Gaussian Beams - Physics

2Beam-Like Solutions of the Wave EquationFor monochromatic 1 fields E(r,t)=E(r)e −iωt ,whereE(r) is any of the Cartesiancomponents of the complex vector field, the corresponding atomic response can bewritten as P at = χ at (ω)e −iωt . To avoid proliferation of symbols, we have denotedthe space dependent part by the same symbol as the total field. Using this in thewave equation (1.10), we find that the space dependent part E(r) satisfies∇ 2 + µω 2 (1 + χ at + iσ/ω) E(r) =0. (1.10*)For χ at =0=σ we get the homogeneous wave equation∇ 2 + ω 2 µ E o (r) =0, (2.1)with a solution E(r) =E o e ik·r , where the magnitude of the propagation constant is(real µ and ) given byk = ω √ µ = ωnc , (2.2)and n is the refractive index of the host medium. Combining this space dependencewith the time dependence e −iωt , we see that the solution E(r,t)=E o e i(k·r−ωt) ,representsa plane wave propagating in the direction κ = k/|k| in a loss-free medium.We now explore beam like solutions of the wave equation.Using this, we write the solution to the full wave equation (1.10*) as E(r) =E o (r)e ik·r , which has a space dependent amplitude E o (r) that satisfies the equation∇ 2 + k 2 (χ at + iσ/ω) E(r) =0. (2.3)2.1 Gaussian BeamsBeam-like solutions should have the following characteristics:(i) a predominant direction of propagation, and(ii) a finite transverse cross-section (finite extent in directions perpendicular tothe direction of propagation).1 We can always decompose a time-dependent field into its Fourier frequency components each withtime dependence of the form e −iωt 59

60 Laser PhysicsFinitetransverse sizeyxTransverse sizechanges slowlyPredominant directionof propagationzFIGURE 2.1A beam has a dominant direction of propagation and finite extent in directionsperpendicular to the direction of propagation.Since a beam has finite transverse size, wave diffraction will cause its cross-sectionto change as the beam propagates. However, we can still speak of a predominantdirection of propagation if the diffractive change of its cross-section is not too rapid(to be made more precise shortly) as the beam propagates. Let us now see how wecan construct a solution with these characteristics.From the introduction we recall that a monochromatic plane wave propagatingin the z-direction will have the formE(r) =E o e i(kz−ωt) , (2.4)where the propagation constant k = nω/c = 2πnλ. Using this plane wave solutionas a guide, we now look for solutions that correspond to a superposition of planewaves propagating in directions making small angles with the z-axis, we can writethe propagation vectors as k = e x k x +e y k y +e z k z with k 2 ≡ kx 2 +ky 2 +kz 2 = n 2 ω 2 /c2and k x ,k y k, k z . For such fields k z ≡ k 2 − kx 2 − ky 2 ≈ k − (kx 2 + ky)/2k 2 2 and wecan write the space dependent part of the field asE(r) = 1 (2π) 2 dk x dk y Ẽ(k x ,k y )e ikxx+ikyy+ikzz ,= eikz(2π) 2 dk x dk y Ẽ(k x ,k y )e ikxx+ikyy−i(k2 x +k2 y )z/2k ,≡ e ikz E o (r) . (2.5)The amplitude |Ẽ(k x,k y )| of off-axis plane wave components decreases rapidly forincreasing values of |k x |, |k y |. This means the dominant direction of beam propagationis along the z−axis. Comparing Eq. (2.5) with the expression for a planewave (2.4), we notice that unlike a plane wave, the field amplitude E o for a paraxialbeam depends on spatial coordinates. Using this in Eq. (1.10*), we find that theequation satisfied by E o (r) is∇ 2 ⊥ + ∂2∂z 2 +2ik ∂ ∂z + k2 (χ at + iσ/ω) E o (r) =0, (2.6a)where∇ 2 ⊥ = ∂2∂x 2 + ∂2∂y 2 .(2.6b)

