Part I: Quantum Electrodynamics of Light-Matter Interaction

1. Introduction: Laser Characteristics and Principles of Amplification

Laser light is defined by a unique set of properties that distinguish it from conventional light sources. These characteristics include exceptional monochromaticity (purity of colour), superior directionality (low beam divergence), immense brightness or intensity, and high coherence (phase predictability).

The technological applications of these extreme characteristics are foundational to modern industry and science. In advanced manufacturing, the focused intensity of high-power fibre lasers is leading advancements in high-speed material cutting and welding due to increased efficiency. Furthermore, the precision afforded by lasers is accelerating the growth of laser-based additive manufacturing (3D printing) for complex and critical components in sectors like aerospace. This integration of high-power sources is increasingly paired with Artificial Intelligence (AI) to optimise cutting parameters, such as laser power and focus, dynamically in real time based on material properties, thereby improving quality and reducing errors.

2. Einstein’s Coefficients: The Quantum Foundation of Stimulated Emission

The interaction between radiation and matter at the atomic level is governed by three transition processes that occur between two energy levels, \(E_1\) (lower) and \(E_2\) (upper). These processes are quantified by Einstein’s coefficients:

1. Stimulated Absorption (\(B_{12}\)): An atom in \(E_1\) absorbs a photon, moving to \(E_2\).
2. Spontaneous Emission (\(A_{21}\)): An atom in \(E_2\) decays spontaneously to \(E_1\), emitting a photon with random phase and direction.
3. Stimulated Emission (\(B_{21}\)): An incident photon induces an excited atom in \(E_2\) to decay to \(E_1\), releasing an identical, coherent photon. This is the fundamental mechanism of light amplification.

By applying the condition of thermodynamic equilibrium (Planck’s law) to balance the rates of upward and downward transitions, two fundamental relationships between these coefficients are established:

1. Symmetry of Stimulated Processes: The probability coefficient for stimulated absorption must equal that for stimulated emission: \(B_{12} = B_{21}\).
2. Ratio of Spontaneous to Stimulated Emission: This crucial ratio is proportional to the cubic power of the transition frequency \((\nu)\): $$\frac{A_{21}}{B_{21}} \propto \nu^3$$

This frequency dependence implies that as the transition frequency (and photon energy) increases, the rate of incoherent spontaneous emission (\(A_{21}\)) grows much faster than the rate of coherent stimulated emission (\(B_{21}\)). Consequently, for short-wavelength lasers (e.g., UV or X-ray), significantly greater population inversion is required to overcome the noise of spontaneous emission.

3. Conditions for Laser Oscillation

Laser action requires two conditions to be met:

1. Population Inversion (\(\Delta N > 0\)): For net light amplification (gain) to occur, the rate of stimulated emission must exceed the rate of stimulated absorption. This requires that the population density of the upper energy level (\(N_2\)) exceeds that of the lower energy level (\(N_1\)), a non-equilibrium state known as population inversion.
2. Threshold Gain Condition (\(\gamma_{th}\)): For sustained oscillation, the amplifying medium must be placed inside an optical cavity (resonator). Oscillation begins when the material gain provided by the active medium (\(\gamma_{th}\)) exactly compensates for the total energy losses incurred during one round trip through the resonator.

The total losses are classified into distributed internal losses (\(\alpha_i\)) and localised mirror losses (\(R_1, R_2\)). The round-trip amplification factor for sustained oscillation must equal unity. The threshold condition for the linear gain coefficient (\(\gamma_{th}\)) required to sustain oscillation, given a gain medium length \(L\) and internal losses \(\alpha_i\), is:

$$\gamma_{th} = \alpha_i + \frac{1}{2L} \ln \left(\frac{1}{R_1 R_2}\right)$$

If the required material gain \(\gamma_{th}\) is reached, the intensity inside the cavity builds up. This buildup increases the stimulated emission rate, causing the population inversion (\(\Delta N\)) and thus the gain to saturate (decrease) until the saturated gain is precisely clamped at the threshold value (\(\gamma_{sat} = \gamma_{th}\)).

Part II: Quantitative Analysis of Spatial and Temporal Coherence

4. Quantitative Metrics for Temporal Coherence

Temporal coherence measures the correlation of a wave’s phase at a single point over different moments in time, determining the maximum path length difference over which interference can be observed.

Coherence Time (\(\tau_c\)): This is the average time interval over which the light wave maintains a predictable phase. Coherence time is inversely related to the spectral bandwidth (\(\Delta\nu\)) of the light source:

$$\tau_c \approx \frac{1}{\Delta \nu}$$

For highly stable single-mode lasers, the linewidth (\(\Delta\nu\)) can be very narrow (a few kHz), resulting in coherence times up to several hundred microseconds.

