1 Laser Basics

1.1 Stimulated Emission

A cross section σ21 for absorption of radiation by a lower state 1 engenders a balancing cross section σ12 for emission stimulated by radiation interacting with an upper state 2. Detailed balance relates these two cross sections according to

$$\displaystyle g_{2}\sigma_{12}=g_{1}\sigma_{21}\;,$$
(44.1)

where g1 and g2 are the statistical degeneracies of their respective states 1 .

For a collection of emitting and absorbing states with densities n2 and n1, amplification may occur when n2σ12 > n1σ21, which leads to a requirement for an inversion of the state populations

$$\displaystyle n_{2}/n_{1}> g_{2}/g_{1}\;.$$
(44.2)

The rate of spontaneous emission at frequency ν can be modeled itself by stimulated emission induced by a noise source of the magnitude of the density of states ρ(ν)

$$\displaystyle\gamma_{12}(\nu)=\sigma_{12}(\nu)\rho(\nu)/c=\sigma_{12}(\nu)8\pi\lambda^{-2}\;.$$
(44.3)

1.2 Laser Configurations

A practical laser combines a population inversion with a means for controlling the radiation.

The basic laser source is the laser oscillator, an amplifier possessing positive feedback. The usual form is simply a piece of active gain medium placed inside a resonant optical cavity (Fig. 44.1). Tunability is produced if the resonant cavity is frequency selective and adjustable (Fig. 44.2). Many laser sources use an amplifier after the oscillator.

Fig. 44.1
figure 1figure 1

Simple laser oscillator and beam parameters. Distances z are generally measured from the minimum beam waist w0. The beam appears with a far-field divergence angle θ0

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Fig. 44.2
figure 2figure 2

Tunable laser oscillator geometries. a Fabry–Perot: tuning is usually done by changing the cavity length, although changing the index of refraction by changing the temperature or current is common with solid state laser diodes. b Littrow prism line selector: typical of atomic ion lasers capable of multiple line output. c Littrow grating tuning: common in pulsed dye lasers with high gain (> 10) per pass. Telescope increases resolution by filling, and reducing angular divergence at the grating. d Grazing incidence, mirror tuned, grating mount

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1.3 Gain

The fundamental gain per pass is given by

$$\displaystyle G=J/J_{0}=\exp[(\kappa-\mu)L]\;,$$
(44.4)

where J ∕ J0 is the ratio of light output to input, κ is the gain coefficient, μ is the nonradiative loss rate, and L is the path length. The gain coefficient

$$\displaystyle\kappa=n^{*}\sigma_{12}$$
(44.5)

depends upon the net inversion n*,

$$\displaystyle n^{*}=n_{2}-(g_{1}/g_{2})n_{1}\;.$$
(44.6)

If λ is the wavelength of radiation, F12 is the emission line shape function normalized over frequency ν, τ2 is the lifetime of the transition, and f12 is the branching ratio for the upper state to undergo this transition, then the stimulated emission cross section is

$$\displaystyle\sigma_{12}=\frac{\lambda^{2}f_{12}F_{12}(\nu)}{8\pi\tau_{2}}\;.$$
(44.7)

For a Lorentzian lifetime-broadened line, the cross section for stimulated emission at the line center becomes

$$\displaystyle\sigma_{12}=\frac{\lambda^{2}f_{12}}{4\pi^{2}\Gamma_{12}\tau_{2}}\;,$$
(44.8)

where Γ12 is the full width at half maximum of the line.