Beam-Like Solutions of the Wave Equation 61Since the amplitude Ẽ(k x,k y ) is dominated by small values of k x and k y , fieldamplitude E o varies slowly with z. This means the following inequalities hold:1∂E ok ∂z = λ∂E o2π ∂z |E o| ,(2.7a)1∂ 2 E o k ∂z 2 = λ ∂ ∂E o 2π ∂z∂z ∂E o∂z .(2.7b)The first inequality says that the change in beam profile over distances of the orderof a few wavelengths λ ∂Eo(r)∂z are a small fraction of |E o (r)|. The second inequalitystates that the rate at which the beam profile evolves does not significantly changeover distances of the order of a few wavelengths, λ ∂ ∂E o(r)∂z ∂z ∂Eo(r) ∂z. In view ofthe inequalities (2.7), the dominant variation of E o with z in Eq.(2.6) is describedby the first derivative term. Neglecting the ∂ 2 E o (r)/∂z 2 term compared to the∂E o (r)/∂z term in Eq. (2.6), we find that the equation governing the variation ofE o (r) withz isor∂E o (r)∂z∂E o (r)∂z= i2k ∇2 ⊥ E o(r)+i kχ at2 E o(r) − kσ= i2k ∇2 ⊥ E o(r)+i kχ at2 E o(r) − k 22ω E o(r) , σω + χ atE o (r) . (2.8)This is one of the basic equations of this course, which tells us that the field amplitudechanges slowly with z due to (i) diffraction (represented by the ∇ 2 ⊥term), (ii)interaction with the atomic medium resulting in a phase shift (represented by theχ at term), and (iii) dissipation of electromagnetic energy (represented by the conductivityterm) because of Joule’s heating and absorption by the atoms (representedby the χ at term).Neglecting the loss and atomic interaction terms in Eq. (2.8) we find that theenvelop function E o (r) satisfies ∂2∂x 2 + ∂2∂y 2 +2ik ∂ E o (r) =0. (2.9)∂zThis equation is known as the paraxial wave equation. By writing this equation as∂E o (r)= i ∂ 2 E o (r)∂z 2k ∂x 2 + ∂2 E o (r)∂y 2 , (2.10)we see that this equation expresses the fact that the beam profile changes withpropagation because of diffraction (due to finite extent in transverse dimensions).If the transverse beam profile does not depend on spatial coordinates, we recoverthe plane wave result E o = const.2.1.1 Fundamental Gaussian SolutionThe paraxial wave equation has many known solutions. These will be introducedin due course. The simplest of these has circular cylindrical symmetry about the

62 Laser Physicsdirection of propagation and is given by (ρ = x 2 + y 2 ) w0E o (ρ, z) =A e i kρ22q(z) −iψ(z)w(z)(2.11a)w(z) =w 01+(z/z R) 2 (2.11b)z R= 1 2 kw2 0 = πw2 0 nλ1q(z) = 1R(z) + i 2kw 2 (z) R(z) =z + z2 Rz = z 1+ z2 Rz 2ψ(z) = tan −1 zz R(2.11c)(2.11d)(2.11e)(2.11f)To see the physical meaning of various terms in this equation, let us first consider1I/I maxequivalent"top hat" beam0.51/e!=w/!221/e 2 !=w02 1 0 1 2x/wFIGURE 2.2Intensity profile of a circular symmetric gaussian beam along the x (or y) axis(ρ = x).the time averaged intensity I(ρ, z) ≡ 1 2 0ncEE ∗ given by 2 w0I(ρ, z) =I 0 exp− 2ρ2w(z) w 2 , (2.12)(z)

