be produced by any single wavelength [1].

Worked Example: Photon Energy at Common Laser Wavelengths

Problem: Compare the photon energy of a HeNe laser (632.8 nm) and an Nd:YAG laser (1064 nm).

Photon energy: E = hc/λ

HeNe at 632.8 nm:

E = (6.626 × 10⁻³⁴)(2.998 × 10⁸) / (632.8 × 10⁻⁹)
E = 3.139 × 10⁻¹⁹ J
E = 1.960 eV

Nd:YAG at 1064 nm:

E = (6.626 × 10⁻³⁴)(2.998 × 10⁸) / (1064 × 10⁻⁹)
E = 1.867 × 10⁻¹⁹ J
E = 1.165 eV

The HeNe photon carries 1.68× more energy — equal to the wavelength ratio (1064/632.8), confirming E ∝ 1/λ. For a given optical power, the Nd:YAG beam delivers 68% more photons per second.

3.4Spectral Regions of Photonics Interest

Several narrow spectral windows are disproportionately important: 193 nm (ArF excimer, semiconductor lithography), 532 nm (Nd:YAG 2ω), 632.8 nm (HeNe alignment and interferometry), 780–980 nm (GaAs/InGaAs diode lasers), 1064 nm (Nd:YAG fundamental), 1310 and 1550 nm (fiber-optic telecom windows), and 10.6 μm (CO₂ laser, industrial cutting).

🔧 Spectral Unit Converter — convert between wavelength, frequency, wavenumber, and photon energy →🔧 Laser Line Reference — searchable database of common laser wavelengths →

4Energy, Momentum & Intensity

4.1Poynting Vector & Irradiance

The instantaneous rate of energy flow per unit area is the Poynting vector [1, 6]:

Poynting vector

\mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B}

The time-averaged magnitude defines the irradiance (W/m²) [1, 2]:

Irradiance

I = \langle S \rangle = \frac{1}{2}\varepsilon_0 c E_0^2 = \frac{E_0^2}{2\eta_0}

where $\eta_0 = \sqrt{\mu_0/\varepsilon_0} \approx 376.73\;\Omega$ is the impedance of free space. For a Gaussian beam (the most common laser profile):

Peak irradiance of a Gaussian beam

I_\text{peak} = \frac{2P}{\pi w^2}

Worked Example: Irradiance and Photon Flux of a HeNe Laser

Problem: A 5 mW HeNe laser (632.8 nm) has a beam diameter of 0.8 mm (1/e²). Calculate peak irradiance and photon flux.

P = 5 × 10⁻³ W
w = 0.4 mm = 4 × 10⁻⁴ m
E_ph = 3.139 × 10⁻¹⁹ J (from WE 3.1)

Peak irradiance:

I_peak = 2(5 × 10⁻³) / [π(4 × 10⁻⁴)²]
I_peak = 1.99 × 10⁴ W/m² ≈ 19.9 kW/m²

This is roughly 15× the solar irradiance at Earth's surface (~1361 W/m²).

Total photon flux:

Φ = P / E_ph = 5 × 10⁻³ / 3.139 × 10⁻¹⁹
Φ = 1.59 × 10¹⁶ photons/s

Even a low-power laser emits roughly 16 quadrillion photons per second. Quantum effects are negligible at typical lab powers because the photon number is enormous.

🔧 Photon Flux Calculator — CW and pulsed beam photon flux, irradiance, and geometry →

4.2Photon Energy

Photon energy

E = hf = \frac{hc}{\lambda}

In electron-volts, the convenient approximation:

E\;\text{(eV)} = \frac{1240}{\lambda\;\text{(nm)}}

Photon energy determines which processes a wavelength can drive. UV photons (3–6 eV) break chemical bonds; IR photons (below ~1.5 eV) excite vibrational modes but cannot drive electronic transitions in most materials.

4.3Photon Momentum

Photon momentum

p = \frac{h}{\lambda} = \frac{E}{c} = \frac{hf}{c}

Practical consequences include radiation pressure on mirrors and spacecraft, optical trapping (optical tweezers), laser cooling of atoms, and recoil shifts in precision spectroscopy.