Coherence Length (\(L_c\)): This is the maximum distance a light beam can travel (\(\Delta L_{max}\)) before its phase correlation is lost and interference fringes vanish.¹⁴ It is related to coherence time by the speed of light \(c\), or by the central wavelength \(\lambda\) and the spectral width in wavelength \(\Delta\lambda\):

$$L_c = c \cdot \tau_c \approx \frac{\lambda^2}{\Delta \lambda}$$

5. Quantitative Metrics for Spatial Coherence

Spatial, or lateral, coherence describes the phase correlation across a plane perpendicular to the direction of propagation at a fixed instant in time. This determines a beam’s ability to be tightly focused.

Lateral Coherence Width (\(L_w\)): This defines the transverse dimension over which the field maintains a strong phase correlation.

Laser beams inherently exhibit high spatial coherence, meaning their coherence width is typically equal to the physical beam diameter, allowing for diffraction-limited focusing.

Part III: Pumping Dynamics and Multi-Level Laser Systems (Qualitative Discussion)

Laser systems utilise multiple energy levels to establish and sustain population inversion, bypassing the fundamental difficulty posed by two-level systems, which cannot achieve stable inversion due to the symmetry of stimulated absorption and emission.

6. Energy Level Architectures

6.1. Three-Level Laser System (e.g., Ruby Laser)

• Architecture: Lasing occurs between the intermediate metastable level (E2​) and the ground state (E1​). Pumping excites atoms from the ground state (E1​) to a broad upper pump band (E3​), from which they rapidly decay to E2​.
• Inversion Challenge: Since the lower lasing level is the heavily populated ground state (E1​), achieving population inversion (N2​>N1​) requires transferring more than 50% of the total atomic population out of the ground state and into the upper laser level (E2​).
• Pumping Requirement: This demanding requirement necessitates extremely high pump power density and typically results in pulsed operation.

6.2. Four-Level Laser System (e.g., Nd:YAG, He-Ne Laser)

• Architecture: Pumping excites atoms from the ground state (E1​) to a pump band (E4​). Atoms then rapidly decay to the upper laser level (E3​). Lasing occurs from E3​ to a lower laser level (E2​), which is an excited state located well above the ground state. An extremely rapid decay clears atoms from E2​ back to the ground state E1​.
• Inversion Advantage: Because the lower lasing level (E2​) is an excited state, its population (N2​) is inherently negligible (N2​≈0) under normal operating conditions due to the rapid clearing process.
• Pumping Requirement: Population inversion (N3​>N2​) is achieved instantly upon accumulating population in E3​. This significantly reduces the required threshold pump power, making the four-level system highly efficient and suitable for Continuous-Wave (CW) operation.

Feature	Three-Level System	Four-Level System
Lasing Levels	E2​→E1​ (Ground State)	E3​→E2​ (Excited State)
Lower Level Population	N1​ (High, Ground State)	N2​≈0 (Low, Excited State)
Inversion Condition	N2​>N1​ (Requires >50% lift)	N3​>N2​ (Requires minimal population lift)
Threshold Power	High (Pth​∝N)	Low (Pth​≪N)
Preferred Operation	Pulsed	Continuous-Wave (CW)
Example	Ruby Laser	Nd:YAG, He-Ne Laser

Part IV: Utilisation of Extreme Laser Characteristics in Advanced Science and Industry

7. Applications of Lasers

The unique characteristics of laser light—high coherence, intensity, directionality, and monochromaticity —make them indispensable across a wide spectrum of applications:   

• Precision Manufacturing: The high directivity of lasers allows for tight focusing, which is critical for material processing such as cutting and welding, enabling the efficient transmission of energy over long distances. High-power fibre lasers are leading this sector due to their efficiency and performance.
• 3D Printing (Additive Manufacturing): Laser technology, particularly through processes using controlled light to solidify materials, is foundational to the growth of 3D printing for complex components in aerospace and healthcare.
• Advanced Chip Fabrication (EUV Lithography): As noted in Section 1, high-power CO2​ lasers are used to generate 13.5 nm EUV light from plasma, a technology critical for producing microchips with structures smaller than 10 nm.   
• Scientific Breakthroughs: Researchers are continuously improving key laser parameters, including peak power and pulse length, with recent developments focusing on generating the strongest ultra-short laser pulses to date. Such high-power, short pulses are enabling new possibilities for precision measurements and advanced materials processing.