Beam-Like Solutions of the Wave Equation 63where I 0 = 1 2 0nc|A| 2 . This has the form of a gaussian distribution in the variableρ. For this reason this beam is called a gaussian beam. In terms of the total powerof the beam 2π ∞P = dϕ ρdρI(ρ, z) =I 0 w02π 2 × 1 4 = I πwo2 o(2.13)200we can express the peak intensity I o =2P/πw 2 o and the intensity asI(ρ, z) =2Pπw 2 (z) e−2ρ2 /w 2 (z) . (2.14)Figure (2.2) shows the intensity profile of the beam as a function of distance ρ(along the x−axis) measured from the z−axis. The intensity attains its peak valueI max =2P/πw 2 (z) at the center of the beam (ρ = 0) and falls to 1/e 2 ≈ 14% ofthis value when ρ = w(z). w(z) is referred to as the 1/e 2 intensity radius or simplyas beam spot size (radius). The intensity of a gaussian beam falls off rapidly as ρincreases beyond the spot radius w(z).The form of the peak intensity 2P /πw 2 (z) suggests another measure of beam size.If we were to imagine a circular cylindrical beam of uniform intensity and the sametotal power as the Gaussian beam, the radius of such a beam will bew TH= w/ √ 2 . (2.15)Such a beam with uniform intensity over its cross sectional area is referred to as a“top hat beam” because the intensity distribution of such a beam has the shape ofa top hat [See Fig. (2.2)]. We may also refer to w TH= w/ √ 2 as the 1/e intensityradius.There are other measures of beam size. For example, we can use a criterion basedon power the transmitted by an aperture. A circular aperture of radius a placed atthe center of a gaussian beam will transmit a fraction of powerP TP = 2 2ππw 2 dϕ0 a0ρdρe −(2ρ2 /w 2) =1− e −(2a2 /w 2) . (2.16)This fraction as a function of aperture radius is plotted in Fig. 2.3.An aperture of radius w THwill transmit only 63% power while an aperture ofradius w will transmit 86% power. An aperture of radius πw/2 will pass 99%power. We find that considerably larger apertures than those with radius w areneeded to pass the power gaussian beam of spot size w.Beyond optimizing power transmission of a gaussian beam, we may also want tominimize diffraction ripples which will significantly distort the intensity distributionof the transmitted beam. Such ripples will be present whenever sharp edgedcircular apertures are used even if they pass a large fraction of the total power. Asharp circular aperture of radius πw/2, which passes 99% of the total power, willcause ripples with intensity variations of 17% in the near field and a peak intensityreduction of the same amount in the far field. To keep diffraction ripples down to1% in the beam transmitted by a sharp edged circular aperture we must employ anaperture of radius ≈ 2.3w. [Siegman, Chapter 18].

64 Laser Physics199%86%P TP0.563%Ripple ±17%Ripple ±1%a = w/!2a = wa =!w/2a =2.3w00 0.5 1 1.5 2 2.5 3a/wFIGURE 2.3Fractional power of a gaussian beam transmitted by a circular aperture of radius ρcentered on the beam.From the preceding discussion, it is clear that w(z) is a measure of the transversesize of the beam, which of course, varies as the beam propagates. Figure (2.4) showsthe variation of spot size as a function of the propagation distance z measured fromthe “beam waist” which is defined to be the plane where the spot size has itsminimum value w 0 . In writing the expression for beam spot size(2.11b) have chosenthis plane to be the z = 0-plane. An examination of Eqs. (2.11a)-(2.11f)shows thata gaussian beam is uniquely determined by the location of its waist and spot sizew o .A gaussian beam spreads in a nonlinear fashion during its propagation. Near thewaist the spread is slow so that the beam remains collimated. Far from the waistthe beam spreads linearly with distance from the waist. The characteristic distancethe beams travels from the waist before the spot size increase to √ 2w o (or the beamspot area doubles) isz R= kw0/2 2 = πnw2 0. (2.17)λThis distance z Ris called the Rayleigh range. Notice that there are two pointslocated√on the opposite sides of the beam waist where the spot size has the value2wo . The distance between these √ 2w o spot size points is the confocal parameterb =2z R= 2nπw2 0λ. (2.18)Confocal parameter b is a measure of the distance over which a beam may beconsidered to have uniform cross-section near its waist. It plays an important rolein the theory of laser resonators which will be discussed shortly.

