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. Figure 3 the nonradiative decay of u should be slow. shows a few of the lowest-order transverse For radiative pumping, it is advantageous mode distributions for a stable resonator. to have e distinct from u and not to have There is generally one transverse mode radiative decay of u to g; for other types of of a cavity that has the largest net pumping this advantage is not obvious. gain (product of the amplifier gain and Not every practical laser satisfies this the transmittance of the remainder of ‘‘ideal’’ model. The Cu-vapor laser violates the cavity). This is the transverse mode the first conclusion listed in the last that oscillates first, and is typically the paragraph, since the lower laser level has lowest-order (TEM00)mode(seeFig.3). a very slow decay rate and the laser self- Single-transverse-mode lasers, in particu- terminates owing to filling of this level with lar, lasers operating in the TEM00 mode, concomitant reduction in the population have nearly optimal ‘‘spatial brightness’’ inversion. Nevertheless, the Cu-vapor laser in the sense that the beam divergence is operates with fair efficiency as a powerful neartheminimumvalueforthespotsize repetitively pulsed laser. of the laser on the output mirror; such a laser is said to have diffraction-limited 2.3 Resonators and Cavity Modes TEM00 TEM01 An important aspect of the laser involves the design of resonators to accommodate the characteristics of the active medium and the diffractive properties of radiation, and at the same time meet requirements such as low angular beam divergence and high efficiency. The electromagnetic field TEM10 TEM11 in a resonator has well-defined modes that have patterns both transverse to and along the cavity axis. Waveguiding with index-of- refraction profiles or reflecting walls can be usefully employed in some cases (most notably in semiconductor and fiber lasers), not only to confine the radiation to the am- plifying medium, but also to force the laser Fig. 3 Lowest-order transverse modes for a ‘‘stable’’ resonator with square symmetry. The to operate in a single transverse mode. TEMmn notation indicates that the modes have More frequently, however, laser resonators nearly transverse electric and magnetic fields are open in the sense that the transverse with m nodes vertically and n nodes horizontally 1034 Laser Physics

output. The optimal resonator for a par- the laser frequency to drift. The influence ticular laser (e.g., ‘‘stable’’ or ‘‘unstable’’) of noise can often be reduced by adjusting is determined by the geometry of the gain controllable parameters such as cavity life- medium, the desired cavity length, and the time. In addition, the frequency drift can single-pass gain. be reduced by measuring the frequency of Cavity modes have an axial periodicity the laser and providing feedback to adjust that is determined by the cavity length. the cavity mode frequency to compensate The frequency spacing between the axial for the drift. The frequency can be mea- (longitudinal) modes is the inverse of the sured in several ways: by using a stable round-trip time for radiation in the cavity. Fabry-Perot´ etalon, by comparing the laser The gain in a laser is peaked at a transition frequency with a narrow resonance in an frequency determined by the energy levels atom or molecule, or by comparing the fre- oftheactivemediumandlaseroperation quency with a nearby frequency reference. tends to occur at the axial-mode frequency (or frequencies) closest to the gain peak. 3.2 Intensity and Directionality (Angular Confinement) – Spatial Brightness 3 Important Characteristics of Laser Radiation As discussed in Sect. 2.3, lasers operat- ing solely in the lowest-order transverse 3.1 mode (TEM00) have optimal spatial bright- Linewidth – Spectral Brightness ness and are described as operating in the diffraction limit. In this case, for propa- Lasers can have very narrow linewidths gation of initially collimated radiation in and, therefore, very high spectral bright- a nonconfining medium, the product A, ness (power per unit spectral interval). where A is the laser output aperture and Linewidths as narrow as a fraction of  is the solid angle into which the power a hertz have been obtained (1-Hz stabil- is confined at long distances, has a mini- ity corresponds to a fractional stability of mum value of order λ2,whereλ = c/ν is − about 2 × 10 15 for visible lasers). the wavelength of the radiation and c is the The first step in achieving narrow-line velocity of light. This aperture–solid-angle operation is to design the laser to oper- product is invariant to transformations of ate in a single longitudinal and transverse the laser beam such as beam expansion. mode (see Sect. 2.3). Once single-mode op- Because of this diffraction-limited charac- eration has been obtained, the linewidth ter, single-mode laser beams can maintain of the output is determined by technical high intensities over long distances and and fundamental noise. Technical noise have a high degree of directionality (small arises from sources that can be con- solid angle). trolled, such as power-supply fluctuations, variations in the thermal environment, 3.3 environmental vibrations, and so on. Fun- Short Pulses – Temporal Brightness damental noise arises from sources that cannot be eliminated, such as spontaneous Short pulses of laser radiation can be emission and fundamental thermal fluctu- made in a variety of ways including: gain ations. The random effect of noise causes switching, Q switching, cavity dumping, Laser Physics 1035 mode-locking, and nonlinear frequency substituted for a small fraction of the Al3+ broadening and pulse compression. These ions) as the active medium. In this laser, techniques are described in following a xenon flashlamp pumps the chromium sections. To obtain the shortest possible ions from their ground state to a broad pulses, mode locking is used to obtain band of states, from which they rapidly de- − pulses in the picosecond (1 ps = 10 12 s) caynonradiativelytoalong-livedstateatan − to femtosecond (1 fs = 10 15 s) range, of- energy of about 14 422 cm−1 (693.4 nm) ten followed by frequency broadening and above the ground state. (In spectroscopy pulse compression. Pulses as short as and laser physics, energy E and frequency 4.5 fs have been obtained at a center wave- ν are often given in terms of wave number length of 800 nm, corresponding to less 1/λ,whereE = hν = hc/λ.) It is the nar- than 2 cycles of light, and very high peak row (≈ 11 cm−1) emission line from this powers are possible. To obtain even shorter level back to the ground state that gives pulses, nonlinear optical techniques are rise to ruby laser emission. When lasing used to frequency convert the output of occurs, the emission narrows in frequency mode-locked lasers into the VUV and soft width to less than 1 cm−1, and leaves the X-ray spectral regions, where pulses as ruby rod with an angular spread of a few − short as 100 attoseconds (1 as = 10 18 s) milliradians. have been reported. The ruby laser is a three-level system High temporal brightness (high peak (see Fig. 4 and Sect. 5.1), and therefore power) competes with high spectral bright- requires depopulation of the ground state ness in the sense that the shortness by more than 50% to obtain population of a pulse is limited by the bandwidth inversion. For this reason, pumping of of the radiation. Said another way, for the laser requires a high-intensity source, a fixed average power in a continuous and continuous-wave (cw) operation of train of bandwidth-limited pulses, peak the system is difficult to achieve. In the power cannot be increased without reduc- free-running mode, as opposed to the Q- ing spectral brightness. switched mode (see below and Sect. 6.2), the output radiation of the ruby laser fluctuates rapidly over a time of about 1 ms 4 and, for a 1-cm-diameter × 10-cm-long Types of Lasers laser rod, is emitted in pulses of about 1 J. The high threshold pump-power re- A rich variety of physical systems have quirement of a three-level laser is greatly been exploited to produce laser radiation relaxed in a four-level system. In the four- over a five-decade range of wavelengths. level system, radiative transitions do not We can only briefly describe here some of terminate on the ground state (see Fig. 2). themostsignificantofthese.Moredetail If the final state is not significantly pop- can be found in the articles on specific ulated at the operating temperature, then types of lasers. population inversion can be maintained 4.1 with only moderate pump power and such Solid-state Lasers lasers can usually be operated on a contin- uous basis. The first laser used a rod of ruby One type of four-level solid-state laser 3+ (single-crystal Al2O3 with the ion Cr is based solely on the electronic levels 1036 Laser Physics