4.4Radiation Pressure

P_\text{rad} = \frac{I}{c}\;\text{(absorbing)} \qquad P_\text{rad} = \frac{2I}{c}\;\text{(reflecting)}

For sunlight at Earth (I ≈ 1361 W/m²), the radiation pressure on a perfect mirror is approximately 9 μPa — negligible for laboratory mechanics but significant for solar sail propulsion and LIGO test masses.

4.5Spectral Energy Density

Planck's law

u(\nu, T) = \frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/k_BT} - 1}

Wien's displacement law

\lambda_\text{max} T = 2.898 \times 10^{-3}\;\text{m·K}

At the Sun's photospheric temperature (5778 K), $\lambda_\text{max} \approx 501\;\text{nm}$ — the green-yellow peak that coincides with peak human visual sensitivity. At room temperature (293 K), $\lambda_\text{max} \approx 9.9\;\mu\text{m}$ — the LWIR band used by thermal imaging cameras.

5Wave-Particle Duality

5.1Blackbody Radiation & Planck's Hypothesis

Classical electromagnetic theory predicted that spectral energy density should increase without bound at short wavelengths — the ultraviolet catastrophe. In 1900, Max Planck resolved this by proposing that electromagnetic energy is emitted and absorbed in discrete quanta: $E = hf$ . Planck regarded this as a mathematical device; it took Einstein to recognize its physical reality [1].

5.2The Photoelectric Effect

When ultraviolet light strikes a metal surface, electrons are ejected. Three observations defied the wave model [1, 2]: (1) below a threshold frequency, no electrons are emitted regardless of intensity; (2) emission is instantaneous; (3) maximum kinetic energy depends on frequency, not intensity. Einstein explained all three by proposing that a single photon is absorbed by a single electron:

Photoelectric equation

K_\text{max} = hf - \phi

Worked Example: Photoelectric Effect: UV on Cesium

Problem: UV light at 254 nm strikes cesium (ϕ = 2.1 eV). Find K_max and threshold wavelength.

Photon energy:

E = 1240 / 254 = 4.88 eV

Maximum kinetic energy:

K_max = 4.88 − 2.1
K_max = 2.78 eV

Threshold wavelength:

λ₀ = 1240 / 2.1
λ₀ = 590 nm (visible orange-yellow)

Cesium's low work function makes it suitable for photomultiplier tube photocathodes sensitive to visible light. Most metals require UV illumination.

Figure 5.1 — The photoelectric effect. A photon of energy hf is absorbed at the metal surface. If hf exceeds the work function ϕ, an electron is ejected with kinetic energy K_max = hf − ϕ.

5.3Compton Scattering

In 1923, Compton measured X-ray scattering by free electrons and found a wavelength shift depending on scattering angle [1]:

Compton wavelength shift

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

The quantity $h/(m_e c) = 2.426 \times 10^{-12}\;\text{m}$ is the Compton wavelength of the electron. This result follows from treating the interaction as a particle-particle collision — direct proof that photons carry momentum $p = h/\lambda$ .

5.4Single-Photon Interference

In the double-slit experiment at extremely low intensities, individual photons arrive at discrete, random positions (particle behavior), yet the ensemble builds the familiar interference pattern (wave behavior) [1, 2]. Each photon is detected at a single point, but the probability of detection is governed by wave interference. This remains one of the most profound demonstrations of quantum mechanics.

5.5When to Use Which Model

Use the wave model for interference, diffraction, polarization, coherence, and beam propagation — the default for photonics engineering.

Include photon effects when signal levels approach the single-photon regime, when shot noise limits precision ( $\sigma = \sqrt{N}$ ), when calculating laser gain and emission rates, or when photon energy thresholds determine system behavior (detector cutoffs, solar cell band gaps).

Rule of thumb: above ~10⁶ photons, the wave model suffices. Below ~10³ photons, photon statistics become important. Section 10 provides a detailed decision framework.

6Speed of Light

6.1The Vacuum Speed

c = 299\,792\,458\;\text{m/s}\;\;\text{(exact by definition)}

Since 2019 this value is a definition: the meter is defined as the distance light travels in vacuum in 1/299 792 458 of a second. Light travels approximately 30 cm in one nanosecond and 0.3 μm in one femtosecond — useful benchmarks for timing in optical systems and for estimating pulse spatial lengths.