Beam-Like Solutions of the Wave Equation 65w(z)Confocal parameter b =2z R!2w o w o!2w oRayleigh range z RzRayleigh range z Rwaist z= 0FIGURE 2.4Variation of spot size with propagation near beam waist.In the far field z z Rthe spot size grows linearly with z as [see Fig. (2.20)]w(z) ≈ w 0zz R. (2.19)The far field divergence angle of the beam may be defined by the ratio of the farfield spot size to the distanceθ = w(z)z= w 0z R= λnπw 0. (2.20)For paraxial approximation to be good we require θ < 1/π which translates tominimum spot size w 0 > λ/n. From the dependence of the confocal parameter [Eq.(2.18)] and the beam divergence angle [Eq. (2.20)] on w 0 we see that a beams withsmaller waist spot size will remain collimated over shorter distances and will spreadmore rapidly in the far zone.The origin of gaussian beam divergence is diffraction, which arises whenever awave is confined to a finite transverse size. In fact, the far field beam divergenceangle is of the same order as the angle associated with the Fraunhofer (far field)diffraction of a plane wave by a circular aperture of radius a ∼ w oθ D=0.61 λ na . (2.21)A Gaussian laser beam thus has the smallest possible divergence allowed by Maxwell’sequations. Since diffractive phenomena cannot be described by ray optics, wave opticsmust always be used when dealing with gaussian beams.The parameter R(z) [Eq.(2.11e)] is the radius of curvature of the very nearlyspherical phase fronts at z. This can be seen by writing the Gaussian beam

66 Laser Physicsw(z)w o0!="/n#w $zBeam waistz=0FIGURE 2.5Far field divergence of a gaussian beam.E o (ρ, z)e i(kz−ωt) [Eq.(2.11a)] in the limit z>>z RE(r,t)=Ae−iψ(z)1+z 2 /z 2 R≈−iAz Re −ρ2 /w 2 (z)where we have used the approximatione −ρ2 /w 2 (z) e ik(z+ρ22R(z) ) e −iωt 1 ρ2eik(z+ 2z ) e −iωt , (2.22)zR(z) =z 1+z 2 R /z2 ≈ z, z z R. (2.23)Let us compare it with a spherical wave emitted by a point source on the z-axis atz = 0. Near the z−axis this wave has the forme i(kr−ωt) e ik√ ρ 2 +z 2E s (r,t)=E o = E o rρ 2 + z 2 e−iωt⎡ ⎤ρ2eik(z+ 2z≈E o⎣ )⎦ e −iωt , ρ 2 z 2 . (2.24)zA comparison of Eqs. (2.22) and (2.24) shows that a gaussian beam has the formof a spherical wave with center of curvature located at the beam waist z = 0. Infact for points close to the z− axis such that ρ 2 w 2 (z), Eq. (2.22), has exactlythe form of a spherical wave with spherical wavefrontskz + kρ22z ≈ k ρ 2 + z 2 ≡ kr = const . (2.25)It is clear that R(z) is the radius of curvature of constant phase surfaces whichcoincide with wavefronts near the z−axis. The sign of R is chosen according to

Beam-Like Solutions of the Wave Equation 67R(z)b0!z Rz Rz!bFIGURE 2.6Variation of the radius of curvature of a gaussian beam with distance.the convention used in discussing ABCD matrices; that is, R(z) is negative for aconverging beam (center of curvature lies in front of the wavefront) and positive fora diverging beam (center of curvature lies behind the wavefront).The radius of curvature R(z) of the wavefronts of a gaussian beam has the followinglimiting forms⎧⎪⎨ ±∞ |z| > z RIts variation with propagation distance z measured from the waist is shown inFigs. (2.6) and (2.7). The wavefront is flat or planar at the waist corresponding toan infinite radius of curvature. As the beam propagates away from the waist thewavefronts become curved, and the radius of curvature drops until z = z R. Beyondz = z Rthe radius of curvature increases again as R(z) ≈ z. The minimum radiusof curvature occurs for the wavefront at z = z Rwith the radius of curvature equalto the confocal parameter R =2z R= b. The center of curvature for the wavefrontat z = z Ris located at z = −z Rand the center of curvature of the wavefront atz = −z Ris located at z = z Ras seen in Fig. (2.7). The curved wavefronts atz = ±z Rhave special significance in stable resonator theory. If these wavefrontsare replaced by two mirrors with matching radii of curvature, we will form a stableresonator. This resonator will have mirrors of radius of curvature R and spacing Lwith R = b =2z R= L. Since the focal length of a mirror of radius R is f = R/2,the focal points of these two mirrors will coincide at the center of the resonator.The two mirrors then form a symmetric confocal resonator, thus giving rise to theconfocal parameter b =2z R.

