Modern Optics

The previous two lessons covered the classical foundations of optics: wave phenomena (interference, diffraction, coherence) and geometric ray tracing (reflection, refraction, lenses). In this lesson, we enter the modern era. Starting in the 1960s with the invention of the laser, optics underwent a revolution that continues today. Lasers, fiber optics, nonlinear optics, and quantum optics have transformed not only physics but also telecommunications, medicine, manufacturing, and fundamental research. Many of these advances connect directly to the quantum mechanics and electromagnetism developed earlier in this course.

Lasers

The word LASER stands for Light Amplification by Stimulated Emission of Radiation. To understand how a laser works, we need three ingredients from quantum mechanics.

Stimulated Emission

Stimulated emission is the key process. It produces a copy of the incoming photon, creating two photons where there was one. If these two photons each stimulate another emission, we get four, then eight, and so on: exponential amplification of light. But there is a catch.

Population Inversion

In a four-level laser (the most common design), the pump excites atoms from the ground state (level 0) to a high-energy state (level 3). Atoms rapidly decay non-radiatively to the upper laser level (level 2). The laser transition occurs from level 2 to level 1, and level 1 rapidly empties back to the ground state. Because level 1 is nearly empty, population inversion between levels 2 and 1 is easily achieved.

The Laser Cavity

To turn an amplifier into an oscillator, we place the gain medium inside an optical cavity (also called a resonator): two mirrors facing each other, one highly reflective and one partially transmitting.

Cavity Modes

The cavity supports standing waves. The condition for a standing wave between two flat mirrors separated by distance $L$ is:

$$L = q \frac{\lambda}{2}, \qquad q = 1, 2, 3, \ldots$$

where $q$ is a (very large) integer. The corresponding frequencies are the longitudinal modes:

$$\nu_q = q \frac{c}{2L}$$

The spacing between adjacent modes is the free spectral range:

$$\Delta\nu_{\text{FSR}} = \frac{c}{2L}$$

For a typical cavity length $L = 30$ cm, $\Delta\nu_{\text{FSR}} = 500$ MHz. The laser oscillates at the longitudinal modes that fall within the gain bandwidth of the medium. A helium-neon laser at 632.8 nm typically operates on 2-5 longitudinal modes simultaneously.

Fiber Optics

An optical fiber is a thin, flexible strand of glass or plastic that guides light over long distances by total internal reflection. Fiber optics is the backbone of modern telecommunications: virtually all internet traffic travels through optical fibers.

Total Internal Reflection in Fibers

A typical fiber consists of a core (refractive index $n_1$) surrounded by a cladding (refractive index $n_2 < n_1$). Light entering the core at a sufficiently shallow angle (relative to the core-cladding interface) undergoes total internal reflection and is trapped inside the core.

The numerical aperture of the fiber determines the maximum acceptance angle:

$$\text{NA} = \sin\theta_{\max} = \sqrt{n_1^2 - n_2^2}$$

For a typical telecom fiber, $n_1 = 1.4475$ and $n_2 = 1.4440$, giving NA $\approx 0.10$ and $\theta_{\max} \approx 5.7°$.

Fiber Modes

Attenuation and Dispersion

Modern silica fibers achieve remarkably low attenuation: about $0.2$ dB/km at $\lambda = 1550$ nm (the telecom C-band). This means that after 100 km, only 1% of the original power remains, but optical amplifiers (erbium-doped fiber amplifiers, or EDFAs) can boost the signal without converting it to electrical form.

Even in single-mode fibers, chromatic dispersion broadens pulses because different wavelengths travel at slightly different speeds. The dispersion is characterized by the parameter $D$ (in ps/(nm$\cdot$km)). Telecom systems manage dispersion using dispersion-compensating fibers or dispersion-shifted fibers designed so that $D = 0$ near 1550 nm.

Nonlinear Optics

At low intensities, the response of a material to light is linear: the polarization $\mathbf{P}$ is proportional to the electric field $\mathbf{E}$. But at high intensities (easily achieved with lasers), nonlinear terms become significant.

Second Harmonic Generation (SHG)

When an intense beam at frequency $\omega$ passes through a crystal with $\chi^{(2)} \neq 0$, the $E^2$ term generates a polarization component at frequency $2\omega$. This oscillating polarization radiates light at twice the input frequency (half the wavelength). For example, a 1064 nm infrared Nd:YAG laser beam can be converted to green 532 nm light by passing it through a potassium titanyl phosphate (KTP) crystal.

Efficient SHG requires phase matching: the generated second-harmonic wave must travel at the same speed as the nonlinear polarization that drives it. This condition is:

$$n(2\omega) = n(\omega)$$

which is generally not satisfied because of dispersion. Phase matching is achieved by exploiting the birefringence of the crystal: choosing a propagation direction where the ordinary wave at $\omega$ has the same index as the extraordinary wave at $2\omega$.