6.2Historical Measurements

Ole Rømer (1676) first estimated c from timing variations in Jupiter's moon eclipses, obtaining ~2.1 × 10⁸ m/s. Armand Fizeau (1849) used a rotating toothed wheel to get ~3.15 × 10⁸ m/s. Léon Foucault (1862) improved this to ~2.98 × 10⁸ m/s with a rotating mirror. Albert Michelson refined these techniques over decades, achieving 299 796 ± 4 km/s in 1926 [1]. By the 1970s, laser interferometry achieved sub-meter-per-second uncertainty, prompting the 1983 meter redefinition.

6.3Speed in Media

Refractive index

n = \frac{c}{v}

Material	n	Notes
Air (STP)	1.000 293	Often approximated as 1
Water	1.333	Visible range
Fused silica	1.458	UV–NIR window material
BK7 glass	1.517	Most common optical glass
Sapphire	1.768	Ordinary ray; birefringent
Diamond	2.417	Highest natural visible n
Silicon	3.48	At 1550 nm; opaque in visible
Germanium	4.00	At 10.6 μm; IR optics

Table 6.1 — Refractive indices of common optical materials (at 589 nm unless noted).

The refractive index is wavelength-dependent (dispersion). Glass typically has higher n at shorter wavelengths (normal dispersion), which is why prisms separate white light.

6.4Group Velocity vs. Phase Velocity

The phase velocity $v_p = \omega/k = c/n$ describes how fast phase fronts advance. A pulse envelope travels at the group velocity [1, 2]:

Group velocity

v_g = \frac{d\omega}{dk} = \frac{c}{n - \lambda\dfrac{dn}{d\lambda}} = \frac{c}{n_g}

where $n_g = n - \lambda(dn/d\lambda)$ is the group index. In normal dispersion ( $dn/d\lambda < 0$ ), the group index exceeds n, so the pulse envelope travels slower than the phase fronts.

Figure 6.1 — Phase velocity vs. group velocity. The carrier wave (navy) advances at v_p while the pulse envelope (copper dashed) advances at the slower v_g in normal dispersion.

Worked Example: Group Velocity in BK7 at 800 nm

Problem: Calculate the group velocity and group index in BK7 glass at 800 nm.

n = 1.5107 at 800 nm
dn/dλ = −0.0178 μm⁻¹ at 800 nm

Group index:

n_g = n − λ(dn/dλ) = 1.5107 − (0.800)(−0.0178)
n_g = 1.5107 + 0.0142
n_g = 1.5249

Velocities:

v_p = c/n = 2.998 × 10⁸ / 1.5107 = 1.984 × 10⁸ m/s
v_g = c/n_g = 2.998 × 10⁸ / 1.5249 = 1.966 × 10⁸ m/s

The pulse travels 0.9% slower than the phase fronts. Over 10 mm of BK7, the pulse arrives ~0.5 ps later than a naive phase-velocity estimate would predict. For 100 fs pulses, group velocity dispersion also broadens the pulse to ~115 fs after 10 mm.

7Polarization Fundamentals

7.1Linear Polarization

The electric field of an EM wave oscillates perpendicular to the propagation direction. In linear polarization, E is confined to a single plane [1, 2]:

\mathbf{E}(z,t) = E_0 \cos(kz - \omega t)\,\hat{x}

Any angle in the x-y plane is possible. Two orthogonal linear states form a complete basis — any polarization state can be decomposed into these two components.

7.2Circular & Elliptical Polarization

Two orthogonal components of equal amplitude with a 90° phase difference produce circular polarization — the E-field tip traces a circle [1]:

\mathbf{E}(z,t) = E_0[\cos(kz-\omega t)\,\hat{x} \pm \sin(kz-\omega t)\,\hat{y}]

The + sign gives left-circular (LCP) and − gives right-circular (RCP) in the optics convention (looking into the beam). The physics convention reverses this — always specify convention when citing handedness. Elliptical polarization is the general case: unequal amplitudes or phase ≠ 0°/90°.

Figure 7.1 — Polarization states viewed along the propagation axis. The E-field tip traces a line (linear), circle (circular), or ellipse (elliptical).