e powers on the order of 10 W. At power levels on the order of 100 mW, they are u in use as amplifiers in long-distance fiber transmission. The availability of high-brightness diode lasers has made some quasi-three-level solid-state systems very attractive for continuous, as well as pulsed, operation. The rare-earth ion Yb3+, which operates as a quasi-three-level system at 1.03 µm, has been found to be very suitable for diode laser pumping because of its g – l relatively broad pump band. Yb3+ also Fig. 4 Schematic representation of a three-level has a paucity of low-lying electronic states, system.Notethatgandlarenowthesamelevel resulting in no absorption of pump light by ions in the excited state. The small of ions in a crystal or other solid-state energy difference between the excitation host, with the laser transition terminating photon and the emitted photon leads on an excited electronic level of the ion. to low thermal loading, but comes with + + + The rare-earth ions Nd3 ,Ho3 ,Er3 , the disadvantage of significant thermal + and Tm3 , which can also be operated as occupation of the lower laser level at quasi-three-level systems (with the lower room temperature, and the Yb-ions must laser level partially occupied), are the most be pumped sufficiently hard to overcome frequently used active ions in such lasers. the consequent absorption. The relatively Extensively developed lasers of this type small amount of heat generated in Yb- are the Nd-glass laser and the Nd:YAG doped materials is attractive for scaling (Y3Al5O12) laser, both of which have their to high average powers, and Yb:YAG most important transitions in the vicinity lasers have been operated with more of 1.06 µm. Nd:YAG lasers have produced than 1 kW of continuous output, making in excess of 5 kW in continuous opera- them competitive with more extensively tion, several joules in low-repetition-rate developed Nd:YAG systems. Double-clad (10 Hz) pulsed operation, and about 10 mJ fiber lasers using Yb have been operated in high-repetition-rate (1 kHz) pulsed op- with output powers over 400 W. In the eration. In order to increase efficiency double-clad structure, the laser radiation and decrease heat loading in the solid- is confined to a single-mode inner core state laser medium, Nd:YAG lasers have while the multimode pump radiation been pumped with arrays of semiconduc- travels through both the inner core and tor diode lasers. Large Nd-glass lasers have a larger multimode region. The shape produced tens of joules at repetition rates of the multimode region can be tailored of several hertz, and several kilojoules at to the shape of the pump beam. For very low repetition rates (≈1perhour). instance, a rectangular region can be used Rare-earth-doped glass fiber lasers are to accommodate the shape of a pump- of use in optical communications. Single- diode array. mode Er3+ fiber lasers at 1.55 µm(aquasi- A second type of four-level solid-state three-level transition) have given output laser involves fewer electronic levels of Laser Physics 1037 the active ion. Instead, laser transitions aretypicallyontheorderof10nsin are employed that terminate on an excited duration and can be obtained with the use vibrational level of the ion in the host lat- of a saturable absorber, an electro-optic tice. The basic optical pumping–emission element, or an acousto-optic element as cycle is similar to that of a molecular dye the switch in the laser cavity. laser (see Sect. 4.3). One important class Because solid-state lasers have spectrally of these solid-state lasers uses transition- broad gain regions, many equally spaced + + + + metalionssuchasCr3 ,Cr4 ,Ti3 ,Ni2 , longitudinal cavity modes can lie within 2+ and Co .Aswiththeruby(Al2O3) laser, the gain bandwidth when cavity lengths the alexandrite (BeAl2O4) laser employs areontheorderoftensofcentimeters. + the Cr3 ion. In the latter case, however, As a result, mode locking (see Sect. 6.5) the laser transition can terminate on a can be used to obtain a train of high- variety of final vibrational states. As a intensity, short pulses. Mode locking has result, this laser is tunable from 700 to been applied to many solid-state laser 818 nm, and has pulsed output energies systems such as Nd:YAG (20-ps pulses) similar to the ruby laser. The Ti:Al2O3 and Ti:Al2O3 (5-fs pulses). Mode locking laser has even broader tunability, covering has also been used in many other types of a range from 660 nm to beyond 1.1 µm. It laser (see, e.g., Sect. 4.3 on dye lasers). has operated continuously at power levels up to 17 W with Ar-laser pumping, and 4.2 pulsed at energies of hundreds of milli- Gas Lasers joules when pumped by doubled Nd:YAG laser radiation. A number of methods can be used to Lasers based on color centers in al- produce population inversion in gaseous kali–halide crystals operate on a principle media. Inversion can exist between some similar to dye lasers (see Sect. 4.3) and of the energy levels of the constituents in transition-metal lasers. Using different a gas discharge (electrical discharge in a types of F-centers in various alkali halides, gas). The first such system, demonstrated wavelength coverage over a range from not long after the announcement of the 0.82 to 3.3 µm can be obtained, with con- ruby laser, was the He-Ne laser, now a tinuous output powers ranging from tens standard item in optics laboratories. This of milliwatts to over a watt. Stability of system makes use of a discharge in He F-centers can be a problem, and low- at a pressure of about 1 torr, with an temperature storage is required for several admixtureofNeatabout0.1torr.The of these lasers. discharge excites He atoms to their first − The energy stored in a pumped solid- excited level, about 160 000 cm 1 above state laser medium can be delivered as their ground state. This excitation is readily a giant pulse in a time much shorter transferred by collisions to a Ne atomic than the spontaneous lifetime of the level with nearly the same excitation energy upper laser energy level through the use (resonant transfer). These excited states of Q switching (quality-factor switching). decay radiatively to lower-energy Ne states, This technique (see Sect. 6.2), which was giving rise to continuous laser emission − first used in the ruby laser, has been in the red at 15 820 cm 1 (632.8 nm) applied to many laser systems. In the case with an output power in the range of the Nd:YAG laser, Q-switched pulses of 10−2 W. Other transitions produce 1038 Laser Physics

strong emission at 8680 cm−1 (1.15 µm) the excitation is resonantly transferred by and 2957 cm−1 (3.39 µm). In a pure Ne molecular collisions to preferentially excite discharge, excitation would occur to many CO2 molecules to a particular vibrational Ne levels, and population inversion would state. These molecules in turn undergo not occur as effectively. radiative transitions to lower vibrational An important gas-discharge laser is levels. The presence of the rotational struc- based on the energy levels of the argon ture gives rise to a cluster of many lines ion (Ar+). Through a complex series of that can lase, grouped near frequencies − − steps, argon-ion–electron collisions in the of 944 cm 1 (10.6 µm) and 1042 cm 1 discharge lead to population inversion and (9.6 µm). The electric-discharge CO2 laser lasing at a number of frequencies near is quite efficient (better than 10% elec- 20 500 cm−1 (488 nm). Continuous output trical power converted to laser power) at power levels of tens of watts in the and is capable of producing continuous- blue–green make this device especially output powers of greater than 1 kW. Other useful as a spectroscopic source in Raman important molecular gas-discharge lasers scattering, and for pumping continuously make use of vibration–rotational or pure operating tunable dye and Ti:Al2O3 lasers. rotational transitions of H2O, CO, and Other intense laser sources arise from HCN, and produce emission at (78 µm, atomic transitions of metal ions in a 119 µm), 5.3 µm, and (337 µm, 311 µm), pulsed He discharge (the Cu- and Cd- respectively. vapor lasers, for example). Gas-discharge A number of other methods of gener- lasers have also been made to operate in ating laser radiation using molecular or the UV, but special problems arise in this atomic energy levels in gases have been de- frequency range; these are discussed below vised. Powerful pulses of laser radiation at in Sect. 4.5. 1.315 µmfromexcitediodineatomshave Gaslaserswithoutputatlonger been produced by flash photolysis (UV wavelengths make use of the vibra- photodecomposition) of CH3I(methylio- tion–rotational energy levels of molecules. dide) and, using energy transfer from 1 In addition to the electronic-state energy- chemically generated O2( ), kilowatts of level structure characteristic of atoms, continuous power have been obtained. In there is vibration–rotational structure as- the gas-dynamic laser, a nonthermal dis- sociated with the relative motion of the tribution of molecular vibrational energy nuclei. The spacing of vibrational en- levels is produced by the rapid expan- ergy levels corresponds to frequencies sion of a hot gas through a nozzle. This in the infrared. It is in this region method has produced continuous emis- of the electromagnetic spectrum that sion of tens of kilowatts at 10.6 µmfrom molecular gas-discharge lasers are espe- CO2 gas. In the chemical laser, two react- cially important. ing molecular species in a gas produce The most efficient and powerful of the a product that is left in an excited vi- molecular gas-discharge lasers is the CO2 bration–rotation state and returns to the laser. One version of this laser makes ground state radiatively. An example is the use of a dilute mixture of CO2 in an HF (DF) laser, which lases in the 2.5- to N2 discharge. The N2 molecules are ex- 3.5-µmregionwhenH2 (D2) and F2 gases cited by collisions with electrons to their combine chemically. Other types of lasers first excited vibrational state, from which produced by excitation of gases are TEA Laser Physics 1039

(transversely excited atmosphere) lasers, be understood in terms of the change e-beam (electron-beam excited) lasers, and of equilibrium internuclear position with UV-preionized, electric-discharge lasers. electronic excitation, the rapid vibrational CO2 laser radiation has been used to relaxation within an electronic state, and pump other gases, yielding far-infrared the Franck-Condon principle, which states emission. If there is a coincidence between that an electronic transition in a molecule aCO2 laser line and a vibration–rotational takes place so rapidly that the nuclear transition in another gas, an excited level coordinates can be regarded as nearly can be populated directly or by collisional fixed. An optical pumping–fluorescence transfer of excitation. Pure rotational cycle for a molecular system is indicated transitions can take place radiatively to an schematically in Fig. 5(a). After absorption unpopulated level, producing far-infrared of a photon and nonradiative vibrational laser radiation. Such gases as NH3 (output cascading to the lowest vibrational level, µ at 291 m) and CH3OH (output at 164 the excited electronic state can decay and 205.3 µm) have been made to lase by by fluorescence at a lower frequency, this method. and the molecule returns to the ground 4.3 state by nonradiative vibrational cascading. Dye Lasers Since the final state in the radiative transition process is unoccupied, the Many organic dyes, when illuminated pumping requirements for such a laser with visible or UV radiation, fluoresce system need not be too severe. Discrete strongly at lower frequencies. This so- vibrational structure is washed out in called Stokes shift of the fluorescence can a liquid, but the general outline of

EE

v = 2 v = 2 v = 1 v = 1 v = 0 v = 0

v = 2 v = 1 v = 0

R R (a) (b) Fig. 5 Light emission from molecular systems (a) bound–bound system and (b) bound–free system 1040 Laser Physics