1.4 Laser Light

Lasers are inherently bright sources of radiation: the radiation field within a practical laser must be high enough for stimulated emission to compete with spontaneous emission. The effective source of spontaneous fluctuations approximates that of the density of states. In terms of the beam parameters photon flux J per solid angle Ω, and frequency ν, this is

$$\displaystyle\frac{\,\mathrm{d}^{2}J}{\mathrm{d}\Omega\mathrm{d}\nu}=\frac{2\nu^{2}}{\epsilon_{\mathrm{r}}c^{2}}\;.$$
(44.9)

Beam quality is given by the product of the angular divergence times the beam width. Highest beam quality is associated with diffraction limited light emitted from a Gaussian spot. For circular laser beams traveling in the z-direction, this corresponds to a solution of the electromagnetic wave equation

$$\displaystyle u(r,\phi,z)=\psi(r,z)\exp(-\mathrm{i}kz)\;,$$
(44.10)

where u is a polarization component of the field. For high values of k = 2π ∕ λ, corresponding to short wavelength, the adiabatic radial solution is also Gaussian

$$\displaystyle\psi(r,z)=\exp\left\{-\mathrm{i}\left[P+kr^{2}/(2q)\right]\right\}\;,$$
(44.11)

where the complex phase shift P, beam parameter q, and beam radius w are functions of z

$$\displaystyle\begin{aligned}\displaystyle P(z)&\displaystyle=-\mathrm{i}\ln\left[1-\mathrm{i}\frac{\lambda z}{\pi w_{0}^{2}}\right]\\ \displaystyle&\displaystyle=-\mathrm{i}\ln\sqrt{1+\left[\frac{\lambda z}{\pi w_{0}^{2}}\right]^{2}}-\tan^{-1}\left[\frac{\lambda z}{\pi w_{0}^{2}}\right]\;,\end{aligned}$$
(44.12)
$$\displaystyle q(z)=\mathrm{i}\pi w_{0}^{2}/\lambda+z\;,$$
(44.13)
$$\displaystyle w^{2}(z)=w_{0}^{2}\left[1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}\right]\;.$$
(44.14)

Here, w0 is the beam waist parameter, the minimum width of the Gaussian beam at a focused spot. For Gaussian beams, the product of the minimum beam waist and beam divergence angle θ0 is given by

$$\displaystyle\theta_{0}w_{0}=\lambda/\pi\;.$$
(44.15)

The beam waist and divergence follow optical imaging according to paraxial ray theory.

Higher order circular modes with p radial nodes and l angular node planes are specified by multiplying Eqs. (44.11) and (44.12) by angular and radial factors to obtain

$$\begin{aligned}\psi_{pl}(r,\phi,z) & =\left(\sqrt{2}r/w\right)^{l}L_{p}^{l}\left(2r^{2}/w^{2}\right)\,\mathrm{e}^{\mathrm{i}l\phi}\psi(r,z)\;,\end{aligned}$$
(44.16)
$$\begin{aligned}P_{pl}(z) & =(2p+l+1)P(z)\;.\end{aligned}$$
(44.17)

Here, the functions L lp (x) are the Laguerre polynomials as defined in Sect. 9.4.2. The radial phase shifts produce a wave front curvature of effective radius

$$\displaystyle R=z+\left[(2p+l+1)/2+w_{0}^{2}\pi/\lambda^{2}\right]w_{0}^{2}\pi/z\;.$$
(44.18)

Modes with the same values of 2p + l have identical axial and radial phase shifts. The two polarization components of the electromagnetic field double the degeneracies of all modes considered here. Often these degeneracies are split in practice by optical inhomogeneities of the medium through which they pass. More details can be found in the summary of Kogelnick and Li 2 , or in the texts by Verdeyen 3 or Svelto 4 .

Some applications require knowledge of the electric field in additionto the flux density J. For purely sinusoidal single mode beams, the rms field is

$$\displaystyle\langle E\rangle=\left(\frac{h\nu J}{c\epsilon_{0}}\right)^{1/2}\;.$$
(44.19)

Nonlinear effects are often expressed in terms of powers of the field by

$$\displaystyle\langle E^{n}\rangle=2^{(n-1)/2}\langle E\rangle^{n}$$
(44.20)

for single mode and multimode radiation of random frequency spacings. For m equally spaced modes, this is increased by m! ∕ (m − n)!.