7.3Unpolarized & Partially Polarized Light

Thermal sources (bulbs, Sun, LEDs) produce light from many independent atomic emissions with random polarization. The ensemble average shows no preferred direction — this is unpolarized light. Partially polarized light is a mixture quantified by the degree of polarization: $\text{DOP} = I_\text{pol}/I_\text{total}$ (0 = unpolarized, 1 = fully polarized).

7.4Polarization in Practice

Polarization affects nearly every optical system: coating reflectance/transmittance differs for s and p polarization, polarizing beam splitters separate components, many lasers emit linearly polarized light, standard single-mode fiber does not preserve polarization (PM fiber is required), and LCDs/SLMs control polarization rotation electrically. Full treatment (Jones calculus, Stokes parameters, Mueller matrices, waveplates) appears in the dedicated Polarization guide.

8Coherence

8.1Temporal Coherence

Temporal coherence describes how well a wave maintains a predictable phase relationship with itself over time. A perfectly monochromatic wave has infinite temporal coherence; real sources have finite spectral width and their phase drifts randomly [2, 7].

Coherence time and coherence length

\tau_c = \frac{1}{\Delta\nu} \qquad L_c = c\tau_c = \frac{c}{\Delta\nu} \approx \frac{\lambda^2}{\Delta\lambda}

The coherence length determines the maximum optical path difference for visible interference fringes. If path difference exceeds L_c, fringes wash out. This is central to interferometer, holography, and OCT system design.

Worked Example: Coherence Length of a Multimode HeNe

Problem: A multimode HeNe laser has a linewidth of 1.5 GHz. Find its coherence length and time.

Δν = 1.5 × 10⁹ Hz

Coherence time:

τ_c = 1/Δν = 1/(1.5 × 10⁹)
τ_c = 0.667 ns

Coherence length:

L_c = c × τ_c = (2.998 × 10⁸)(6.67 × 10⁻¹⁰)
L_c = 20.0 cm

Fringes are visible only when the path difference is less than 20 cm — sufficient for small interferometers but not large holograms. A single-mode HeNe (linewidth < 500 kHz) achieves L_c > 500 m.

Figure 8.1 — Temporal coherence comparison. Top: narrow linewidth source produces a long, uniform wave train (high coherence). Bottom: broad linewidth source produces short segments with random phase jumps (low coherence).

🔧 Coherence Length Calculator — compute L_c and τ_c from linewidth or bandwidth →

8.2Spatial Coherence

Spatial coherence is the phase correlation between two points in a beam cross-section at the same instant. A point source produces perfect spatial coherence; an extended source (finite size) reduces it because different parts emit independently [2, 7]. The van Cittert-Zernike theorem relates the coherence area to the source geometry: $A_c \approx (R\lambda/d)^2$ where R is the distance from the source and d is the source diameter. Laser beams in the fundamental TEM₀₀ mode are fully spatially coherent across the entire cross-section.

8.3Coherence of Common Sources

Source	Linewidth Δν	Coherence Length L_c	Spatial Coherence
Incandescent	~300 THz	~1 μm	Very poor
LED	~10–30 THz	~10–30 μm	Poor
Sodium lamp	~510 GHz	~0.6 mm	Moderate
Multimode HeNe	~1.5 GHz	~20 cm	Excellent
Single-mode HeNe	< 500 kHz	> 500 m	Excellent
Single-mode diode	~1–100 MHz	3–300 m	Good
DFB laser	~1 MHz	~300 m	Excellent
Single-freq. fiber laser	~1–10 kHz	30–300 km	Excellent

Table 8.1 — Coherence properties of common optical sources.

8.4Why Coherence Matters

Interferometry: Path difference must be less than L_c. OCT deliberately uses short-L_c sources (~10 μm) for depth resolution. Holography: Object-to-reference path difference may be centimeters to meters; a multimode HeNe (L_c ≈ 20 cm) works for small holograms, single-mode for large scenes. Fiber sensing: Coherence length determines range and resolution in FBG interrogation and distributed sensing. Speckle: Highly coherent illumination produces speckle on rough surfaces — a nuisance in imaging but useful in metrology. Low coherence is not always undesirable: broadband sources are chosen for OCT and white-light interferometry precisely because short L_c provides micrometer depth resolution.

