UNITS AND NOTATION The units and dimensions in this book are almost entirely mks,or SI,except for a few concessions to long-established habits such as expressing atomic densities N in atoms/cm3 and cross sections o in cm2.Such non-mks values should of course always be converted to mks units before plugging them into formulas. In general,lower-case symbols in bold-face type such as E(r,t),b(r,t), h(r,t),and so on refer to electromagnetic field quantities as real vector functions of space and time,while E(r,t),b(r,t),h(r,t),etc.,refer to the scalar counter- parts of the same quantities.Bold-face capital letters E,B,H,etc.,refer to the complex phasor amplitudes of the same vector quantities with et variations, while E,B,H,etc.,are the complex phasor amplitudes of the corresponding scalars.As illustrated here,complex quantities are sometimes,but not always, identified by a superposed tilde. In writing sinusoidal signals and waves,waves propagating toward positive z are written in the "electrical engineer's form"of exp j(wt-Bz)rather than the "physicist's form"of expi(kz-vt).(This of course does not imply thati!) Linewidths△f,△w,△A and pulsewidths△t,TorT,unless specifically noted,. always mean the full width at half maximum(FWHM). In contrast to much of the published literature,an attenuation or gain co- efficient a in this book always refers to an amplitude or voltage growth rate, such as for example E(z)=E(0)exptoz.Signal powers or intensities in this book,therefore,always grow or attenuate with exponential growth coefficients 2a rather than o. The notation in the book has a few other minor idiosyncrasies.First,we are often concerned with signals and waves inside laser crystals,in which the host crystal itself has a dielectric constant e and an index of refraction n even without any atomic transition present.To take the dielectric properties of a possible host medium into account,the symbols e,c and A in formulas in this text always refer to the dielectric permeability,velocity of light and wavelength of the radiation in the dielectric medium if there is one.We then use co and Ao in the few cases where it is necessary to refer to these same quantities specifically in vacuum. The advantage of this choice is that all our formulas involving e,c and A remain correct with or without a dielectric host medium,without needing to clutter these formulas with different powers of the refractive index n. The other special convention peculiar to this book is the nonstandard manner in which we define the complex susceptibility xat associated with a resonant atomic transition.In brief,we define the linear relationship between the induced polarization Pat on an atomic transition in a laser medium and the electric field E that produces this polarization by the convention that Pat =ateE where c is the dielectric permeability of the host laser crystal rather than the vacuum value co usually used in this definition.The merits of this nonstandard approach are argued in Chapter 2. XV
LIST OF SYMBOLS Throughout this text we attempt to follow a consistent notation for subscripts, using the conventions that: a either atomic,as in atomic transition frequency wa or homogeneous atomic linewidth Awe;or sometimes absorption,as in absorption coef- ficient oa. c=cavity,as in cavity decay time Te or cavity energy decay rate Ye;also, carrier,as in carrier frequency we. d=doppler,as in doppler broadening with linewidth Awd,and by extension any other kind of inhomogeneous broadening. e external,as in cavity external coupling factor 6e or external decay rate Ye;also,sometimes,effective,as in effective lifetime or pumping rate. m molecular or maser,generally used to refer to atomic or maser or laser quantities,e.g.,laser gain coefficient am or laser growth rate Ym. o ohmic,referring generally to internal ohmic and/or scattering losses,as in the ohmic loss coefficient oo or ohmic cavity decay rate Yo.Also used in several other ways,generally to indicate an initial value;a thermal equilibrium value;a small-signal or unsaturated value;a midband value; or a free-space (vacuum)values,as in co,co,and Ao. p=pump,as in pumping rate Rp or pump transition probability Wp. We also frequently use ax axial;avail available;circ circulating; eff≡effective;eg=equivalent;inc≡incident;opt≡optimum;out 三output;ref=reflected;rt≡round-trip;sat≡saturation;sp≡ spontaneous or spiking;ss small-signal or steady-state;and th threshold as compound subscripts. A partial list of symbols used in the text then includes: a=exponential gain or loss coefficient for amplitude (or voltage);also,am- plitude parameter for gaussian optical pulse a"=second derivative of a(w)with respect to w n=complex amplitude of n-th order Hermite-gaussian mode am maser/laser/molecular gain (or loss)coefficient oo ohmic and/or scattering loss coefficient B=propagation constant,including host dielectric effects,but usually not loss or atomic transition effects;also,chirp parameter for gaussian pulse;relaxation-time ratio in multilevel laser pumping systems;Bohr magneton Br Nuclear magneton ,"=first and second derivatives of B(w)with respect to w Am added propagation constant term due to reactive part of an atomic transition xvii
