1029 Laser Physics John J. Zayhowski MIT Lincoln Laboratory, Lexington, MA 02420–9108, USA Phone: 781-981-0701; Fax: 781-981-0602; e-mail: [email protected] Paul L. Kelley Department of Electrical Engineering, Tufts University, Medford, MA 02155, USA e-mail: [email protected] Abstract This article concerns the physics of lasers, including population inversion, pumping processes, stimulated emission, and resonators. Important laser characteristics such as narrow spectral linewidth, high spatial collimation, and ultrashort pulses are discussed. A review is given of several types of laser media, laser dynamics, and modes of laser operation. Methods for control of laser output and practical considerations in laser design are considered. Finally, there is a discussion of nonlinear optical techniques for frequency conversion of laser radiation. Keywords laser; coherent radiation; laser, solid-state; laser, gas; laser, semiconductor; laser dynamics; rate-equation model; tunable lasers; mode-locked lasers; optical amplifiers; frequency conversion. 1 Introduction 1030 2 Basic Theory of Operation 1031 2.1 Population Inversion and Stimulated Emission 1031 2.2 Pumping and Relaxation Processes 1031 2.3 Resonators and Cavity Modes 1033 3 Important Characteristics of Laser Radiation 1034 3.1 Linewidth – Spectral Brightness 1034 3.2 Intensity and Directionality (Angular Confinement) – Spatial Brightness 1034 3.3 Short Pulses – Temporal Brightness 1034 1030 Laser Physics 4 Types of Lasers 1035 4.1 Solid-state Lasers 1035 4.2 Gas Lasers 1037 4.3 Dye Lasers 1039 4.4 Semiconductor Lasers 1040 4.5 UV and X-ray Lasers 1044 4.6 Free-electron Lasers 1045 5 Laser Dynamics 1045 5.1 Rate-equation Model 1045 5.2 Buildup from Noise 1047 5.3 Threshold 1047 5.4 Gain Saturation 1047 5.5 Laser Efficiency 1048 5.6 Multimode Operation 1048 5.6.1 Spatial Hole Burning 1049 5.6.2 Spectral Hole Burning 1049 5.6.3 Single-frequency Operation 1050 6 Types of Pulsed Operation 1051 6.1 Long-pulse Operation 1051 6.1.1 Relaxation Oscillations 1051 6.2 Q-switched Operation 1052 6.3 Gain-switched Operation 1052 6.4 Cavity-dumped Operation 1053 6.5 Mode-locked Operation 1053 6.5.1 Active Mode Locking 1054 6.5.2 Passive Mode Locking 1054 7 Control of Laser Output 1055 7.1 Frequency Tuning 1055 7.2 Amplitude Modulation 1055 8 Oscillator–Amplifier Systems 1056 9 Issues in Laser Design 1056 10 Frequency Conversion and Nonlinear Control of Laser Radiation 1057 Glossary 1058 Further Reading 1062 1 a large number of frequencies. The word Introduction laser is an acronym for ‘‘light amplification by stimulated emission of radiation.’’ The The laser is a device that generates co- principle of operation of lasers is similar herent, highly directional electromagnetic to that of the maser, which is somewhat radiation somewhere in the wavelength arbitrarily defined as a device operating range from submillimeter through X-ray. in the range from the radio or microwave Lasers can operate at a single wavelength region down to millimeter wavelengths. (and frequency) or, when mode locked, on Since the first laser was operated in 1960, Laser Physics 1031 the laser has come to play an important Laser transitions in the optical region role through its revolutionary impact on are most often electric dipole in character. applied optical technology, including fiber- In the dipole approximation to the Hamil- optical communications and optical data tonian, the transitions arise from a term storage. of the form erE,wheree is the electronic charge, r is the quantum-mechanical co- ordinate operator defined relative to the 2 center of coordinates of the material sys- Basic Theory of Operation tem (such as an atom or molecule), and E is the electric field of the optical wave at the 2.1 center of coordinates. The transition rate Population Inversion and Stimulated and gain cross-section are proportional to Emission the square of this interaction term. The transition-matrix element of the coordi- Quantum theory shows that matter can nate operator between upper and lower exist only in certain allowed energy levels laser levels ranges from about one hun- or states. In thermal equilibrium, lower- dredth of a Bohr radius (≈ 0.5 × 10−8 cm), energy states of matter are preferentially for vibrational transitions in molecules populated, with an occupation probabil- and for local-field-induced transitions of −E/kT ity proportional to e ,whereE is the rare earths in solids, to several hundreds state energy, T the temperature, and k the of Bohr radii, for highly excited Rydberg Boltzmann constant. An excited state can atoms. decay spontaneously (i.e., with only zero- A laser generally consists of three com- point electromagnetic radiation present) to ponents: (1) an active medium with energy a lower-energy state, emitting a quantum levels that can be selectively populated, or wave packet of electromagnetic radi- (2) a pump to produce population in- ation (photon) with transition frequency version between some of these