the cycle indicated in Fig. 5(a) is still and cw dye lasers. Dye-jet fluctuations nor- preserved in dye fluorescence in a liquid. mally limit the stability of cw dye lasers, This process is virtually identical to although stabilities of hundreds of kilo- that occurring in tunable transition-metal hertz are readily obtained, and stabilities 3+ solid-state lasers such as Ti :Al2O3, of a fraction of a hertz have been achieved. 3+ 4+ Cr :BeAl2O3,andCr :Mg2SiO4.Inthe Thus, the dye laser became an important solid, the substitutional ion’s coordinates source of radiation for spectroscopy, but with respect to neighboring host ions recently the tunable solid-state laser has play the same role as the internuclear proven more practical. coordinates in the dye molecules. Since the dye laser has a spectrally A problem that arises with the use of broad gain region and many equally spaced dyes in a laser is the existence of long- longitudinal cavity modes falling within lived electron spin-triplet states into which the gain bandwidth, it is well suited the usual excited electron spin-singlet state for mode-locked operation (see Sect. 6.5). can relax nonradiatively. (For rhodamine Using continuous mode-locked dye lasers, 6G, a commonly used dye, this nonra- pulsetrainshavingpulsesoflessthan10-fs −7 diative relaxation time is ≈10 s.) This duration have been obtained. These short process interferes with laser action, but pulses are used for studies of fast electronic can be largely circumvented by circulating processes in solids and organic molecules. the dye solution through the laser cav- ity. Dye lasers can be flashlamp pumped, 4.4 or pumped with radiation from Ar ion, Semiconductor Lasers frequency-doubled Nd:YAG, or N2 lasers. Dye systems fluoresce over a wide band In 1962, it was reported that a forward- of frequencies from the near-infrared biased semiconductor diode of GaAs −1 through the visible, and are well adapted radiated efficiently at about 11 800 cm for use in a cw laser, with wavelength tun- (850 nm). In the following year, a number ingoverasmuchas40nmforasingle of groups reported the observation of laser dye. A battery of dyes placed in optical cav- emission in this frequency region from ity structures gives tunable laser radiation suitably prepared diode structures. While over a range from roughly 10 000 cm−1 these semiconductor devices are solids, (1 µm) to 25 000 cm−1 (400 nm). they are differentiated from those lasers Because of the broad bandwidth of involving optically active ions in ionic hosts dye fluorescence lines and, in the case on account of their markedly different of pulsed dye lasers, the low Q of the physical and technological characteristics. resonant structure, the dye laser emis- To understand how semiconductor sion line is fairly broad. It can, how- lasers function, it is necessary to consider ever, be greatly narrowed without much the nature of the electronic energy states loss in output power through the use in a semiconductor. A periodic crystal has of a diffraction grating in place of one bands of allowed energy levels separated of the mirrors in the cavity. Even nar- by forbidden energy gaps. In an intrinsic rower linewidths may be obtained by semiconductor at low temperatures, there using intracavity frequency-selective ele- are just enough electrons present to fill ments (etalons). Single-longitudinal-mode the uppermost-occupied energy band (va- operation can be obtained in both pulsed lence band), leaving the next higher band Laser Physics 1041

E Electrons

Conduction Conduction band band Electron current E Radiative Eg Holes flow g recombination

Valence Valence band band

n p x Junction Fig. 6 Schematic showing the emission of radiation from a forward-biased light-emitting diode

(conduction band) empty. In an n-type pair density. The x coordinate shown in semiconductor, impurity atoms (donors) Fig. 6 corresponds to the vertical direction are present that contribute electrons to the in Fig. 7. conduction band; in a p-type semiconduc- In many diode (or injection) lasers, a tor, there are impurity atoms (acceptors) cavity structure (typically a fraction of a present that can bind electrons, leaving millimeter in dimensions) is provided by behind missing electrons (holes) in the plane-parallel, cleaved facets at right angles valence band. to the junction plane. Laser emission Figure 6 shows a schematic of a p–n is perpendicular to the cleaved facets, junction, fabricated by forming p-and once the diode injection current reaches n-type semiconductor layers in intimate a threshold value. Multimode behavior contact. When a voltage is applied in the often occurs in the plane of the junction. forward direction, electrons are injected Since stimulated emission occurs in a from the n region into the depletion region narrow area near the junction (a few of the junction (a region of the junction to several hundred micrometers in the about 1 µm thick). At the same time, holes plane of the junction and a fraction of a are injected from the p region. As electrons micrometer perpendicular to the junction), drop into hole states, they may emit the angular spread of emitted radiation is radiation (electron–hole recombination fairly large, as expected from consideration radiation) at a frequency in the vicinity of diffraction. of the energy gap (ν = Eg/h). When the The technology of diode lasers has un- injection current density is sufficiently dergone considerable development, with high, population inversion and gain will the primary goals of achieving room- be induced. temperature operation, low thresholds, A typical semiconductor diode-laser high output powers, improved mode qual- heterostructure is shown schematically in ity, wavelength diversity, and long life- Fig. 7. The dielectric film serves to guide times. Progress has included improve- the current into a narrow stripe region ments in electrical and optical confine- in order to concentrate the electron–hole ment, and closer coupling of the gain 1042 Laser Physics

Mirror facets

p-side electrode

Dielectric film

p-type GaAs cap layer Current p-type AlGaAs

GaAs or AlGaAs active layer

n-type AlGaAs

n-type GaAs n-side electrode substrate Light Fig. 7 Typical semiconductor diode-laser heterostructure. The material compositions shown produce lasers operating in the 750- to 850-nm wavelength range. The device length is typically 250 µm, the width 50 to 100 µm, and the height on the order of 50 µm. The epitaxial layers (i.e., excluding the substrate and the contacts) contribute only a few micrometers to the height. For thermal and electrical connection, the device is soldered to an electrically conducting heat sink, and an electrical lead is attached to the topside of the device. The width of the emitting region is 10 to 20 µm, while the height is approximately 0.5 µm; this causes the beam to diverge rapidly in the vertical direction

region to the heat sink. Much of the im- radiative gain cross section. In strained provement has involved advances in ma- quantum well devices, the density of states terials growth, including molecular-beam in the valence band is reduced, leading to a epitaxy (MBE) and metal-organic chem- further reduction in the transparency cur- ical vapor deposition (MOCVD). These rent and reducing parasitic effects such as growth techniques have resulted in ma- Auger recombination, which is important terial of exceptionally high quality as well at long wavelengths. as structures with quantum confinement In 1969 (before the advent of quantum- of carriers (electrons and holes). well devices), room-temperature continu- Quantum-well diode lasers have ex- ous operation was achieved in a GaAlAs ceptional optical and electrical properties double-heterostructure laser. This laser including very low laser thresholds. By structure consists of a small region of GaAs reducingthevolumeoftheactivere- sandwiched between p-andn-type layers of gion through the use of quantum wells, the wider-bandgap alloy AlxGa1−xAs (x < the amount of current required to bleach 1). With further development, device life- the absorption of the semiconductor (the times of tens of years were obtained, with transparency current) is reduced, with a output powers in the tens of milliwatts. commensurate reduction in threshold. In These improvements opened up applica- addition, the quantum confinement of car- tions of considerable significance – in par- riers in the active region increases their ticular, fiber-optical communication and Laser Physics 1043 optical data storage. GaAlAs diode lasers mid-infrared wavelengths and longer. The have also been operated continuously at active region of a quantum-cascade laser room temperature with output powers in features several epitaxially grown layers of the range of several watts. Linear arrays of semiconductor. The device generates radi- diodes, in the form of multiple stripes in ation based on electronic transitions that a 1-cm bar, have given a total output of occur in the stacked layers. The thickness over 50 W. In addition, electrical-to-optical of the layers, rather than the fundamental power conversion efficiencies of greater bandgap Eg of the materials, determines than 50% have been obtained. the frequency of the emitted radiation. Continuous laser operation at room Quantum-cascade lasers have been oper- temperature has also been obtained in ated at wavelengths from 3.4 to 67 µm, the quaternary alloy system InGaAsP. with the potential to extend the range from Because laser operation is much less 1 to 100 µm. Continuous operation with sensitive to dislocation effects in this output powers in excess of 0.5 W has been system, long-lifetime devices were readily obtained at liquid-nitrogen temperature, achieved. When operating at 1.3 and and pulsed operation with peak powers 1.55 µm, these lasers are matched to low- approaching 1 W has been demonstrated loss (<1dbkm−1), low-dispersion fiber at room temperature. These systems have optics, and are currently used in high- also demonstrated tunability in excess data-rate, long-distance communication. of 150 nm. InGaAsSb/AlGaAsSb quantum-well la- In the visible region, using the qua- sers have produced 1 W of continuous ternary alloy AlGaInP, continuous room- output (5 W pulsed) at wavelengths as temperature operation has been ob- long as 2.5 µm, at near room temperature. tained at 635 nm and pulsed oper- Cooled to liquid-nitrogen temperature, ation at 603 nm; continuous liquid- this material system can be used to nitrogen-temperature operation has been produce laser output at wavelengths as achieved at 583.6 nm, which is in the long as 4 µm. yellowregionofthespectrum.Ini- The compositionally tuned lead-salt tial attempts to obtain operation in lasers (PbxSn1−xTe, PbSxSe1−x) operate the blue–green spectral region focused at cryogenic temperatures. As with other on Zn1−xCdxSe/ZnSySe1−y quantum-well semiconductor lasers, changes in com- heterostructure diodes. These lasers have position change Eg and, therefore, the been operated continuously at room tem- frequency of laser emission. Since Eg and perature, but have not yet demonstrated hence the dielectric constant at near-band acceptable lifetimes for commercial appli- gap wavelengths are sensitive functions of cations. Operation of SiC diode lasers at temperature in these small gap semicon- 403 nm, in the blue–violet spectral region, ductor systems, cavity-mode frequencies has faced a similar problem. More recently, and gain peaks are tunable, giving rise to heterostructure diode lasers based on In- temperature-tunable laser output. These AlGaN alloys have demonstrated in excess tunable sources have been used extensively of 10 000 h lifetime for continuous room- for high-resolution infrared spectroscopy temperature operation at wavelengths as in the 5- to 20-µmregion. short as 375 nm, and this system has Quantum-cascade lasers offer an alter- the potential to be tailored for outputs native to lead-salt lasers for operation at throughout the visible and near-ultraviolet. 1044 Laser Physics