9Light-Matter Interaction

9.1Absorption

Photons absorbed by a medium transfer energy to electronic, vibrational, or rotational transitions. The transmitted intensity falls exponentially with path length [1, 2]:

Beer-Lambert law

I(z) = I_0\,e^{-\alpha z}

where α is the absorption coefficient (m⁻¹ or cm⁻¹), specific to material and wavelength. In spectroscopy, the absorbance form $A = \varepsilon \ell c$ (molar absorptivity × path length × concentration) is the foundation of quantitative analysis.

Worked Example: Transmission Through Colored Glass

Problem: A 10 mm colored glass filter has α = 1.15 cm⁻¹ at 532 nm. Find transmission including surface reflections (n ≈ 1.5).

Absorption transmission:

T_abs = e^(−1.15 × 1.0) = e^(−1.15)
T_abs = 0.317 (31.7%)

Surface reflection losses:

R = [(1.5 − 1)/(1.5 + 1)]² = 0.04 per surface
T_surfaces = (1 − 0.04)² = 0.922

Total transmission:

T_total = 0.317 × 0.922
T_total = 0.292 (29.2%)

AR coatings reduce reflection losses from ~4% per surface to less than 0.5%, nearly doubling throughput in high-absorption systems.

9.2Reflection & Refraction

At an interface between media with different refractive indices, the beam splits into reflected and refracted components. The law of reflection (angle in = angle out) and Snell's law govern the geometry [1]:

Snell's law

n_1\sin\theta_1 = n_2\sin\theta_2

When $n_1 > n_2$ , total internal reflection occurs above the critical angle $\theta_c = \arcsin(n_2/n_1)$ — the basis of optical fiber waveguiding. At normal incidence, the reflectance is $R \approx [(n_1 - n_2)/(n_1 + n_2)]^2$ . For air-glass (n = 1.5): R ≈ 4%. A system with 10 uncoated surfaces loses roughly 34% of the light to reflection alone.

9.3Scattering

Rayleigh scattering (particle size d ≪ λ): $I \propto \lambda^{-4}$ — short wavelengths scatter more efficiently, producing the blue sky and red sunsets. Mie scattering (d ~ λ): strongly forward-peaked with complex wavelength dependence — responsible for white clouds and fog. Raman scattering (inelastic): the scattered photon gains or loses energy via molecular vibration. Frequency shifts are characteristic of the molecule, enabling chemical identification. Raman is extremely weak (~1 in 10⁷ photons).

9.4Emission

Spontaneous emission: An excited state decays randomly, emitting a photon of random direction and phase — the mechanism behind LEDs and lamps. Stimulated emission: An incident photon triggers an excited atom to emit a second photon identical in frequency, phase, direction, and polarization — the foundation of laser operation, requiring population inversion. Fluorescence: The emitted photon has lower energy than the absorbed photon, with the difference lost to heat. Phosphorescence: Similar but involving longer-lived triplet states, so emission persists after excitation is removed.

9.5Practical Perspective

Every optical component exploits or manages these interactions. A lens transmits (minimize absorption and scattering). A mirror reflects (minimize transmission and absorption). A filter selectively absorbs or reflects wavelengths. A detector converts absorbed photons to electrical signal. Understanding which interactions dominate at which wavelengths is essential for material selection: glass transmits visible but absorbs UV and mid-IR; silicon absorbs visible (excellent detector below 1100 nm) but transmits mid-IR; germanium is opaque in the visible but transmits the 8–12 μm LWIR thermal imaging band.

Process	Mechanism	Key Equation	Application
Absorption	Energy transfer to medium	I = I₀ exp(−αz)	Spectroscopy, filtering
Reflection	Boundary condition mismatch	n₁ sin θ₁ = n₂ sin θ₂	Mirrors, beam splitters
Refraction	Speed change at interface	n₁ sin θ₁ = n₂ sin θ₂	Lenses, prisms, fiber
Rayleigh scattering	Dipole re-radiation (d ≪ λ)	I ∝ λ⁻⁴	Blue sky, attenuation
Mie scattering	Particle ~ λ	Complex	Fog, clouds, turbidity
Spontaneous emission	Excited state decay	E = hf	LEDs, lamps
Stimulated emission	Photon-triggered decay	Gain	Laser amplification

Table 9.1 — Light-matter interaction summary.