energy ν = E/h,whereE is the energy dif- levels, and (usually) (3) a resonant elec- ference between the two states and h is tromagnetic cavity that contains the active Planck’s constant. In the presence of radi- medium and provides feedback to main- ation at frequency ν, a transition from tain the coherence of the electromagnetic the upper state to the lower state can field (see Fig. 1). In a continuously operat- be induced, with the simultaneous emis- ing laser, coherent radiation will build up sion of a photon in phase (coherent) with in the cavity to the level required to balance the stimulating radiation. This stimulated the stimulated emission and cavity losses emission process is the reverse of the ab- (see Sect. 5.3). The system is then said to sorption process. If matter can be forced be lasing, and radiation is emitted in a out of thermal equilibrium to a sufficient direction defined by the cavity. degree, so that the upper state has a higher population than the lower state (popula- 2.2 tion inversion), more stimulated emission Pumping and Relaxation Processes than absorption occurs, leading to co- herent growth (amplification or gain) of A material system can become excited the electromagnetic wave at the transition and displaced from normal thermal frequency. equilibrium when driven by processes 1032 Laser Physics Cavity mode e u Laser material Mirror Mirror Fig. 1 Simplified schematic of a laser oscillator. The mirrors at the ends of the laser form an open resonator. Stable modes that consist of electromagnetic waves that travel back and forth in the resonator are amplified by the active laser material. In the radiative steady state, the gain l due to amplification balances the loss due to intracavity absorption, mirror reflection losses, g and diffraction beyond the edges of the mirrors. Fig. 2 Schematic representation of a four-level The pumping system is not shown, nor are system. Population is pumped from g to e and ancillary intracavity elements that are often used laser operation occurs on the transition between for temporal (including frequency selection) and uandl spatial control of the laser output. Usually, one of the mirrors is partially transmitting so that some of the highly directional radiation leaves the cavity through the mirror. The dashed lines Let us try to understand pumping and are approximately characteristic of the transverse relaxation in an ‘‘ideal’’ four-level laser extent of the lowest-order transverse mode with the aid of Fig. 2. The pumping process, indicated by the upward arrow, such as chemical reactions or under is assumed to excite the system from sufficiently strong external influence. Ex- the lowest energy level, denoted by g for ternal influences include electron beams ground state, to the highest level, denoted and optical fields that selectively excite en- by e for excited state. Pumping might ergy levels of the material. Applied voltages occur in a variety of ways, one of which can create electrical currents, also result- could be through radiative excitation using ing in disequilibrium. Disequilibration, if light whose frequency coincides with the carried out by a sufficiently selective pro- transition frequency between g and e. cess, can result in population inversion The state e is assumed to relax to the and laser operation. This ‘‘pumping’’ can upper laser level u. The population of the be carried out continuously, with single upper laser level is radiatively transferred, pulses, or with multiple pulses of ex- either through spontaneous or stimulated citation. The inversion and its duration emission, to the lower laser level l. Finally, depend on the relaxation rates for the the lower laser level can either relax to the different energy levels and the degrees ground state or absorb the laser radiation of freedom of the system, as well as and repopulate the upper laser level. on the rate of stimulated emission. The Several conclusions concerning optimum energy-level scheme of the laser plays an operation can be made from this model. important role in obtaining inversion; in First, the relaxation rates from e to u and Sect. 5.1 we will discuss the difference from l to g should be as rapid as possible in the operation of three-level and four- in order to maintain the maximum level lasers. population inversion between u and l. Laser Physics 1033 Second, the pumping rate between g and structure is defined only by axial mirrors eshouldbesufficientlyrapidtoovercome or lenses (see Fig. 1). Open resonators the spontaneous emission from u to l. formed with convergent optics (‘‘stable’’ Third, the thermal equilibrium population resonators) generally have the lowest of l should be as small as possible. Fourth, diffraction losses, while planar resonators decay of e to any level other than u have higher losses, and resonators formed should be as slow as possible (for optical with divergent optics (‘‘unstable’’ res- pumping, e can decay radiatively to g) and onators) have the highest losses.