Operating at an output power of a few mil- 4.5 liwatts, InAlGaN diode lasers are now in UV and X-ray Lasers use in CD/DVD recorders. VCSELs (vertical-cavity surface-emitting To produce stimulated emission in the lasers) have become an important class UV and X-ray region, special problems of semiconductor lasers. As with cleaved- must be addressed. Short lifetimes of inverted populations become important, cavity semiconductor lasers, these devices since spontaneous radiative lifetime varies are constructed in layers using MBE or with frequency as ν−3. For example, MOCVD. The peculiarity of the VCSEL the nitrogen gas-discharge laser, which is that some of the layers are used to radiates in the near UV (337 nm), can only form distributed Bragg mirrors on either operate in a pulsed mode, and must be side of the junction region. Radiation pumped by a powerful intermittent source. builds up in the direction normal to the Another problem is associated with the junctioninsteadofintheplaneofthe difficulty of devising resonant structures, junction. Careful cleaving does not have since the reflectivity of materials becomes to be done and the output beams are very small in the vacuum-UV and X-ray circularly symmetric. regions. A different conceptual scheme Semiconductor lasers with one of the must frequently be used in these spectral radiation feedback elements external to regions, involving directionally amplified the semiconductor material are also tech- spontaneous emission from an inverted nologically significant. Wavelength-tuned population (superradiance), where the external-cavity lasers are flexible sources directional amplification is achieved by the for optical communications and other geometry of the pumped region. applications. Both conventional cleaved- The nitrogen and hydrogen discharge cavity diodes and VCSELs have been used lasers make use of radiative transi- to make external-cavity lasers. In order tions between two bound electronic lev- to reduce the effect of the internal cavity els (bound–bound transitions). On the (formed by the interface of the semicon- other hand, there are molecular systems ductor to air), very good antireflection (excimers) in which radiative transitions coatings must be used, having reflectivities occur between a bound excited state and of the order 0.0001. a free or very weakly bound ground state Semiconductor optical amplifiers (for bound–free transitions – see Fig. 5b). (SOAs) are being developed to provide Xe and Kr form the excited molecular high-power outputs and booster amplifi- ∗ ∗ states Xe2 and Kr2, although the di- cation in fiber-communication systems. atomic molecules are unstable in their Tapered-waveguide designs are used to ground states. At high pressure these prevent filamentation or ‘‘hot spots’’ at gases, when pumped by powerful electron- the end of the amplifier. beam sources, emit superradiantly at 172 In addition to injection pumping of and 145.7 nm, respectively. At somewhat semiconductor lasers, a number of other longer wavelengths, rare-gas–halide ex- methods not requiring the fabrication of cimers, such as ArF∗ at 248 nm, KrF∗ at a junction have been used successfully. 193 nm, and XeCl∗ at 308 nm, have been These include electron-beam pumping operated both by electron-beam pump- and optical pumping. ing and by transversely excited discharge. Laser Physics 1045

These lasers can produce multijoule pulses the wavelength restrictions imposed on at 100-Hz repetition rates and are commer- other lasers, and do not have the inho- cially available. mogeneities characteristic of many laser X-ray lasers at wavelengths shorter than media, they require electron beams of a a few tens of nanometers present a more current level and quality that are difficult difficult challenge. Nevertheless, several to achieve simultaneously. Further, the groups have achieved laser operation in electron-beam and wiggler requirements this region. A multijoule visible laser can tend to become more demanding as wave- directly pump the gain medium through length decreases. These systems have the inversion produced in a laser-generated large sizes that are associated with rela- plasma, or indirectly by using the X-rays tivistic electron-beam sources. from a laser plasma to pump a separate X- ray laser medium. The first approach has 5 been used to achieve operation near 21 nm Laser Dynamics in Se XXV, which has a 1s22s22p6 Ne-like electron configuration. Current research 5.1 involves efforts to improve the very low Rate-equation Model efficiency of X-ray lasers as well as studies of potential new X-ray laser systems. Many of the properties of the laser can be determined from a rate-equation model for 4.6 the population of the laser levels and the Free-electron Lasers photon number in the laser cavity. The rate equations provide a simple and intuitive, Relativistic electrons traveling in a peri- yet accurate, picture of the behavior of odically alternating transverse magnetic many lasers. In the most simplified form, field (wiggler) can be stimulated to give theincreaseinphotonnumberwithinthe up radiation to a copropagating electro- laser cavity is balanced by the decrease 2 magnetic field of wavelength λ = λw/2γ , in the population difference between the where λw is the wiggler period and γ upper and lower laser levels. In addition, is the ratio of the electron energy to its the population difference increases on rest-mass energy. This phenomenon can account of the pumping process, while be pictured as stimulated inverse Comp- the photon number decreases because of ton scattering from the electromagnetic absorption, diffraction of the beam out field of the wiggler seen in the electron of the cavity, and transmission through rest frame. To obtain efficient conversion the mirrors. over reasonable distances with high out- The rate-equation model can be derived put power, the electron-beam must be very as an approximation to the fundamen- monoenergetic, constitute a high current, tal equations relating the electromagnetic and have very low angular divergence. Effi- field, the material polarization, and the cient operation also requires either the use populations. The validity of the rate equa- of an electron storage ring or a tapered- tionsrequiresthatthepolarizationcanbe period wiggler. Free-electron lasers have accurately approximated by assuming that been operated at several wavelengths rang- it instantaneously follows the field; this is ing from the visible to millimeter waves. a situation that applies to most lasers. In While free-electron lasers do not have order to describe the problem in terms of 1046 Laser Physics

total population and total photon number cavity is thus within the laser cavity, it is necessary that qNσc q thegainofthelaserbesmallduringone q˙ = q˙stim − q˙dec = − .(1) V τ pass through the cavity and that the laser c operate in a single longitudinal mode (see We will now derive the rate equations Sect. 5.6). for the population inversion of both a four- As stated above, the number of photons level and a three-level laser. The energy- q within the laser cavity is affected by two level diagrams of a four- and a three-level types of events: the emission of a photon laser are shown in Figs. 2 and 4. In both cases, the pump excites the active medium by the gain medium (q˙em) and the escape of a photon from the cavity or absorption from the ground level (with population ˙ N ) to level e (N ). It is then assumed by unpumped transitions (qdec). Photon g e that the excited state quickly decays to emission can be either stimulated (q˙stim)or spontaneous (q˙spon). Once a laser is above the upper laser level, level u (Nu), so that ≈ threshold, the stimulated-emission rate is Ne 0. Lasing occurs between levels u much greater than the spontaneous rate and l (Nl). The difference between a four- and, to first order, spontaneous emission and a three-level laser is that for the three- can be ignored. We will return to the level laser the lower laser level is also the issue of spontaneous emission in the next ground state. section when we discuss the buildup of a To derive the rate equation for the pop- = − laser from noise. ulation inversion N Nu Nl,westart The stimulated-emission rate is propor- by considering the population of the up- tional to the number of photons within per laser level Nu. The population of the the cavity, the total population inversion upper level is affected by pumping Pr, ˙ N, and the probability per unit time B stimulated emission Nu,stim,andsponta- ˙ that a given photon will interact with a neous decay Nu,spon. For most pumping given inverted site. The interaction prob- schemes, the pump rate is proportional to ability B is the product of the probability the number of ions in the ground state = that a photon will pass within the gain and can be written as Pr WpNg.The cross section σ of a given inverted site stimulated-emission process decreases the as it traverses the laser cavity and the population of the upper laser level by number of times per second the cavity one for every photon created, so that ˙ =−˙ is traversed. Mathematically, this reduces Nu,stim qstim. Spontaneous decay is to B = σ/A × c/l = σc/V,whereA is the characterized by the spontaneous lifetime ˙ cross-sectional area of the laser mode, l is τ, corresponding to Nu,spon =−Nu/τ.The the cavity length, and V isthevolumeof rate equation for the upper-level popula- the lasing mode. The stimulated emission tion is thus rate is, therefore, q˙stim = qNσc/V. ˙ ˙ ˙ Nu = Pr + Nu,stim + Nu,spon The escape of photons from the laser σ cavity and their absorption within the qN c Nu = WpNg − − .(2) cavity are characterized by the cold-cavity V τ lifetime τc (cavity lifetime in the absence In an ideal four-level laser there is a very of any inversion) and corresponds to rapid decay of the lower laser level to ˙ q˙dec = q/τc. The total rate of change of the ground state, so that Nl and Nl are the number of photons within the laser approximately equal to zero. Since the Laser Physics 1047 total number of active ions Nt is constant, of noise stimulates optical transitions and Nt ≈ Ng + Nu. The rate equations for a initiates lasing. four-level laser are, therefore,
 5.3 Nσc 1 q˙ = − q,(3) Threshold V τc ˙ qNσc N The threshold inversion required for lasing N = Wp(Nt − N) − − .(4) V τ is derived by requiring that the photon rate equation have a nontrivial solution Since, in an ideal three-level laser, the in steady state. In steady state q˙ = 0, lower laser level is the ground state, resulting in the condition qNσc/V − N ≈ N + N and N˙ ≈−N˙ .Asaresult, t l u l u q/τ = 0. Physically, this states that the the rate equations for a three-level system c number of photons leaving the cavity reduce to
 must be balanced by the number of σ ˙ = N c − 1 ,() photons generated through the stimulated q τ q 5 V c emission process. The threshold inversion σ ( + ) ˙ 2qN c Nt N is, therefore, N = Wp(Nt − N) − − . V τ = V .() ( ) Nthresh 8 6 σcτc The photon rate equations (Eqs. 3 and 5) The pump rate Wp required to reach ˙ = ˙ = for the four-level and three-level lasers are threshold is obtained by setting N 0, q = the same. However, the rate equations 0, and N Nthresh. For a four-level laser for the population inversion are slightly Nthresh V Wp,thresh = τ ≈ , different. In particular, the stimulated (Nt − N ) Ntσcτcτ thresh ( ) emission term for a three-level laser is 9  twice that of a four-level laser. wherewehaveassumedthatNthresh Nt. For a three-level laser 5.2 ( + ) = Nt Nthresh τ ≈ 1 .() Buildup from Noise Wp,thresh 10 (Nt − Nthresh) τ τ In the photon rate equation derived For the same value of ,thethresh- in Sect. 5.1, the term corresponding old pump rate for a four-level laser to spontaneous emission was left out. is smaller than the threshold pump Laser action is initiated by sponta- rate for a three-level system by a / σ τ neous emission, or noise. As a re- factor of V Nt c c, which is usually sult, these rate equations cannot ac- quite large. This is the basis of the count for the onset of lasing, as is superior performance of a four-level seen by setting q = 0attimet = 0. system over a three-level system in When spontaneous emission is properly cw operation. taken into account, the photon rate equa- 5.4 tion becomes σ Gain Saturation ˙ = ( + ) c − q .() q qN Nu τ 7 V c The gain g of an active medium is The net effect is as if there were initially defined as the fractional change in optical one photon in the cavity. This one photon intensity per unit length as a light beam 1048 Laser Physics