2 Laser Designs

2.1 Cavities

The simple Fabry–Perot cavity consists of two spherical mirrors facing one another. The surfaces are chosen to be constant phase surfaces for the desired modes in Eq. (44.18). Stability criteria are shown in Fig. 44.3. A cavity is stable when initial angles θ and displacements r of paraxial rays transform during a round trip into θ′ and r′ satisfying

$$\displaystyle-2<\frac{\partial\theta^{\prime}}{\partial\theta}+\frac{\partial r^{\prime}}{\partial r}<2\;.$$
(44.21)

At frequencies for which the round-trip phase change per passage

$$\displaystyle\begin{aligned}\displaystyle\delta(\nu)&\displaystyle=2\pi(z_{2}-z_{1})\nu/c+2\left[P_{pl}(z_{2})-P_{pl}(z_{1})\right]\\ \displaystyle&\displaystyle\approx 2\pi(z_{2}-z_{1})/\lambda+\pi(2p+l+1)\end{aligned}$$
(44.22)

is an integer multiple of 2π, the phases from different passages interfere constructively, giving longitudinal modes. (Here Ppl(zi) gives the additional phase shift for higher transverse modes at mirrors i = 1,2.)

Fig. 44.3
figure 3figure 3

Stability parameters for simple two-mirror laser cavities of length L and mirror radii of curvature R1 and R2. Here, \(x=\frac{1}{2}(L/R_{1}-L/R_{2})\) is the mean curvature difference of the two mirrors; \(y=\frac{1}{2}(L/R_{1}+L/R_{2})\) is the mean curvature of the two mirrors. Cavities with parameters in the unshaded region are stable

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For a particular radial mode structure in an empty Fabry–Perot cavity, the ratio of the maximum cavity decay time for these standing waves to the minimum cavity decay time for frequencies between longitudinal modes is

$$\displaystyle(1+r)^{2}/(1-r)^{2}\;.$$
(44.23)

Here r is the reflectivity of the end mirrors; for cavities with mirrors having different reflectivities, one may use the square root of the product of their reflectivities. A simple Fabry–Perot cavity may be tuned by changing the cavity length or by changing the index of refraction of the cavity material. Since both of these may be properties of temperature, temperature tuning may be possible. The index of refraction of a material may also be sensitive to the intensity of excitation. Diode lasers consisting of a semiconductor die with polished, reflecting faces are often tuned by changing temperature and pumping current.

For lasers with a dispersive optical element within the cavity, highest selectivity is obtained when the light has low angular divergence and impinges upon the dispersive element as nearly plane waves. A beam expander may reduce the angular spread while simultaneously increasing the beam width. For a grating used as a mirror in a Littrow mount, the dispersion equation is

$$\displaystyle\Updelta\lambda=(d/n)\cos\phi\,\Updelta\phi\;,$$
(44.24)

where n is the diffraction order, d is the distance between lines, and ϕ is the angle of incidence off normal.

Cavities with prisms or gratings can be conveniently tuned by rotating the angle of the dispersive element. For a grating used as a mirror in a Littrow mount, the grating equation is

$$\displaystyle\lambda=(2d/n)\sin\phi\;.$$
(44.25)

Often, more than one longitudinal cavity mode operates within the selected frequency band, and tuning consists of hopping from mode to mode, rather than smoothly sweeping a single line across a band of frequencies. Smooth frequency tuning can be achieved, for example, in a design by Wallenstein and Hänsch 5 , in which the grating and Fabry–Perot cavity are placed together inside a chamber. The whole laser is then tuned by changing the index of refraction of the gas inside by varying its pressure.

The current trend with pulsed lasers is to use a very lossy, short oscillator cavity in which the longitudinal modes are nearly absent, making up for the cavity losses with a very high gain lasing medium. The front, output mirror of such a cavity may actually be antireflection coated, with a reflectivity of only a percent or less.

