NE 345: Photonic Materials and Devices

Estimated study time: 12 minutes

Table of contents

Sources and References

Saleh and Teich, Fundamentals of Photonics (Wiley)
Kasap, Optoelectronics and Photonics: Principles and Practices (Pearson)
Coldren, Corzine, and Mashanovitch, Diode Lasers and Photonic Integrated Circuits (Wiley)
Yariv and Yeh, Photonics: Optical Electronics in Modern Communications (Oxford)
Online: RP Photonics Encyclopedia, MIT OCW 6.007 Applied Electromagnetics

Chapter 1: Nature of Light and Material Response

1.1 Plane Waves, Index, and Dispersion

A monochromatic plane wave propagating in a dielectric satisfies

\[ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0 \exp\!\left[ i(\mathbf{k}\cdot\mathbf{r} - \omega t) \right], \]

with wavevector magnitude \( k = n\omega/c \). The refractive index \( n(\omega) \) is frequency dependent because material polarisation lags the driving field. Near a single Lorentz oscillator of resonance \( \omega_0 \) and damping \( \gamma \),

\[ \varepsilon(\omega) = 1 + \frac{\omega_p^{2}}{\omega_0^{2} - \omega^{2} - i\gamma\omega}, \]

producing normal dispersion \( dn/d\omega > 0 \) away from resonance and anomalous dispersion near \( \omega_0 \) accompanied by absorption.

1.2 Group Velocity and Irradiance

Optical pulses propagate at the group velocity

\[ v_g = \frac{d\omega}{dk} = \frac{c}{n + \omega\,dn/d\omega}, \]

which differs from the phase velocity \( v_p = c/n \) whenever the medium is dispersive. Group-velocity dispersion stretches pulses and limits bit rates in fibre-optic links.

The time-averaged energy flow is the Poynting vector magnitude

\[ I = \tfrac{1}{2}\,n\,c\,\varepsilon_0\,|E_0|^{2}. \]

This irradiance is what photodetectors measure.

Chapter 2: Reflection, Refraction, and Coatings

2.1 Snell’s Law and Fresnel Equations

At a planar interface between media of indices \( n_1 \) and \( n_2 \),

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2. \]

Fresnel coefficients for the electric-field amplitude are, for s-polarisation,

\[ r_s = \frac{n_1\cos\theta_1 - n_2\cos\theta_2}{n_1\cos\theta_1 + n_2\cos\theta_2}, \]

and analogously for p-polarisation. The reflectance is \( R = |r|^2 \).

At normal incidence between air and silicon (\( n\approx 3.5 \) at 633 nm), \( R = ((3.5-1)/(3.5+1))^2 \approx 0.31 \): about 31% of incident light is reflected.

2.2 Antireflection Coatings

A quarter-wave layer of refractive index \( n_c = \sqrt{n_1 n_s} \) thickness \( \lambda/(4 n_c) \) interferometrically cancels reflection at the design wavelength. For a silicon solar cell, a silicon-nitride AR coating with \( n \approx 2 \) and thickness \( \approx 75 \) nm eliminates reflection near 600 nm, lifting short-circuit current by tens of percent. Multilayer dielectric stacks achieve broadband or high-reflectance mirrors by stacking alternating high- and low-index quarter-wave films.

Chapter 3: Absorption, Emission, and Coherence

3.1 Absorption in Semiconductors

In a direct-gap semiconductor, the absorption coefficient above the gap grows as

\[ \alpha(h\nu) = A\,(h\nu - E_g)^{1/2}, \]

reflecting the joint density of states. Indirect-gap materials like silicon require phonon assistance, giving weaker absorption

\[ \alpha(h\nu) \propto (h\nu - E_g \mp \hbar\Omega)^{2}. \]

This is why silicon photodetectors need tens of micrometres of active material while GaAs detectors need only a micron or two.

3.2 Coherence

Temporal coherence quantifies the correlation of a field with a delayed copy of itself. The coherence time is \( \tau_c \approx 1/\Delta\nu \) where \( \Delta\nu \) is spectral width.

Spatial coherence measures correlation between different transverse points of the wavefront. The van Cittert–Zernike theorem links it to the angular size of the source.

Lasers exhibit near-ideal spatial coherence and very narrow spectra; incoherent sources such as LEDs have coherence times of femtoseconds.