68 Laser PhysicsSSR=!bR=bz=0RConfocal parameter bz=!z Rz =z RFIGURE 2.7Wavefronts and ray trajectories in the waist region for a wave moving from leftto right. Far from the beam waist the wavefronts are part of spheres with centerof curvature at the center of the waist. The hyperbolas are rays indicating thedirection of energy flow in a gaussian beam.The direction of energy flow in a gaussian beam is indicated in Figure (2.7) for agaussian beam traveling from left to right. Energy is transported along rays whichare the directed curvesρ ≡ x 2 + y 2 = ρ 01+z 2 /z 2 R , (2.27)where ρ 0 is the ray coordinate at the waist. The Poynting vector - S which describesthe flow of energy (energy flux density, W/m 2 ) is tangential to the rays. To the leftof the waist, Poynting vector has a small radial component pointing toward the axiscorresponding to a converging (focusing) beam. To the right of the waist it has asmall radial component pointing away from the axis corresponding to a diverging(defocusing) beam. Energy flow across the waist is from left to right as in a planewave moving in the z-direction.Finally, we note that in passing the waist the phase angle ψ(z) [Eq. (2.11f)]changes from −π/2 (far to the left of the waist) to π/2 (far to the right). Thusthe fundamental gaussian beam acquires an extra phase of π in passing through itsfocal region. This phase shift, called the Guoy phase shift, is in fact a special caseof the Guoy effect which says that a wave with a reasonably simple cross-sectionwill acquire an extra phase shift of π in passing through a focal region. Higherorder gaussian beams acquire larger Guoy phase shifts on account of their morecomplicated cross-section.RefrencesDeviations from paraxial theory become significant for beams that are focused so tightlythat beam divergence angles exceed θ>1/π. Corrections to these solutions are discussedin M. Lax, W. H. Louisell, and W. B. McKnight, From Maxwell to paraxial optics, Phys.

Beam-Like Solutions of the Wave Equation 69Rev. A 11, 1365 (1975); L. W. Davis, Theory of electromagnetic beams, Phys.Rev.A19,1177 (1975); G. P. Agarwal and D. N. Pattanayak, Gaussian beam propagation beyond theparaxial approximation, J. Opt. Soc. Am. 69, 575 (1979).A paper dealing with Gaussian beam solution of Maxwell’s equations including correctionsto paraxial theory and describing experimental observations of these corrections is W. L.Erikson and S. Singh, Polarization properties of Maxwell-Gaussian beams, Phys. Rev. E49, 5778 (1994).2.2 ABCD Law for Gaussian BeamsDuring the propagation of a gaussian beam in a homogeneous medium beam spotsize w(z) and the radius of curvature of phase front R(z) change according to Eqs.(2.11b) and (2.11e) but the beam retains its gaussian shape. We have seen that ageometrical ray picture fails to account for this evolution.The evolution of R(z) and w(z) during propagation is equivalent to an evolution ofthe complex beam parameter q(z). We have seen that geometric ray picture cannotbe used to describe the propagation of a gaussian beam. It turns out, however, thatgaussian beam propagation is described in terms of a very simple propagation lawfor the q-parameter of a gaussian wave.When a gaussian beam passes through an optical element whose ray matrix isgiven by (ABCD), its q-parameter changes. Complex beam parameter q o just afterthe beam emerges from the optical element is related to the complex parameter q ijust before the beam enters the optical element byq o = q i A + Bq i C + DThis important relation is known as the ABCD law of a gaussian beam.relation follows from paraxial diffraction theory (paraxial wave equation)(2.28)ThisReferences1. H. Kogelnik and T. Li, Laser Beams and Resonators, Proc. of IEEE 54,1312-1329 (1966).2. A. Siegman, Lasers (University Science Books, Mill Valley, CA 1986), Chapter20.Example 1: Gaussian beam propagation in a homogeneous medium starting at itswaist.Let us take the beam waist location to be the z = 0 plane. At the waist, thewavefronts are planar, so that the complex beam parameter is pure imaginary givenbyq i = −iz R= −i πw2 0λ . (2.29)