Figure 9.1 — Light-matter interactions. An incident beam encountering an optical material can be reflected, refracted (transmitted), absorbed (converted to heat), scattered, or trigger fluorescence emission at a longer wavelength.

10Selecting the Right Model

10.1Ray Optics Regime

Ray optics applies when all dimensions — apertures, elements, beam diameters, features — are much larger than the wavelength. Diffraction is negligible and light is traced as rays obeying Snell's law and the law of reflection. Computationally simple and intuitive, this model is correct for lens/mirror system design, first-order optical layout, catalog component selection, fiber coupling geometry, and prism specifications [1, 3].

Ray optics fails silently — it does not warn when diffraction becomes significant. The Fresnel number provides a check: $N_F = a^2/(\lambda L)$ where a is the aperture radius and L the propagation distance. When $N_F \gg 1$ , ray optics is valid. When $N_F \lesssim 1$ , wave effects dominate.

10.2Wave Optics Regime

Wave optics is required when feature sizes approach the wavelength or when interference and diffraction are central to function: thin-film coatings (layer thickness ~λ/4), diffraction gratings (groove spacing ~1 μm), single-mode fiber (core diameter ~8 μm at 1550 nm), Gaussian beam propagation near the diffraction limit, interferometry, holography, and laser cavity mode analysis [2, 7]. Wave optics includes ray optics as a limiting case, so it is never wrong to use — only sometimes unnecessarily complex.

10.3Quantum Optics Regime

Quantum optics is required when the discrete photon nature affects observables: photon counting (signal < 10³ photons per measurement, Poisson statistics dominate), detector cutoff wavelengths (photon energy vs. band gap), laser physics (spontaneous/stimulated emission rates, Einstein A/B coefficients), nonlinear quantum processes (parametric down-conversion, entangled photon generation), and solar cell/photodiode efficiency (photon energy vs. band gap) [2, 7].

10.4Quick-Reference Decision Table

Scenario	Model	Why
Lens system design / f/#	Ray	All elements ≫ λ
Mirror alignment / beam steering	Ray	Geometric path tracing
Multimode fiber coupling	Ray	Core ≫ λ
AR coating design	Wave	Film thickness ~ λ/4
Diffraction grating spectrometer	Wave	Groove spacing ~ λ
Single-mode fiber propagation	Wave	Core ~ 5–10λ
Gaussian beam focusing	Wave	Diffraction-limited spot
Interferometer fringe analysis	Wave	Phase relationships central
Low-light detector SNR	Quantum	Shot noise ∝ √N
Solar cell efficiency vs. λ	Quantum	Band gap / photon energy threshold
Laser gain medium design	Quantum	Einstein coefficients, transition rates
Quantum key distribution	Quantum	Individual photon states matter
Entangled photon sources	Quantum	Non-classical correlations

Table 10.1 — Model selection guide for common photonics scenarios.

In practice, many systems span regimes. A fiber-coupled spectrometer uses ray optics for the collection lens, wave optics for the diffraction grating, and photon statistics for detector noise. The skill is knowing which model governs each subsystem.

References

[1]E. Hecht, Optics, 5th ed. Pearson, 2017.
[2]B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. Wiley, 2019.
[3]F. L. Pedrotti, L. M. Pedrotti, and L. S. Pedrotti, Introduction to Optics, 3rd ed. Cambridge University Press, 2017.
[4]F. G. Smith, T. A. King, and D. Wilkins, Optics and Photonics: An Introduction, 2nd ed. Wiley, 2007.
[5]M. Born and E. Wolf, Principles of Optics, 7th ed. Cambridge University Press, 1999.
[6]D. J. Griffiths, Introduction to Electrodynamics, 4th ed. Cambridge University Press, 2017.
[7]A. E. Siegman, Lasers, University Science Books, 1986.
[8]CIE, “Commission Internationale de l'Éclairage Proceedings,” Vienna, 1931.

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