Chapter 4: Waveguides and Optical Fibres

4.1 Planar Dielectric Waveguides

A symmetric slab of index \( n_1 \) clad by \( n_2 < n_1 \) guides modes whose transverse field satisfies the eigenvalue equation

\[ \tan\!\left(\frac{k_x d}{2} - \frac{m\pi}{2}\right) = \frac{\gamma}{k_x}, \]

with \( k_x^{2} + \beta^{2} = n_1^{2} k_0^{2} \) inside and \( \beta^{2} - \gamma^{2} = n_2^{2} k_0^{2} \) outside. Only discrete values of \( \beta \) give bound modes. The number of supported modes scales with the normalised frequency \( V = k_0 d \sqrt{n_1^{2} - n_2^{2}} \); a single-mode waveguide requires \( V < \pi \) for symmetric slabs.

4.2 Optical Fibres

Step-index fibres are cylindrical analogues. Single-mode fibre at 1550 nm typically has a 9 \(\mu\)m core and index contrast near 0.003. Modal, material, and waveguide dispersion contribute to pulse broadening. Chromatic dispersion

\[ D = -\frac{\lambda}{c}\frac{d^{2}n}{d\lambda^{2}} \]

crosses zero near 1310 nm in standard silica fibre. Dispersion-shifted fibres move this zero to 1550 nm, aligning with the minimum-loss window.

Silica fibre loss bottoms at roughly 0.2 dB/km near 1550 nm — the physical basis for long-haul telecom deployment at that wavelength.

Chapter 5: Light-Emitting Devices

5.1 The pn Junction Under Forward Bias

Under forward bias \( V \), minority-carrier injection produces excess carriers that recombine. Radiative recombination rate is \( R_{rad} = B n p \), and total recombination also includes Shockley-Read-Hall (non-radiative) and Auger terms. Internal quantum efficiency is

\[ \eta_{i} = \frac{R_{rad}}{R_{rad} + R_{nr}}. \]

5.2 LED Materials and Extraction

Visible LEDs use direct-gap III-V alloys: AlGaInP for red-yellow and InGaN for green-blue. White light is typically produced by blue InGaN pumping a YAG:Ce phosphor. External quantum efficiency is limited by extraction: Snell’s law restricts the escape cone from a high-index semiconductor (\( n \approx 3.5 \)) to a half-angle \( \theta_c = \arcsin(1/n) \approx 16° \). Surface texturing, shaped dies, and encapsulants raise extraction efficiency beyond 80% in modern devices.

5.3 Laser Diodes

A semiconductor laser is an LED with optical feedback and gain above threshold. Stimulated emission occurs when population inversion is maintained via strong injection into a narrow active region, often a quantum well. The threshold current density satisfies

\[ g_{th} = \alpha_i + \frac{1}{2L}\ln\!\left(\frac{1}{R_1 R_2}\right), \]

balancing modal gain against internal loss and mirror loss. Above threshold, output power grows linearly with current with slope \( \eta_d h\nu/q \).

Chapter 6: Photodetectors and Photovoltaics

6.1 Photodetector Figures of Merit

Responsivity \( \mathcal{R} = I_{ph}/P_{opt} \) measures photocurrent per watt. Ideal quantum efficiency gives \( \mathcal{R} = q/h\nu \) so at 1550 nm the ideal responsivity is 1.25 A/W. Avalanche photodiodes multiply primary photocurrent by an internal gain \( M \) at the cost of excess noise.

6.2 Solar Cells

A solar cell is a large-area photodiode operated in the fourth quadrant under illumination. The diode equation with photogeneration is

\[ I = I_0 \left[ \exp\!\left( \frac{qV}{n k_B T} \right) - 1 \right] - I_{ph}. \]

Open-circuit voltage and short-circuit current define the power curve; maximum power is extracted at the knee. The fill factor

\[ FF = \frac{V_{mp} I_{mp}}{V_{oc} I_{sc}} \]

captures the squareness of the curve. The Shockley–Queisser limit places single-junction efficiency at about 33% under AM1.5 illumination. Multijunction, tandem, and perovskite-silicon devices push toward higher limits by better matching the solar spectrum.

A silicon cell with \( I_{sc} = 40 \) mA/cm², \( V_{oc} = 0.72 \) V, \( FF = 0.82 \) gives 23.6 mW/cm², a 23.6% efficiency under the AM1.5 reference 100 mW/cm².

Across emitters and detectors, the common thread is engineering the interaction of photons with charge carriers: confining modes, matching indices, tuning band gaps, and managing radiative versus non-radiative pathways so that optical energy flows predictably into and out of electronic systems.