Low gain, continuous wave (cw) lasers often use a combination of cavity length tuning along with dispersive element rotation, such as a prism or Lyot filter. Often, lasing on a traveling wave in a ring configuration is used to avoid longitudinal modes, as illustrated in Fig. 44.4. Some commonly used gain media are listed in Tables 44.1 and 44.2.

Fig. 44.4
figure 4figure 4

Ring laser. The Faraday rotator and half-wave plate permit circulation of the cavity fields in only one direction

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Tab. 44.1 Fixed frequency lasers
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Tab. 44.2 Approximate tuning ranges for tunable lasers
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2.2 Pumping

Many methods, including electrical discharges and flashlamps, have been used to pump the gain media of lasers. Generally, the best pump is another laser.

Two notable pump lasers have dominated the field of tunable, visible lasers: pulsed neodymium YAG, frequency doubled to ≈ 503 nm, and cw Ar II ion at 514.5 nm. Both are extremely effective at exciting the highly efficient rhodamine class dyes in the red–yellow portion of the visible spectrum. The typical pump beam of a Nd∕YAG laser enters the amplifying dye cell transversely along one on the faces of the cell. The typical cw pump beam enters the dye almost collinearly with the laser axis.

By temporarily spoiling the Q by making the laser cavity lossy, lasing can be held off until the gain medium stores a greater energy density than the minimum required for lasing. Rapidly switching off the loss mechanism releases this energy in one giant pulse. An electronic optical shutter, such as a Pockels cell, or a saturable dye inside the laser cavity, designed to photobleach from the spontaneous emission just before the laser reaches threshold, are commonly used.

Periodically spoiling the laser gain or the cavity Q at the period of a round trip produces intense, short pulses. Viewed from the frequency domain, the phases of individual longitudinal modes are locked together to produce a light packet circulating at the frequency of the reciprocal of the mode spacing. Extremely short pulses (< 1 ps) can be produced by mode locking. Practically, mode locking can be achieved by using a thin, saturable absorber near one of the cavity mirrors 6 , or by acoustically modulating an optical element in the cavity – even an end mirror itself. One of the best ways of mode locking is to pump a short lifetime gain medium, such as a dye, with mode locked laser light, such as from a mode locked argon ion laser 7 .

3 Interaction of Laser Light with Matter

3.1 Linear Absorption

The absorption cross section σfi for transition to the final state | f⟩, when integrated over frequency ν, is given theoretically by the leading first-order perturbation of the initial state | i⟩ in the electric dipole approximation

$$\displaystyle\int\sigma_{fi}(\nu)\,\mathrm{d}\nu=4\pi^{2}\alpha\bar{\nu}\left|\langle f|\sum_{e}\boldsymbol{r}_{e}\boldsymbol{\cdot}\hat{\boldsymbol{\epsilon}}|i\rangle\right|^{2}\;,$$
(44.26)

where the sum is over all charges e at distances re, \(\hat{\boldsymbol{\epsilon}}\) is a unit polarization vector, and \(\bar{\nu}\) is the averaged transition energy. Averaged over orientations and summed over possible final states, each electron contributes to the total integrated absorption cross section one electron oscillator, πr0c ≈ 0.03 cm2 s−1; here, r0 is the classical electron radius. Electronic absorption bands typically contain an oscillator strength f = 0.010.5 of an electron oscillator, while weaker vibrational transitions have \(f=E-6E-4\) in each band.

For narrow lines with radiation of broader band width Δν at flux density J, the linear absorption rate constant can be usefully estimated as (πr0c)fJ ∕ Δν.