passes through. From the discussion in incident optical energy may be reflected Sect. 5.1, it follows that g = Nσ/V.In by the gain medium and part may be the presence of a strong optical field, the transmitted. Both of these effects decrease population inversion is reduced and the the pump efficiency. The area efficiency gain is saturated. The rate equation for the ηa is a measure of how well the pumped population inversion (of a three- or four- volume is used by the oscillating mode. If level gain medium) in steady state can be the cross section of the pumped volume is rewrittenintheform much larger than the cross section of the g lasing mode, only a small portion of the g = 0 ,(11) (1 + I/Is) pumped volume contributes to the gain of the system, and the area efficiency where g0 is the unsaturated gain (the will be low. The intrinsic efficiency ηi gain in the absence of an optical field), is simply the ratio of the energy of the = ν / ν I h qc V is the optical intensity (h is photon created during lasing to the energy the energy of one photon), and Is is known required to create one excitation. In an as the saturation intensity. optically pumped system, the intrinsic The steady-state photon rate equation efficiency is the ratio of the energy of ˙ = = (q 0) predicts that above threshold (q a photon at the oscillating frequency to 0) the inversion density and the gain of the energy of an absorbed pump photon. a laser are clamped at their threshold Finally, the output coupling efficiency ηo values; the round-trip gain of the cavity is is the ratio of the output coupling to the equal to the round-trip loss. With increased total round-trip loss of the laser cavity. pumping, the gain remains fixed while The total efficiency η of a laser (power the photon number and the output of the out divided by power in) is dependent laser increase. on the slope efficiency and the laser threshold. Using the rate equation model, 5.5 the slope efficiency is constant and the Laser Efficiency total efficiency is given by
 The efficiency of a laser is often discussed η = η − Pthresh ,() η s 1 13 in terms of the slope efficiency s.Slope P efficiency is defined as the ratio of the change in output power to the change where P is the total pump power and in pump power of a laser once it has Pthresh is the pump power required to reached threshold, and is determined by reach threshold. four factors: the pump efficiency ηp,the 5.6 area efficiency ηa, the intrinsic efficiency Multimode Operation ηi, and the output coupling efficiency ηo. Mathematically, For most lasers, the frequency spacing

ηs = ηpηaηiηo.(12) between adjacent longitudinal modes is much less than the gain bandwidth. As The pump efficiency ηp is the ratio of a result, lasers tend to oscillate at several the energy absorbed by the gain medium frequencies simultaneously. to the energy of the pump source. In Although the above statement is true, the an optically pumped laser, part of the reasons are subtler than they may initially Laser Physics 1049 seem. In the early days of lasers, it was believed that lasers with homogeneously I1(z) broadened gain spectra should operate in a single longitudinal mode. The reasoning (a) behind this can be understood from the rate equations. If we assume a uniform optical intensity within the laser cavity, N(z) the steady-state solution to the photon rate equation fixes the inversion density at its (b) threshold value. The first cavity mode to lase (the one with the highest net gain) clamps the inversion density and no other I (z) mode can reach threshold. The flaw in 2 this reasoning lies in the assumption of (c) uniform optical intensity. Fig. 8 Illustration of spatial hole burning, Experimentally, lasers with both homo- showing (a) the intensity profile of the first ( ) geneously and inhomogeneously broad- longitudinal mode to lase, I1 z ;(b)the population inversion in the presence of the first ened gain media tend to oscillate in several oscillating mode, N(z); and (c) the intensity longitudinal modes as a result of spatial profile of the second longitudinal mode to and spectral hole burning. lase, I2(z)

5.6.1 Spatial Hole Burning 5.6.2 Spectral Hole Burning In standing-wave laser cavities, the co- Homogeneous gain broadening occurs herent superposition of the optical fields when each excited state of the gain traveling in two directions within the cavity medium has exactly the same energy distri- results in a sinusoidal intensity distri- bution.Thisisoftenthecaseincrystalline bution. At positions where the intensity solid-state gain media at room tempera- distribution is at its maximum, there is ture. In materials such as glasses, each strong gain saturation and the population excited ion sees a slightly different envi- inversion is depleted. However, at nulls ronment, resulting in a slightly different in the optical field the oscillating mode energy spectrum. The ensemble effect is is unable to deplete the inversion. As a inhomogeneous gain broadening. Another result, the inversion density is no longer example of an inhomogeneous system is a uniform, but has ‘‘holes’’ at the positions gas, where each atom or molecule is mov- corresponding to the peaks in the opti- ing at a slightly different speed and has cal intensity. This phenomenon is known its energy spectrum Doppler shifted by a as spatial hole burning. The gain at the different amount. nulls in the optical field will continue to In an inhomogeneous system, only increase as the gain medium is pumped those excitations with gain at the lasing harder. Since other cavity modes have a dif- frequency are able to participate in the ferent spatial profile than the first mode, stimulated emission process. As a result, and can use the population inversion at only those excitations become depleted, these positions, this will lead to the onset producing a gain spectrum that has a dip of multimode operation. These ideas are at the lasing frequency. This is known as illustrated in Fig. 8. spectral hole burning, and is illustrated in 1050 Laser Physics

frequency selectivity to ensure single- frequency operation. Single-mode operation has also been obtained by reducing the length of the cavity so that the longitudinal mode spacing is comparable to, or less than, the (a) (b) gain bandwidth. This is most easily done with gas lasers, which have a narrow gain bandwidth, but has also been achieved in very short solid-state lasers. Alternatively, single-mode operation can be achieved in a laser with a homoge- neously broadened gain medium by re- (c) ducing or eliminating the effects of spatial Fig. 9 Illustration of spectral hole burning, hole burning. A unidirectional ring cavity showing (a) several closely spaced has a uniform optical intensity within the homogeneously broadened spectra; (b) the cavity, rather than the sinusoidal inten- inhomogeneously broadened spectrum resulting sity distribution of a standing-wave cavity. from the sum of several homogeneously Spatial hole burning is therefore elimi- broadened spectra; and (c) the inhomogeneously broadened spectrum with a nated, and such a laser may operate at a spectral hole burnt in the center as a result of the single frequency well above threshold. A saturation of one of the homogeneously variation of the same idea involves plac- broadened components ing a quarter-wave plate on either side of the gain medium. As a result, the optical fields traveling in opposite directions in the gain medium are orthogonally polar- Fig. 9. Excitations that cannot contribute ized and do not interact coherently. The to the lasing process for the first mode can optical intensity within the gain medium contribute to the onset of lasing for other is, therefore, uniform and there is no spa- modes, resulting in multimode oscillation. tial hole burning. In some gain media, there is a large 5.6.3 Single-frequency Operation amount of energy diffusion. Energy diffu- There are many techniques for obtaining sion moves some of the excited states away single-frequency operation from a laser. from the peaks in the population inversion, Several of these involve introducing an toward the minima. This smoothes out the element into the cavity such that the cavity population-inversion profile, reducing the sees a frequency-dependent loss, thereby effects of spatial hole burning. One impor- decreasing the net gain bandwidth and tant example of such a gain medium is a selecting an individual longitudinal cavity semiconductor. mode. Examples of such elements are a Finally, the effects of spatial hole burn- prism, a grating, a Fabry-Perot´ etalon, ing are reduced if the gain medium is and the combination of a birefringent located very close to a cavity mirror. At filter and a polarizing element. In some a mirror, the phase of the optical spa- cases, a cavity may require more than tial intensity distribution is pinned at zero one device in order to obtain enough for all of the cavity modes. The phase Laser Physics 1051 difference between longitudinal modes in- creases gradually as one moves toward the center of the cavity. Quite close to the mirror, the peaks and nulls of the optical intensities for each of the modes occur N(t ) in approximately the same place, and that portion of the gain that is not depleted by one mode will not be in a good position to contribute to any other. (a) t