70 Laser PhysicsThe matrix for free propagation over a distance z in a homogeneous medium is AB 1 z= . (2.30)CD 01Using this matrix, we find that the beam parameter at a distance z from the waistwill be given byorq(z) = q i A + Bq i C + D = q i + z1q(z) ≡ 1R(z) + i 2kw 2 (z) = 1q i + z = 1−iz R+ z = iz + z R. (2.31)+ z2z 2 REquating the real and imaginary parts from the two sides, we obtain the familiarexpressions [ 1 2 kw2 = 1 2 kw2 0 (w/w 0) 2 = z R(w/w 0 ) 2 ] for the wavefront radius ofcurvature and spot sizeR(z) = z2 + z 2 Rzw(z) =w 0z 2 R + z2z 2 R= z + z2 Rz ,= w 0Example 2: Passage of a gaussian beam through a lens.The ray transfer matrix for the lens is(2.32a)1+ z2z 2 . (2.32b)R AB 1 0=CD − 1 f 1 . (2.33)Let q 1 be the incident beam parameter just before the lens. Then the output beamparameter q 2 (just after the lens) is given byoror1R 2 (z) + i 2kw22q 2 = q 1 A + Bq 1 C + D = q 1−(q 1 /f)+1 ,1= − 1 q 2 f + 1 q 1= − 1 f + 1R 1 (z) + i 2kw 2 1(2.34)Equating the real and imaginary parts on the two sides gives1R 2= 1 R 1− 1 f ,w 2 = w 1 .(2.35a)(2.35b)Thus, a lens changes the curvature of the phase front but leaves the spot sizeunaffected. A related problem is the focusing of a gaussian beam by a mirror offocal length f = R/2.

Beam-Like Solutions of the Wave Equation 712.2.1 Gaussian Beam FocusingConsider a gaussian beam incident from left on a lens of focal length f. Let theincident beam waist be located a distance d 1 from the lens and let the spot radiusthere be w 01 . After passing through the lens the beam has a new waist at d 2 andspot size w 02 at the new waist. We are interested in finding d 2 and w 02 .The ray transfer matrix for beam passage from the first waist to the second waistis AB=CD 1 d2 1 0 1 d1 1 −d 20 1 − 1 f 1 = fd 1 + d 2 − d 1d 2f0 1 − 1 f1 − d 1fThe complex beam parameter q 2 at the second waist is then given by(2.36)q 2 = Aq 1 + BCq 1 + D = (1 − d 2/f)q 1 +(d 1 + d 2 − d 1 d 2 /f), (2.37)−q 1 /f +(1− d 1 /f)where q 1 is the beam parameter at the first waist. At the two waists, the complexbeam parameters are pure imaginary,q 1 = −iz R1≡−iπnw 2 01/λ, q 2 = −iz R2≡−iπnw 2 02/λ. (2.38)Using these in the transformation equation (2.37) we obtain−iz R2= −iz A + BR1−iz R1C + D = (−iz A + B)(iz A + B)R1 R1(z R1C) 2 + D 2= −iz (AD − R1 BC)+(z2 AC + BD)R1(z R1C) 2 + D 2 (2.39)Equating the real an imaginary parts of the expression on the right hand side tothe corresponding terms on the left we findRe[q 2 ] ≡ 0= z2 AC + BDR1(z R1C) 2 + D 2 , (2.40a)Im[q 2 ] ≡−z R2= −z R1(AD − BC)(z R1C) 2 + D 2 . (2.40b)Using the fact AD − BC = 1 and z 0i = πnw0i 2 /λ we find from Eq. (2.40b) that thenew waist spot size is given byw 2 02 =w012 λf 2(z R1/f) 2 +(1− d 1 /f) 2 = 1πnw 01 1+(f/z R1) 2 (1 − d 1 /f) 2 . (2.41)From Eq. (2.40a) we find, since the denominator is not zero, z 2 AC + BD = 0,R1which leads us toz 2 R11 − d 2f− 1 + d 1 + d 2 − d 1d 21 − d 1=0. (2.42)ff f