3.2 Multiphoton Absorption

Second-order perturbation theory gives the theoretical two-photon contribution to the absorption. An absorption cross section σ(2)J1 for a photon of frequency ν2 is induced by an off-resonance monochromatic field of frequency ν1 and photon flux density J1. When integrated over the frequency of the second photon, the second-order cross section can be related to the dipole matrix elements

$$\displaystyle\begin{aligned}\displaystyle\int\sigma_{fi}^{(2)}(\nu_{1},\nu_{2})J_{1}\,\mathrm{d}\nu_{2}&\displaystyle=\frac{4\pi^{2}\alpha^{2}}{\nu_{1}\bar{\nu}_{2}}J_{1}\\ \displaystyle&\displaystyle\quad\times\left|\sum_{m}\frac{\nu_{fm}\nu_{mi}\langle r_{fm}\rangle\langle r_{mi}\rangle}{\nu_{mi}-\nu_{1}}\right|^{2}\;,\\ \displaystyle\end{aligned}$$
(44.27)

where

$$\displaystyle\begin{aligned}\displaystyle\langle r_{fm}\rangle&\displaystyle=\langle f|\sum_{e}\boldsymbol{r}_{e}\boldsymbol{\cdot}\hat{\boldsymbol{\epsilon}}_{2}|m\rangle\;,\\ \displaystyle\langle r_{mi}\rangle&\displaystyle=\langle m|\sum_{e}\boldsymbol{r}_{e}\boldsymbol{\cdot}\hat{\boldsymbol{\epsilon}}_{1}|i\rangle\;,\end{aligned}$$
(44.28)

and the sum is taken over intermediate states | m⟩ having frequencies νmi and νfm for transitions to the initial and final states. The energy for the overall transition comes from two photons; hence the two-photon resonance condition νfi = ν1 + ν2. A special case often occurs when only one radiation frequency is used: then ν1 = ν2, and two-photon resonance is achieved when the energy of the transition corresponds to twice the frequency of the radiation field.

This integrated, induced cross section is roughly of the order \(\alpha(\pi cr_{0})^{2}J_{1}/(\nu_{1}\bar{\nu}_{2})\). Two-photon absorption becomes comparable to the one-photon absorption with off-resonance fields on the order of

$$\displaystyle J_{1}\approx\frac{\nu_{1}\nu_{2}}{\alpha\pi cr_{0}}\;,$$
(44.29)

or about 1021 photons s−1 cm−2 (1017 W m−2 s−1 for typical green light).

Typical dipole-allowed molecular multiphoton absorption cross sections are ≈ 10−58 m4 s for two photons and 10−94 m6 s2 for three photons.

Multiphoton absorption was one of the first effects explored with lasers. Two-photon absorption was first reported for inorganic crystals containing europium ions 8 . Blue and ultraviolet fluorescence appeared in the interaction of red ruby laser light with organic compounds 9 ; 10 ; 11 ; 12 . Others observed two-photon absorption directly 13 ; 14 . Selection rules for multiphoton absorption are summarized by McClain 15 ; recent work is reviewed by Ashfold and Howe 16 .

Highly excited states may subsequently ionize in the intense fields in a multiphoton ionization (MPI) process. The ionization signal is often detected in a proportional ionization cell. A low pressure cell containing the vapor of a transition metal organometallic, such as iron carbonyl or ferrocene which photodissociates to give the metal atom, may be used for wavelength calibration by MPI.

3.3 Level Shifts

High radiation power causes resonant frequencies to shift, responding to the ac Stark effect 17 ; 18 . Even moderate fields, tuned near resonance, interact strongly with an atom or small molecule, which undergoes rapid excitation and stimulated emission at the characteristic power dependent Rabi frequency,

$$\displaystyle\nu_{\text{Rabi}}=\frac{2\pi g_{1}}{g_{1}+g_{2}}\sigma_{21}J\;.$$
(44.30)

Fluorescence from such an interacting system has a characteristic head and shoulders spectrum best understood as radiation at the fundamental frequency amplitude modulated at the Rabi frequency 19 .