6 Types of Pulsed Operation

6.1 q(t ) Long-pulse Operation

Long-pulse or quasi-cw operation refers to apulsedlaserwithapulsedurationlong (b) t enough for all relevant parameters within Fig. 10 Computer solutions to the rate the system to come to their steady state equations for a laser with a step-function pump value. Although the behavior of the system source, showing (a) the population inversion is cw-like at the end of the pulse, it will, in N(t) and (b) the photon number q(t) general, be quite different at the beginning of the pulse. Let us consider a laser with a step- Figure 10 shows the computer solutions function pump source. The pump may to the rate equations for a laser under quickly create a population inversion. the conditions just described. In this It will take some time, however, for a computer simulation, the spiking was lasing mode to buildup from spontaneous heavily damped and cw-like behavior was emission. During this time, the inversion quickly obtained. In a multimode laser density may greatly exceed threshold. The the interaction between modes often leads large inversion density eventually results to mode hopping, mode beating, and in a large optical intensity, well in excess very irregular spiking, which may never of the cw value. This optical intensity, in damp out. turn, drives the inversion density below threshold, substantially reducing the laser 6.1.1 Relaxation Oscillations intensity. The entire process then starts Relaxation oscillations occur whenever the again. For a single-mode laser, this often population inversion of a laser is disturbed leads to regular spiking at the beginning of from its steady-state value. It is a result thepulse.Theprocessisdamped,however, of the coupling between the population and with time the intensity of the spikes inversion and the photon density within decreases. Spiking eventually gives way to the laser cavity, as described above. From damped oscillations (known as relaxation the rate equations, it can be shown that for oscillations) in the optical intensity and a single-mode laser a small disturbance in finally cw-like behavior. the inversion density results in damped 1052 Laser Physics

oscillations with an oscillation frequency the output pulse also relies on the presence   / of a strong intracavity optical field, which (N/N − 1) 1 2 ω = thresh ( ) argues in favor of a high cavity Q.Inorder τ τ 14 c to obtain a short output pulse, however, and a damping constant the intracavity intensity must also decay quickly after the peak of the pulse. Since 2τNthresh t0 = .(15) N the Q of the cavity is constant for the duration of the pulse (it is difficult to Note that if 1/t >ω the oscillations are 0 change the Q of the cavity significantly overdamped and spiking will not occur. during the duration of a short output Although this condition is not satisfied pulse, although it can be changed in in solid-state lasers, it is common in the relatively long pulse buildup time gas lasers. preceding the pulse), this would argue 6.2 for a low cavity Q. Solutions to the rate Q-switched Operation equations show that the minimum-width output pulse is obtained when the total For many applications it is desirable to round-trip loss of the cavity during the obtain short, high-peak-power pulses from output pulse is adjusted so that the initial a laser. This can be achieved by creating inversion density is about three times the a large population inversion and then threshold value. quickly decreasing the cavity loss so that Methods for Q switching a laser include the inversion density is well in excess the use of electro-optic shutters, acousto- of its new threshold value. The large optic Q switches, and mechanical devices. inversion density allows an intracavity Passive Q switching can be obtained optical field to rapidly develop. This field through the use of an intracavity saturable then depletes the population inversion absorber with a long recovery time. and turns itself off. The cavity loss is subsequently increased to prevent the 6.3 development of a second pulse. This Gain-switched Operation technique is known as Q switching, since the quality factor, or Q,oftheopticalcavity Gain switching is another way to obtain is changed. short, high-peak-power pulses from a laser. Q switching relies on the fact that the The idea is to rapidly increase the pump lifetime of the population inversion is power so that the population inversion of much longer than the output pulse. The the laser is well in excess of the threshold gain medium is therefore able to store value by the time the first output spike energy that can be quickly released in the develops. The optical pulse then drives the form of a short output pulse. population inversion below its threshold The length of the Q-switched output value. The pump power is subsequently pulse is dependent on several factors. reduced so that the inversion remains In order to obtain a rapid buildup of below threshold and only a single pulse the optical pulse, it is desirable to have is obtained. a large gain cross-section and a large The factors that are important for short population inversion when the Q of the gain-switched pulses are the same as for cavity is switched. The rapid buildup of short Q-switched pulses. Laser Physics 1053

6.4 carrier frequency ω0 whose amplitude Cavity-dumped Operation A(t) is time-dependent. The intensity of this wave is given by A2(t), which consists Cavity dumping allows the energy in a laser of a train of pulses whose peak intensity is cavity to be output in a time comparable to 2( ) = ( + )2 2 τ = A 0 2n 1 E0,pulsewidthis p the cavity round-trip time. The concept is 2π/(2n + 1)ω, and separation between to rapidly (within a cavity round-trip time) pulses is τp = 2π/ω.Sincethetotal introduce a large output coupling (nearly oscillating bandwidth νosc is given by 100%) into a cavity that previously had (2n + 1)ω/2π, the pulse width can also no output coupling. Methods for cavity by written as τp = 1/νosc.Usingthe dumping include the use of an electro- relationship ω = πc/l,wherel is the optic Pockels cell and polarizing beam length of the laser cavity, the time between splitter, and acousto-optic devices. pulses is τp = 2l/c. In words, a mode- locked laser produces a train of output 6.5 pulses whose pulse width is given by the Mode-locked Operation inverse of the oscillating bandwidth and whose separation between pulses is equal Mode locking refers to the situation when to the round-trip time of the laser cavity. the phases of several cavity modes are Figure 11 shows A2(t) for n = 5. fixed (or locked) with respect to each other In the above example, the phases of all of such that the electric fields add coherently the modes were locked so that the output and constructively for a short period of pulse had its minimum possible duration. time. This allows the generation of a Such a pulse is referred to as a transform- train of high-intensity, ultrashort pulses. limited pulse, since its temporal profile To understand this, consider the case of is the Fourier transform of its spectral 2n + 1 equally spaced longitudinal modes profile. This need not be the case – it is oscillating with the same amplitude E . 0 possible to obtain longer pulses, but not Assume that the phases φ of the modes m shorter pulses. Also, it is worth noting that, are locked according to φ − φ − = φ, m m 1 unlike the other pulsed schemes described where φ is a constant. The total electric in this section, a mode-locked laser is a field is the sum of all of these modes: n E(t) = E0 exp[i(ω0 − mω)t + mφ] t = p 2l/c m=−n  = A(t ) exp(iω0t), (16) where ω0 is the frequency of the center mode, ω is the frequency difference  = + between two adjacent modes, t t A2(t ′) ∆t φ/ω,and p   sin[(2n + 1)ωt /2] A(t ) = E0 .(17) sin[ωt/2]

Equation (16) shows that E(t) can be Fig. 11 Train of mode-locked pulses, made up represented in terms of a wave with a of 11 modes of equal intensity 1054 Laser Physics

cw device and there is phase coherence Modulation of the refractive index at some between pulses. point in the cavity has the same effect. The above discussion tells us what mode locking is, but sheds little light on why or 6.5.2 Passive Mode Locking how it occurs. In general, mode locking Passive mode locking can occur when a will occur if the net gain for a mode-locked laser cavity contains a nonlinear optical train of pulses is greater than the net gain element, such as a saturable absorber. of any other combination of cavity modes. In this case, the more intense the light incident on the saturable absorber, the 6.5.1 Active Mode Locking less the total absorption. The total loss Mode locking can be obtained actively of the cavity is therefore minimized by or passively. Active mode locking can be putting all the energy into short pulses. broken into two categories, ‘‘AM mode This is essentially self-induced AM mode locking’’ (produced using an amplitude locking. A similar effect is obtained by modulator within the laser) and ‘‘FM mode putting a Kerr lens and an aperture within locking’’ (produced using a frequency or the cavity. Other techniques include the phase modulator within the laser). In AM use of interferometric elements containing mode locking, the loss of some element in nonlinear media. the laser cavity is modulated at the round- Passive mode locking must be initiated trip cavity frequency. Light circulating in bythepresenceofapulsewithinthe the cavity will see less loss, and therefore cavity. If the optical intensity within the more net gain, when it is incident at the cavity is uniform in time, there is no loss element during the time of minimum loss or gain element that is modulated loss. This encourages short-pulsed opera- at the round-trip cavity frequency in order tion and mode locking can be induced. The to induce mode locking. Noise, however, is same result occurs if the gain of the cavity capable of introducing a small amplitude is modulated. Gain modulation through modulation on the optical field. In some modulation of the pump source is known lasers, this small modulation is sufficient as synchronous pumping. to start the mode-locking process. Such In FM mode locking, the optical length lasers are referred to as self-starting. In of the laser cavity (length or refractive other systems, a pulse (or AM modulation) index) is modulated at the round-trip cavity must be intentionally introduced into the frequency. For simplicity, let us consider cavity to start the mode-locking process. the case where one of the mirrors is Once started, however, mode locking can moved sinusoidally along the direction persist for a long time. of the cavity axis. Light incident on the The introduction of an appropriate mirror during its motion will be Doppler nonlinear optical element into a laser shifted. As a result, it will not reproduce cavity is not sufficient to guarantee mode itself after one round trip, and will not locking. In order for passive mode locking produce a coherent oscillating mode. Light to work, the relative phases of all of the incident on the mirror at its turning points longitudinal modes must remain constant. (maximum or minimum cavity length) Oneeffectthatcandestroythephase sees a stationary mirror and will not relationship, and therefore prevent mode experience a Doppler shift. The net result is locking, is dispersion. Passively mode- that mode-locked pulses will tend to form. locked lasers must be dispersion free if Laser Physics 1055 mode locking is to occur. To accomplish cavity mode, by changing the optical this, prisms are often introduced into the length of the cavity. Since the cavity laser cavity to compensate the dispersion length can be changed continuously, of other intracavity elements, such as the this leads to continuous tuning. This gain medium. In a properly compensated type of tuning is often limited by the cavity, non-mode-locked operation can free spectral range of the cavity. Once be unstable. the cavity modes are shifted by a full The other effect that can destroy the free spectral range, an adjacent cavity phase relationship between the spectral mode is positioned at the frequency components in a mode-locked pulse train where the initial mode started. For the is spontaneous emission, or noise. The same reasons that the initial mode was phaseofthenoiseisunrelatedtothephase originally favored, the adjacent mode is of the oscillating mode. The net phase will now favored, and the laser will have be shifted when the two are combined. a tendency to mode-hop back to the original frequency.