The Kerr Effect and Self-Phase Modulation

The third-order susceptibility $\chi^{(3)}$ causes the refractive index to depend on intensity:

$$n = n_0 + n_2 I$$

where $n_2$ is the nonlinear refractive index (typically $\sim 10^{-20}$ m$^2$/W for glass). This optical Kerr effect has several consequences:

• Self-focusing: A beam with a Gaussian intensity profile experiences a higher refractive index at its center than at its edges, creating a self-induced lens that focuses the beam.
• Self-phase modulation (SPM): A pulse traveling through a Kerr medium accumulates an intensity-dependent phase, broadening its spectrum.
• Optical solitons: In optical fibers, the interplay between SPM (which broadens the spectrum) and anomalous dispersion (which compresses the pulse) can create solitons, pulses that propagate without changing shape. Solitons are governed by the nonlinear Schrodinger equation, connecting optics to the mathematics of integrable systems.

Quantum Optics Preview

Classical electromagnetism treats light as a continuous wave. Quantum mechanics (Lessons 6-11) tells us that light is made of discrete photons. Quantum optics is the field where these two descriptions meet, and where the quantum nature of light produces effects that have no classical analog.

Photon Statistics

Squeezed States

The electromagnetic field has two quadratures (like the position and momentum of a harmonic oscillator), satisfying an uncertainty relation:

$$\Delta X_1 \cdot \Delta X_2 \geq \frac{1}{4}$$

A coherent state (laser light) saturates this bound with equal uncertainties in both quadratures. A squeezed state reduces the uncertainty in one quadrature below the coherent-state level, at the cost of increased uncertainty in the other:

$$\Delta X_1 = \frac{1}{2}e^{-r}, \qquad \Delta X_2 = \frac{1}{2}e^{r}$$

where $r$ is the squeezing parameter. Squeezed states are produced using parametric down-conversion (a $\chi^{(2)}$ process) and are used in gravitational wave detectors (LIGO uses squeezed light to improve sensitivity below the quantum shot noise limit) and in quantum information protocols.

Ultrafast Optics

One of the most remarkable capabilities of modern lasers is the generation of extremely short pulses. A mode-locked laser produces pulses as short as a few femtoseconds ($1$ fs $= 10^{-15}$ s).

Mode Locking

Mode locking can be achieved by placing a modulator inside the cavity that periodically attenuates the light (active mode locking) or by using a saturable absorber that transmits high-intensity peaks and absorbs low-intensity continuous-wave light (passive mode locking). Kerr-lens mode locking, which exploits self-focusing in the gain medium, is the most common technique for generating the shortest pulses.

Applications of Ultrafast Pulses

Femtosecond pulses have opened entirely new frontiers:

• Femtochemistry: Ahmed Zewail (Nobel Prize 1999) used femtosecond pulses to observe chemical bonds breaking and forming in real time, capturing the "transition state" of chemical reactions.
• Attosecond science: Through high-harmonic generation (a highly nonlinear process), femtosecond laser pulses produce attosecond ($10^{-18}$ s) pulses of extreme ultraviolet light, enabling observation of electron dynamics within atoms (Nobel Prize in Physics 2023).
• Optical frequency combs: The spectrum of a mode-locked laser consists of a comb of precisely equally spaced frequencies. This frequency comb can serve as an "optical ruler" for measuring frequencies with extraordinary precision (Nobel Prize 2005, John Hall and Theodor Hansch). Frequency combs are now used in precision spectroscopy, atomic clocks, exoplanet detection, and tests of fundamental physics.
• Laser machining and surgery: Ultrashort pulses can ablate material with minimal heat damage to surrounding tissue, enabling precise eye surgery (LASIK) and micromachining of delicate structures.

Laser Types and Applications

Different applications demand different laser characteristics. Here is a summary of common laser types:

Laser Type	Wavelength	Key Properties	Applications
He-Ne	632.8 nm	Low power, excellent beam quality	Alignment, interferometry, teaching
Nd:YAG	1064 nm	High power, pulsed or CW	Machining, surgery, SHG to 532 nm
Ti:sapphire	700-1000 nm	Broadband, ultrafast	Femtosecond spectroscopy, frequency combs
CO$_2$	10.6 $\mu$m	High power, infrared	Cutting, welding, surgery
Semiconductor	Various (400-1600 nm)	Compact, efficient, direct modulation	Telecom, optical storage, laser pointers
Fiber laser	~1060 nm	High power, excellent beam quality	Industrial cutting, defense