3.4 Hole Burning

Radiation at a particular frequency generally moves population out of states that absorb that radiation. Molecules that interact resonantly with the radiation may also spontaneously emit at different frequencies, thereby ending in nonabsorbing states. This optical pumping effect can make resonance features in an absorption spectrum disappear at high power levels 20 . Depletion may appear as a hole in the absorption spectrum 21 . Recently, interest has shifted to permanent hole burning as a method of information storage in materials.

Hole burning is the basis for Doppler-free Lamb-dip spectroscopy, in which only absorbing atoms or molecules having little or no velocity component along the axis of two counterpropagating beams are temporarily depleted 22 . This technique is commonly used for laser frequency stabilization, such as with the iodine-locked He–Ne laser.

3.5 Nonlinear Optics

3.5.1 Multiplying

Nonlinear susceptibility of an optical medium can generate radiation at frequencies which are multiples of the frequency of laser radiation passing through. Phenomenologically, the second order polarization

$$\displaystyle\boldsymbol{P}_{2\nu}=\epsilon_{0}\chi^{[2]}\boldsymbol{E}_{\nu}^{2}$$
(44.31)

is given in terms of the second order nonlinear susceptibility χ[2], a third rank tensor, and the electric field at the fundamental frequency (see Chap. 76). This nonlinear susceptibility can range from 0.55 pm ∕ V for typical materials used for frequency doubling. Typical materials and their use are reviewed by Bordui and Fejer 23 .

For a nonlinear process occurring over a length l in a cylindrical region with Gaussian waist w0 with polarization P0 on axis, the far field flux is given by

$$\displaystyle\begin{aligned}\displaystyle J(R,\theta)&\displaystyle=\frac{\pi^{4}\nu^{3}n^{3}w_{0}^{4}P_{0}^{2}}{4hc^{3}\epsilon_{0}R^{2}}\left(\frac{\sin^{2}[(k_{0}-k\cos\theta)l/2]}{(k_{0}-k\cos\theta)^{2}}\right)\\ \displaystyle&\displaystyle\quad\times\exp\left[-k^{2}w_{0}^{2}\sin^{2}(\theta)/2\right]\;.\end{aligned}$$
(44.32)

Here, n is the index of refraction and k the propagation constant for the induced radiation, while k0 is that for the induced polarization, the vector sum of those of the original radiation. When phase matched, k0 − kcosθ =  0, and the term in the brackets maximizes to (l ∕ 2)2.

The greatest difficulty is in selecting materials which can be phase matched such that the relative phases of the fundamental and overtone radiation propagate together through the material; otherwise radiation at the higher frequency generated at different places inside the material destructively interfere. Phase matching is usually achieved by either angle tuning of a birefringent crystal, or by temperature tuning.

Only materials without a center of inversion in their crystal structures have a second order nonlinear susceptibility. All materials, including gases, will have a third order nonlinear susceptibility χ[3]. This can be used to generate third harmonics, especially in the vacuum ultraviolet (VUV), where doubling materials are not available 24 ; 25 .

3.5.2 Mixing

The same materials that permit frequency doubling and tripling also allow 3-wave and 4-wave frequency mixing. The frequency matching conditions are, respectively,

$$\displaystyle\begin{aligned}\displaystyle\nu_{1}\pm\nu_{2}&\displaystyle=\nu_{3}\,,\\ \displaystyle\nu_{1}\pm\nu_{2}\pm\nu_{3}&\displaystyle=\nu_{4}\;.\end{aligned}$$
(44.33)

Tunable UV radiation may be generated by adding the frequencies of a fixed and tunable visible outputs. Tunable IR has been obtained by differencing fixed and tunable visible lasers. Mixing of radiation from an Ar ion laser with that from a tuned R6G dye laser in lithium iodate to produce tunable 22004600 nm radiation is noteworthy.