7 7.2 Control of Laser Output Amplitude Modulation 7.1 Frequency Tuning The output power of a laser can be controlled by changing the pump power, Frequency tuning of a laser can occur the output coupling, or the intracavity in one of two ways. If the longitudinal loss. This type of amplitude modulation mode spacing of the laser cavity is much is usually limited to frequencies below less than the gain bandwidth, the cavity the frequency of the relaxation oscillations. is capable of supporting several modes, The relaxation frequency characterizes the each at a different frequency. A single response time of the cavity. Near the frequency is then selected through the relaxation frequency there is resonant en- insertion of an element into the cavity such hancement of the modulation response; that the cavity sees a frequency-dependent above the relaxation frequency the re- loss, as discussed in Sect. 5.6.3. In most sponse rolls off. of the examples listed in that section, Methods used for direct amplitude mod- a small repositioning of the frequency- ulation of a laser may have the side effect selective element would result in a new of introducing frequency modulation as longitudinal mode (and hence a new well. For example, changing the pump operating frequency) being selected. The power affects the thermal load on the gain frequency-selective element is used to medium, and therefore the temperature. select one of the several cavity modes, This, in turn, affects the refractive in- and discrete tuning is obtained. Fast dex, changing the optical length of the tuning can be obtained through the cavity and the oscillating frequency. For use of electro-optic frequency-dependent amplitude-modulation applications where components. frequency stability is critical, it is often bet- The other way a laser can be tuned ter to modulate the laser power external to is to change the frequency of a given the cavity. 1056 Laser Physics

8 to the amplifier material. Instead, a dis- Oscillator–Amplifier Systems persive element can be used to chirp and stretch the pulse in time. This low- As the required output power of the laser ers the peak power so that the pulse can increases, the need for amplification stages be passed through a broadband amplifier, becomes more apparent. Master oscilla- such as Ti:Al2O3,andlaterthroughan- tors (lasers) that operate at low powers other dispersive element that un-chirps can be easily controlled to produce a and compresses the amplified pulse. desired output. Successive amplification stages can be designed to provide increas- ingly higher output powers by increasing 9 the pumping power and the amplifier Issues in Laser Design aperture. Limiting the gain of each stage can eliminate parasitic oscillation. Isola- The design of a laser is dependent on many tors are used to prevent feedback from interdependent factors, including the re- subsequent amplification stages. The de- quirements placed on the output beam sign of each stage can also be optimized (wavelength, spectral purity, tunability, di- for heat removal and low optical distortion. vergence, polarization, power, and power Beam cleanup between stages can be used stability), the operating environment (tem- to suppress unwanted spatial frequencies perature, humidity, vibration, acceleration, and, thereby, maintain single-transverse- and externally applied forces), and practi- mode operation. Amplifiers can also be cal considerations (size, cost, and available paralleled by splitting the master oscillator power). There is an increasingly large beam; if a coherent output is desired, the number of gain media, cavity designs, and outputsoftheamplifiersmustbephased. pump configurations that have been em- There are two basic types of ampli- ployed in lasers, and several texts have fiers, regenerative and traveling wave. A been written on the subject of laser de- regenerative amplifier provides a feedback sign. No one design is well suited for all loop and can oscillate without the laser applications; every laser is optimized for (master oscillator) input. Control of the operation at one point in the multidimen- regenerative amplifier is achieved by injec- sional parameter space outlined above. tion seeding it with the master oscillator A very important issue in the design signal. In a traveling-wave configuration, of many lasers is the extraction of heat the amplifier simply boosts the signal from the gain medium. In the process injected by the master oscillator. Regener- of pumping the gain medium, heat is ative amplifiers are, by nature, multi-pass. generated. As the temperature of the gain Traveling-wave amplifiers can be designed medium changes, so too does its physical so that the radiation travels through the length and refractive index. Each of these amplifying medium more than once with- contributes to a change in the optical out retracing its path. length and resonant frequencies of the Laboratory-scale oscillator–amplifier laser cavity. Nonuniform heating results systems can be designed to produce in thermal lensing and internal stress. petawatt (1015 W) pulses. Very short (fem- Thermal lensing changes the confocal tosecond) pulses cannot be directly ampli- parameters of the laser cavity and can fied to very high powers without damage destabilize an otherwise stable cavity Laser Physics 1057

(or vice versa). Internal stress leads Finally, it should be mentioned that laser to stress birefringence and, eventually, oscillators and amplifiers are nonlinear op- stress fracture. tical devices in that there is a reduction in Other issues that must be considered optical gain caused by partial depletion of in high-power lasers are nonlinear optical the population inversion by the laser radi- effects and optical damage. The electri- ation. This partial depletion, or saturation, cal field within the optical beam of a stabilizes the laser output amplitude in cw high-power laser can be large enough to lasers and plays a vital role in determin- damage optical components. This is par- ing the operating characteristics of pulsed ticularly important in high-power pulsed lasers,aswehaveseeninSect.6. lasers. At optical intensities below the op- The most frequently used frequency- tical damage level, deleterious nonlinear conversion techniques are second-harmo- optical interactions can still degrade the nic and sum-frequency generation. Typi- performance of the laser, and even destroy cally, the output of infrared lasers in the the device. One example is stimulated Bril- 1-µm region, such as Nd:YAG or Nd:glass, louin (acoustic wave) scattering in fiber is frequency doubled into the green. De- lasers. In this case, nonlinear interactions pending on the application, shorter wave- create acoustic waves that can blow off the lengths may be obtained (for example, by ends of the fiber. summing the green radiation with the in- frared). There are a variety of uses for this short-wavelength output, including 10 pumping short-wavelength lasers and op- Frequency Conversion and Nonlinear tical parametric oscillators (OPOs). For Control of Laser Radiation second-harmonic generation to the green, conversion efficiencies as high as 75% Nonlinear optical techniques can be used have been reported and average powers to extend the frequency coverage of lasers of greater than 50 W have been obtained. as well as to modify other characteristics MaterialssuchasKTiOPO4 (KTP), periodi- of laser radiation. Frequency conversion cally poled LiNbO3 (PPLN), LiB3O5 (LBO), is a very important adjunct, converting and β-BaB2O4 (BBO) are used. the output of practical lasers to regions Difference-frequency generation and where primary laser sources may not ex- OPOs have been used as tunable sources ist or may not be very practical. Harmonic of radiation for various spectroscopic generation, frequency mixing, optical para- applications, primarily in the infrared. metric oscillation, and stimulated Raman Continuous-output difference-frequency scattering have been used for frequency generation, while producing very low aver- conversion. Nonlinear processes have also age power with low efficiency, has been been used to produce mode locking (see used for ultrahigh-resolution molecular Sect. 6.5.2), to improve transverse beam spectroscopy in the mid-infrared. OPOs quality (i.e. to produce output closer to the are useful sources of tunable pulsed output diffraction limit), and to dampen relaxation with high peak power and high efficiency. oscillations in pulsed lasers. The article Stimulated Raman scattering has been NONLINEAR OPTICS describes the nonlinear used to generate large pulse energies with processes in detail; here, we will only dis- essentially unity quantum efficiency at a cuss their general significance for lasers. variety of wavelengths. However, because 1058 Laser Physics

energy is deposited in the Raman process, back and forth between the population in- moving Raman media are often required version and the radiation field. Conversion to obtain high-average-power conversion. to the second harmonic can clip the high For high energies and powers, gases such initial peaks in the laser power, an action as H2 have been used. With a very that reduces the amount of depletion in the high finesse cavity, a diode-laser-pumped population inversion, thereby damping the hydrogen Raman laser has been operated oscillations. continuously. Fiber Raman lasers have produced tens of watts of continuous power, while fiber Raman amplifiers have Glossary been used to increase the useful extent of the fiber-communications spectrum. Active Medium: Material that amplifies The optical Kerr effect, which is the radiation. change in refractive index proportional to the optical intensity, can cause short Active Mode Locking: Mode locking using pulses, typically 0.1 to 10 ps in dura- a modulator. tion, to acquire a frequency sweep, or chirp. By sending the chirped pulse AM Mode Locking: Active mode locking through a frequency-dispersive delay line, using amplitude modulation. which might be a grating or prism pair, even shorter pulses, typically 5 to 50 fs, Amplitude Modulation: Temporal control can result. of the amplitude of an electromagnetic Nonlinear phase conjugation can reverse field. the phase distortion acquired from the active medium in a laser (for example, an Axial Mode: Radiation pattern satisfying optically imperfect laser crystal or a laser a resonance condition defined by the material that has large thermal gradients number of waves along the principal due to high-average-power operation). On axis of a cavity; same as longitudi- a second pass through the distorting nal mode. medium, the reversed phase distortion is Bandwidth: Width, in frequency or wave- canceled. Most often, backward stimulated length, of a radiative transition or the Brillouin scattering is used to reverse the output of a laser; same as linewidth. phase; the small acoustic frequency shift produces an optical wave that remains Bandwidth Limited: Having as short a within the gain bandwidth of most lasers duration as allowed by the bandwidth of and has linear propagation characteristics the radiation. that match the input to the Brillouin cell. Other techniques used take advantage of Brightness: Measure of the spatial or the photorefractive effect or absorptively spectral quality of a laser beam, see spatial induced nonlinearities. brightness and spectral brightness. Harmonic generation has been used to moderate the amplitude of the re- Cavity: Region of space where electro- laxation oscillations that often occur in magnetic radiation is confined, usually long-pulse lasers. Relaxation oscillations consisting of two or more reflecting (see Sect. 6.1.1) involve the flow of energy surfaces. Laser Physics 1059

Cavity Dumping: Using an active optical Frequency Conversion: Nonlinear conver- element to allow radiation to rapidly leave sion of the frequency of laser radiation, acavity. includes harmonic generation, sum and difference frequency generation, paramet- Cavity Mode: Radiation pattern satisfying ric generation, and stimulated scattering. the boundary conditions of a cavity. Frequency Tuning: Changing the fre- Q π Cavity : 2 times the damping time of quency of a laser. the cavity divided by the period of the wave. Gain Cross Section: Parameter of an active medium that, when multiplied by the Coherent: Having a definite phase rela- population inversion density, gives the tionship between any two points in the gain per unit length. radiation field.