xviⅷi LIST OF SYMBOLS y=in general,an energy or population decay rate Ye decay rate for cavity stored energy (=1/Te) Yi=total downward population decay rate from energy level Ei Yij =population decay rate from upper level Ei to lower level E Ynr=nonradiative part of total decay rate for a classical oscillator or an atomic transition Yrad=radiative decay rate for classical electron oscillator or real atomic tran- sition y=complex eigenvalue for optical resonator or lensguide Ym-complex eigenvalue for mn-th order transverse eigenmode T=a+jp complex propagation constant for an optical wave T=a-jB complex gaussian pulse parameter 6=coefficient of(logarithmic)fractional power gain or loss,per bounce or per round trip 6e total(round-trip)power loss coefficient due to cavity losses plus exter- nal coupling 6e cavity loss coefficient due to external coupling only 6m=power gain coefficient due to laser atoms 6o cavity loss coefficient due to internal (ohmic)losses only AmAM or FM modulation index e=dielectric permeability of a medium eo dielectric permeability of free space (vacuum) n=efficiencies of various sorts;also,characteristic impedance vu/e of a dielectric medium no characteristic impedance of free space (vacuum) A optical wavelength (in a medium);also,eigenvalue for optical ray matrix Ao optical wavelength in vacuum Aa,A=eigenvalues of periodic lensguide or ABCD matrix A spatial period of optical grating u=electric or magnetic dipole moment;also,magnetic permeability of a magnetic medium Le electric dipole moment m =magnetic dipole moment uo magnetic permeability of free space P amplitude reflection or transmission of optical mirror or beamsplitter; also,distance between two points;p(w)=cavity mode density p=complex amplitude reflection or transmission of optical mirror or beam- splitter o ohmic conductivity;also,transition cross section,standard deviation cross section for stimulated transition from level Ei to E; 7=lifetime or decay time Te cavity decay time due to all internal losses plus external coupling total lifetime (energy decay time)for energy level E B,中，中=phase shifts and phase angles of various sorts (r,)Schrodinger wave function
LIST OF SYMBOLS Xix mn=Guoy phase shift for an mn-th order gaussian beam =susceptibility of a dielectric or magnetic medium=x'+jx" x,x"=real and imaginary parts of Xat susceptibility of a resonant atomic transition Xe,Xm=electric (magnetic)dipole susceptibilities w=frequency (in radians/second) in general,a frequency that has been shifted,pulled,or modified in some small manner wa=atomic transition frequency w=a beat frequency (between two signals) we cavity or circuit resonant frequency;also,carrier frequency wi(t)=instantaneous frequency of a phase-modulated signal wm generally,a modulation frequency of some sort wa=resonant frequency of g-th axial mode WR Rabi frequency on an atomic transition =Spiking or relaxation-oscillation frequency wa=frequency pulling of axial mode frequency wa Aw linewidth,or frequency tuning,in radians/sec Awa atomic linewidth (FWHM)nin radians/sec Awa axial mode spacing between adjacent axial modes solid angle;also,radian frequency or rotation rate ai,b=normalized wave amplitudes A=area Aj=Einstein A coefficient on EjE transition ABCD=matrix elements for optical ray matrix or paraxial optical system b=magnetic field as real function of space and time;also,confocal param- eter for gaussian beam b=magnetic field as real vector function of space and time;also,confocal parameter for gaussian beam B=magnetic field;also,pressure-broadening coefficient or "B integral"for nonlinear interaction B=phasor amplitude of sinusoidal B field c=velocity of light in a material medium co=velocity of light in vacuum C=in general,an unspecified constant;also,electrical capacitance;coupling coefficient in mode competition analysis CC complex conjugate (of preceding term) CEO=classical electron oscillator model d electric displacement as real function of space and time;also,distance or displacement d electric displacement as real vector function of space and time D dimensionless dispersion parameter D=phasor amplitude of sinusoidal electric displacement e magnitude of electronic charge E=electric field;usually,real field E(x,t)as function of space and time
x LIST OF SYMBOLS E=phasor amplitude of sinusoidal E field En(t)=amplitude of n-th mode in a normal mode expansion f frequency in Hz (cycles/sec);also,lens focal length f#=lens f-number Af linewidth,or frequency detuning,in Hz Afa=atomic transition linewidth (FWHM)in Hz Afa=doppler or inhomogeneous linewidth(FWHM)in Hz F=oscillator strength for an atomic transition;also,lens f-number F=finesse,of interferometer or laser cavity F(z)=Fresnel integral function Fi oscillator strength of EjEi atomic transition Yrad,ji/3Yrad.ceo g=amplitude(or voltage)gain,as a number;also,gaussian stable resonator parameter;magnetic resonance g value g(v),g(w)=normalized lineshapes g=complex amplitude (or voltage)gain,as a (complex)number gi,9;=degeneracy factors for quantum energy levels Ei and Ej gr=nuclear magnetic resonance g value grt=round-trip voltage gain inside an optical cavity G=power gain (as a number);also,electrical conductance GaB power gain in decibels h=magnetic intensity as real function of space and time;also,Planck's constant h=h/2m h=magnetic I field as real vector function of space and time hn n-th order polynomial function H=phasor amplitude of sinusoidal H field Hn=n-th order hermite polynomial I=intensity (power/unit area)of an optical wave;also sometimes,loosely, total power in the wave Im modified Bessel function of order m Isat=amplifier (or absorber)saturation intensity j=current density as real function of space and time;also,v-I j=current density as real vector function of space and time J=phasor amplitude of sinusoidal current density Jm=Bessel function of order m k propagation vector of optical wave =w/c K scalar constant in various equations (especially coupled rate equations); also,spring constant in classical oscillator model L=length;electrical inductance m electron mass;also,magnetization (magnetic dipole moment per unit volume)as real function of time m magnetization (magnetic dipole moment per unit volume)as real vector function of space and time m,m half-trace parameter for ray or ABCD matrix M proton mass;molecular mass