3.5.3 Optical Parametric Oscillator

It is possible to reverse 3-wave mixing, generating two frequencies whose sum is that of the input radiation. In parametric generation, the output frequencies are given by the phase match conditions. Both the desired frequency and the secondary idler frequency must be allowed to build up in the nonlinear medium. The idler radiation is not present initially, but results from the frequency mixing process itself. The process has many of the characteristics of a laser oscillator, including that of a gain threshold. This makes tuning of an optical parametric oscillator similar to that of a laser, but with more degrees of freedom: now oscillation at two frequencies must be attained simultaneously, along with the correct phase matching of the nonlinear material 26 .

3.6 Raman Scattering

It is possible to have one or more of the fields in a mixing process belong to just polarization, rather than radiation. The frequency additive case of multiphoton absorption has already been consisdered; the frequency subtractive case is Raman scattering.

3.6.1 Incoherent Raman Scattering

Radiation at a higher frequency can excite a lower frequency vibration or rotation within a material, with appearance of radiation at the frequency of the incident radiation minus that of the absorption. The integrated cross section for this effect is given by Eq. (44.27). However, in this case the cross section is for emission of the second photon. The rate of spontaneous Raman emission is obtained by multiplying Eq. (44.27) by the spectral flux density of the zero-point field, 8πν2 ∕ ϵrc. Typical vibrational Raman cross sections for transparent molecules are about 10−34 m2 sr−1 in the blue–green (488 nm) 27 .

Alternatively, energy can be extracted from an excited state, with the inelastically scattered photon departing with the sum of the incident frequency and that of the deexcitation. The term anti-Stokes distinguishes it from the more usual Stokes type of Raman scattering.

3.6.2 Coherent Raman Emission

The integrated cross section for emission stimulated at the Raman frequency is given directly by Eq. (44.27).

A fourth-order mixing process that is less susceptible to saturation involves coherent anti-Stokes Raman scattering (CARS). Two beams excite the material: the difference of their frequencies corresponds to an excitation of the material. Stimulated Raman scattering excites the material, which then deexcites through an anti-Stokes process, giving rise to a third, higher frequency radiation field. The phase is determinate, and the radiation leaves the region of scattering as a beam 28 .

If the incident radiation induces a Raman process over a sufficiently long path, the stimulated Raman process can be used for gain at both the Stokes and anti-Stokes frequencies. Since spontaneous Raman processes are proportional to the integrated cross section, while gain in the stimulated Raman process is proportional to the peak cross section, simple materials with sharp, simple line spectra are most suitable for Raman gain media. While Raman lasers have been produced using vibrational excitation of organic liquids, currently the most important technical application is for Raman shifting the output of lasers, tunable or otherwise, to frequencies which otherwise would be inaccessible. The high pressure H2 Raman shifter produces, at low powers, beams consisting almost entirely of well separated, sharp lines shifted by 4160 cm−1 from the pump beam; at high powers, a series of Stokes and anti-Stokes bands appear, each separated by 4160 cm−1 from each other.

4 Recent Developments

One of the most exciting advancements in the past decade in laser physics has been the generation of optical frequency combs; and, more specifically, their applicability in the domain of high-resolution laser spectroscopy. Basically, through a superposition process of many continuous wave modes, a short train of frequency spikes may be produced from a mode-locked laser 29 (see also Sect. 30.1.5). These spikes are equally spaced and are referred to as a frequency comb. The frequency ωn of the n-th cavity mode may be expressed as

$$\displaystyle\omega_{n}=n\omega_{r}+\omega^{\prime}\,,{}$$
(44.34)

where ωr is characteristic of the laser and ω′ is a frequency offset due to the difference between the phase and group velocity of the superposed waves.

The microwave frequencies ωr and ω′ are determined through the use of nonlinear optics. Once these two parameters are determined, any unknown optical frequency ωo may be measured by recording the beat frequency between it and the closest comb frequency. This technique gives experimenters a high-precision method for the spectroscopic determination of such fundamental quantities as the fine structure constant, the Rydberg constant, and the Lamb shift 30 ; 31 .