CW Operation: Continuous-wave opera- Gain Peak: Frequency or wavelength tion; steady-state operation. where the gain of a laser transition is greatest. Diffraction Limited: Having the minimum divergence allowed by diffraction for a Gain Switching: Increasing the gain of given focal spot size. a laser cavity sufficiently fast that the population inversion is temporarily out of Diode Laser: Semiconductor junction la- equilibrium with the optical field in the ser, where population inversion is created cavity resulting in pulsed laser output. by injection of electrons and holes into the junction region; same as injection Harmonic Generation: Nonlinear genera- laser. tion of radiation at a multiple of the input frequency. Energy Gap: Energy range where there are no allowed states in a pure material. Homogeneously Broadened: Having a ra- diative transition where all contributors to Fabry-Perot:´ Cavity consisting of two par- the transition have the same environment allel (but not necessarily flat) reflect- and characteristics. ing surfaces. Inhomogeneously Broadened: Having a ra- Feedback: Transfer of some of a system’s diative transition where all contributors optical output back to its input. to the transition do not have the same environment or characteristics. FM Mode Locking: Active mode locking using frequency modulation. Injection Current: Current that injects elec- trons and holes into the junction region of Franck-Condon Principle: Transitions be- a semiconductor diode. tween electronic states occur without changing nuclear coordinates. Injection Laser: Semiconductor junction laser, where population inversion is cre- Free-running Mode: Laser operation with- ated by injection of electrons and holes into out active control. the junction region; same as diode laser. 1060 Laser Physics

Junction: Interface region between n-type Output: Optical radiation from a laser. and p-type semiconductor materials. Parasitic Oscillation: Laser oscillation aris- Laser Dynamics: Temporal behavior of ing from spurious reflections in a laser a laser. oscillator or amplifier.

Laser Transition: Change of material quan- Passive Mode Locking: Mode locking us- tum state that occurs during the laser ing a nonlinear element in the laser cavity emission process. such as a saturable absorber or a Kerr lens.

Linewidth: Width, in frequency or wave- Phase Conjugation: Reversal of the phase length, of a radiative transition or the of an electromagnetic field. output of a laser; same as bandwidth. p-n Junction: Interface region between n- Longitudinal Mode: Radiation pattern sat- type and p-type semiconductor materials. isfying a resonance condition defined by the number of waves along the principal Population Inversion: State of a mate- axis of a cavity; same as axial mode. rial in which a level with an energy greater than another level also has a Long-pulse Operation: Operation of a larger occupation probability (popula- pulsed laser with a pulse long enough tion). for the population inversion in the active medium to reach equilibrium with the p-type: Semiconductor material where optical field in the cavity. valence-band holes are the primary current carriers. Mode Locking: Forcing a phase relation between the axial (longitudinal) modes of Pulsed Operation: Laser operation where a laser. the radiation is produced in pulses.

Multimode Operation: Laser operation Pumping: Producing a population inver- with multiple cavity modes oscillating at sion in an active medium. thesametime. Q Switching: Reducing the loss in a laser Nonlinear Optics: Optical processes that cavity sufficiently fast that the population have a nonlinear dependence on the inversion is temporarily out of equilibrium amplitude of the optical field. with the optical field in the cavity resulting in pulsed laser output. n-type: Semiconductor material where conduction-band electrons are the primary Quantum Well: Semiconductor structure current carriers. that spatially confines carriers and pro- duces a new energy spectrum. Optical Kerr Effect: Quadratic electro- optical effect induced by an optical field. Radiative Transition: Transition that can produce electromagnetic radiation. Optical Pumping: Producing a population inversion by pumping an active medium Rate Equations: Equations characterizing with an optical field. the number of photons in a laser cavity Laser Physics 1061 and the populations of the various energy saturation of a transition in an inhomo- levels of the active material system. geneously broadened material.

Regenerative Amplifier: Amplifier in which Spontaneous Emission: Emission that oc- radiation is fed back through the active curs absent excitation by a radiation field. medium so that an input signal is not required for significant extraction of op- Spontaneous Lifetime: Exponential time tical power, also called an oscillator or constant associated with spontaneous ra- laser oscillator. diative decay.

Relaxation Oscillation: Oscillatory condi- Stable Resonator: Optical cavity with mir- tion that occurs when energy flows back rors configured to generate modes that and forth between an active material and converge on the cavity axis. a radiation field as a system approaches steady state. Stimulated Emission: Emission induced by a radiation field. Saturation: Reduction of laser gain (satur- able-absorber loss) that occurs when Stimulated Scattering: Scattering induced the difference between upper and lower by an output (scattered) field. laser-level (saturable-absorber-level) popu- lations approaches zero. Stokes Shift: Shift of output radiation to- ward longer wavelength (lower photon Single-mode Operation: Laser operation energy), the energy difference is trans- with only a single cavity mode oscillating; ferred to the material system responsible same as single-frequency operation. for the wavelength shift.

Single-frequency Operation: Laser opera- Stored Energy: Energy stored in an active tion with only a single cavity mode os- medium that can be extracted optically. cillating; same as single-mode operation. Threshold: Onset of laser operation that Slope Efficiency: Change in output power occurs when the pump power is sufficient of a laser per change in pump power. to achieve an amplifier gain equal to the cavity loss. Spatial Brightness: Power per beam area at the focal point per beam solid Transition: Change in the quantum state angle. of a material system.

Spatial Hole Burning: Spatial variation in Transition Frequency: Energy difference the saturated inversion density in a between the two quantum states involved standing-wave laser cavity. in a transition divided by Planck’s con- stant. Spectral Brightness: Power per unit fre- quency. Transition Strength: Magnitude of the in- teraction between the radiation field and Spectral Hole Burning: Variation of pop- the material system, proportional to the ulation inversion in frequency due to square of the transition dipole. 1062 Laser Physics

Transverse Mode: Radiation pattern trans- Detailed theoretical discussion of lasers is verse to the principal axis of a cavity. contained in:

Traveling-wave Amplifier: Amplifier in Haken, H. (1986), Laser Theory,(reprinted.),New which radiation travels along a non- York: Springer. Sargent, M. III, Scully, M. O., Lamb, W. E. Jr. repeating path. (1974), Laser Physics. Reading, MA: Addison- Wesley. Tunable Laser: Laser whose output wave- length can be varied. The underlying physics of laser resonators and its application to the design of many types of Unstable Resonator: Optical cavity with lasers in use today is the topic of: mirrors configured to generate modes that do not converge on the cavity axis. Hall, D. R., Jackson, P. E. (Eds.) (1989), Physics and Technology of Laser Resonators.Bristol,UK: Adam Hilgar. Further Reading

Specific types of lasers are discussed in: Popular discussions of lasers and their applica- tions and history are given in: Brown, D. C. (1981), High-Peak-Power Nd: Glass Laser Systems.NewYork:Springer. Hecht, J. (1992), Laser Pioneers,(reviseded.), Casey, H. C. Jr, Panish, M. B. (1978), Heterostruc- Boston, MA: Academic Press. ture Lasers, Parts A and B, New York: Academic Hecht, J., Teresi, D. (1998), Laser: Light of a Press. Million Uses. Mineola, New York: Dover Coldren, L., Corzine, S. W. (1995), Diode Lasers Publications. and Photonic Integrated Circuits.NewYork: Wiley. Duarte, F. J. (Ed.) (1995), Tunable Lasers Hand- A general discussion of laser science and tech- book. San Diego, CA: Academic Press. nology, accessible to anyone with a background Duarte, F. J., Hillman, L. W. (Eds.) (1990), Dye of electromagnetic theory, basic quantum me- Laser Principles. Boston, MA: Academic Press. chanics, and calculus, can be found in any of Garrett, C. G. B. (1967), Gas Lasers.NewYork: the following: McGraw-Hill. Kaminskii, A. A. (1990), Laser Crystals,(2nded.), New York: Springer. Eastham, D. A. (1986), Atomic Physics of Lasers. Kapon, E. (Ed.) (1999), Semiconductor Lasers London: Taylor & Francis. I: Fundamentals.SanDiego,CA:Academic Kaminow, I. P., Siegman, A. E. (1973), Laser Press. Devices and Applications. New York: IEEE. Koechner, W. (1999), Solid-State Laser Engineer- Shimoda, K. (1991), Introduction to Laser Physics, ing, (5th ed.), New York: Springer. (2nd ed.), New York: Springer. Koechner, W., Bass, M., Roth, H. (2003), Solid Siegman, A. E. (1986), Lasers. Mill Valley, CA: State Lasers: A Graduate Text.NewYork: University Science. Springer. Silfvast, W. T. (2004), Laser Fundamentals,(2nd Mollenauer, L. F., White, J. C., Pollock, C. R. ed.), Cambridge: Cambridge University Press. (Eds.) (1992), Tunable Lasers,(2nded.),New Svelto, O. (1998), Principles of Lasers,(4thed.), York: Springer. New York: Plenum. Schafer,¨ F. P. (Ed.) (1990), Dye Lasers,(3rded.), Verdeyen, J. T. (1995), Laser Electronics,(3rded.), New York: Springer. Englewood Cliffs, NJ: Prentice Hall. Willett, C. S. (1974), Introduction to Gas Lasers: Yariv, A. (1989), Quantum Electronics,(3rded.), Population Inversion Mechanisms. Oxford: Perg- New York: Wiley. amon Press. Laser Physics 1063

A concise source of data in tabular and graphical (1987), Handbook of Laser Science and Technol- form is provided for workers in the areas of laser ogy, Vol. V; Weber, M. J. (Ed.) (1991), Hand- research and development in: book of Laser Science and Technology, Suppl. 1; Weber, M. J. (Ed.) (1995), Handbook of Laser Weber, M. J. (Ed.) (1982), Handbook of Laser Sci- Science and Technology, Suppl. 2, Boca Raton, ence and Technology, Vols. I and II; Weber, M. J. FL: CRC Press. (Ed.) (1986), Handbook of Laser Science and Weber, M. J. (Ed.) (2003), Handbook of Lasers. Technology, Vols. III and IV; Weber, M. J. (Ed.) Boca Raton, FL: CRC Press.