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Optics & Photonics
1. Refractive index
\[n=\frac{c}{v_p}.\]For a nonmagnetic low-loss dielectric, approximately $n\approx\sqrt{\epsilon_r}$, though real materials are dispersive and absorptive.
2. Snell’s law
Tangential phase continuity at an interface gives
\[n_1\sin\theta_1=n_2\sin\theta_2.\]Total internal reflection occurs for propagation from higher to lower refractive index above the critical angle.
3. Fresnel reflection
At normal incidence,
\[r=\frac{n_1-n_2}{n_1+n_2},\qquad R=|r|^2\]for nonmagnetic lossless media. At oblique incidence, TE and TM polarizations behave differently; TM reflection vanishes at the Brewster angle for ideal dielectrics.
4. Interference
For coherent fields,
\[I\propto|E_1+E_2|^2.\]The cross term is the origin of fringes, interferometers, standing waves and coherent detection.
5. Diffraction and aperture
Finite apertures spread angular spectrum. A larger aperture produces a narrower diffraction-limited beam. This is the optical version of antenna aperture/directivity.
A recurring bridge is
\[\boxed{\text{antenna far field}\leftrightarrow\text{Fourier optics}.}\]6. Gaussian beams
For waist $w_0$,
\[z_R=\frac{\pi w_0^2}{\lambda},\] \[w(z)=w_0\sqrt{1+(z/z_R)^2}.\]The beam waist, Rayleigh range and wavefront curvature determine focusing and atom-light interaction volume.
7. Optical fibers
Total internal reflection provides a first picture, while waveguide modes provide the rigorous description. The normalized frequency of a step-index fiber is
\[V=\frac{2\pi a}{\lambda}\sqrt{n_1^2-n_2^2}.\]Single-mode operation requires the appropriate $V$ range (approximately $V<2.405$ for the standard step-index case).
8. Lasers
A laser combines:
gain medium + population inversion/pump + optical resonator + feedback/selective loss.
Key practical quantities are linewidth, output power, frequency noise, relative intensity noise, mode structure, polarization and beam quality.
For a detailed treatment of laser linewidth, frequency-noise spectra, heterodyne/self-heterodyne linewidth measurement, Fabry-Perot reference cavities, Pound-Drever-Hall locking, cavity-stabilized linewidths, SHG systems, and linewidth transfer through frequency doubling, see the dedicated Laser Systems reference.
9. Integrated photonics
Waveguides, ring resonators, Mach–Zehnder interferometers, modulators and photodetectors move optical functions onto chips. The same concepts of impedance/mode matching, scattering matrices, resonances and coupling reappear in optical form.
10. Atom–light interface
Photon energy is
\[E=hf=\frac{hc}{\lambda}.\]The electric-dipole interaction is
\[H_{int}=-\mathbf d\cdot\mathbf E.\]This connects classical optical fields to Rabi frequency, EIT, AC Stark shifts and quantum sensing. See Quantum Technologies and Rydberg Semiclassical Optics.
Worked example — Rayleigh range
For $w_0=100\ \mu$m and $\lambda=852$ nm,
\[z_R=\frac{\pi(100\times10^{-6})^2}{852\times10^{-9}}\approx3.69\ \text{cm}.\]This sets the length scale over which the beam radius remains near its waist.
11. How optics is measured
- optical power meter / calibrated photodiode;
- optical spectrum analyzer;
- wavemeter / frequency comb;
- beam profiler / knife-edge scan;
- polarimeter;
- interferometer;
- fast photodetector + RF analyzer for beat notes/noise.
Engineering reality
References
- E. Hecht, Optics.
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics.
- A. E. Siegman, Lasers.
- R. W. Boyd, Nonlinear Optics.
Related: Laser Systems · Foundations · Quantum Technologies · Measurements
Advanced topic — waveguide dispersion and loaded resonators
For a translationally invariant optical waveguide, fields can be decomposed into eigenmodes proportional to (e^{j(\beta z-\omega t)}). Their propagation constants (\beta(\omega)) depend on geometry and dispersive permittivity. Group velocity is (v_g=(d\beta/d\omega)^{-1}), while group-velocity dispersion is
\[\beta_2=\frac{d^2\beta}{d\omega^2}.\]A finite-bandwidth pulse broadens when propagating through a dispersive waveguide. The relevant parameter is the mode’s total dispersion, including both material and waveguide effects.
An optical resonator has loaded quality factor (Q_L), intrinsic (Q_i), and coupling (Q_c):
\[\frac1{Q_L}=\frac1{Q_i}+\frac1{Q_c}.\]Measured linewidth generally reveals the loaded response. Separate coupling loss from intrinsic attenuation before inferring absorption or material loss.
Advanced exercise: numerically calculate effective index and dispersion of a slab waveguide, then estimate pulse broadening and sensitivity to waveguide dimensions.
Further reading: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics; MIT Advanced Electromagnetism.
Space-optical and detector applications
Continue with Radiometry & Calibration, Optical Space Communications, Cryogenic Detectors and Inside a Scientific Instrument.