72 Laser PhysicsOn simplifying and solving this equation for d 2 we obtaind 2 = z2 /f − d R11(1 − d 1 /f)(z R1/f) 2 +(1− d 1 /f) 2 = f (1 − d 1 /f)1 −(z R1/f) 2 +(1− d 1 /f) 2 . (2.43)A plot of exit waist position d 2 /f as a function of the incident waist position d 1 /fis shown in Figure (2.8). To see the variation of the exit waist spot size with d 1 ,we find it is convenient to plot the Rayleigh range z R2/f = πw02 2 /λf which is ameasure of the spot size in units of f as a function of d 1 /f. This is shown in Fig.(2.9).2.52Geometrical opticsz R1/f=0.41.5d 2/f12.50.510!0.5!4 !2 0 2 4d 1 /fFIGURE 2.8Beam waist location d 2 /f after a gaussian beam passes through a positive lens offocal length f as a function of the incident beam waist location d 1 /f for differentvalues of the ratio z R1 /f.Let us compare these results with the predictions of geometrical optics. If weconsider the outgoing beam waist as the image of the incident beam waist, then

Beam-Like Solutions of the Wave Equation 73geometrical optics gives the location of new beam waist d 2 and spot size w o2 to bed 2 = fd 1d 1 − f = f 11 −(2.44)1 − d 1 /fw 2 o2 = w 2 01 ×− d 2d 1 2=w 2 01(1 − d 1 /f) 2 (2.45)The prediction of geometrical optics for the second beam waist location d 2 is shownby the dashed curve in Fig.(2.8). We see that gaussian beam and geometrical opticspredictions agree when |d 1 /f| 1 and z R1/f 1, that is, when the lens is locatedin the far zone of the incident beam waist. The disagreement between the twopredictions is complete as d 1 → f. In this case, geometrical optics predicts d 2 →∞and w02 2 →∞, whereas gaussian beam results are λf 2d 2 → f and w02 2 =. (2.46)πnw 01A noteworthy features of Fig. 2.8 is that the distance d 2 for the second waist fromthe lens has a maximum. The maximum occurs for d 1 /f ≈ 1.5. Similarly, thespot size for a gaussian beam after passing through has a finite maximum which isattained for d 1 /f ≈ 1.A question of practical importance when discussing applications such as lasertraps, cutting, drilling, and laser fusion is how small focal spots are possible toboost power density. For a given focal length f, we can reduce the size of w 02 bymaking z R1/f large and, since z R1= πnw01 2 /λ, this means we need to make w 01as large as possible. But w 01 cannot be larger than the lens aperture if significantbeam power loss is to be avoided and may need to be even smaller if we allow forbeam spreading from the first waist to the lens. One way to address this is tocollimate the incident beam with a confocal parameter many focal lengths long andplace the lens in the near zone so that the input beam spot size does not changesignificantly from the first beam waist to the lens. Under these conditions w 01 islimited by the lens aperture. If we want the lens to transmit 99% of the incidentpower, then incident beam spot radius w 01 must satisfy the condition12 πw 01 = 1 D, (2.47)2where D is the lens aperture (diameter). Under these conditions (z R1/f 1 andthe constraint πw 01 = D), Eq. (2.41) leads to the following expression for the focalspot radius (spot size at the second waist)w 02=λf = λ fnπw 01 n D = λ n × (f # ) , (2.48)where f # is the f-number of the lens. A small f # implies a fast lens (high lightgathering capability) and a large f # ⇒ a slow lens (low light gathering capability).The best lenses have f # ≈ 1, while most have f # > 1. Thus the smallest spotradius of focal spot is about the size of a wavelength.

74 Laser Physics2.52z R1/f=0.41.5z R2/f12.50.50!4 !3 !2 !1 0 1 2 3 4d 1/fFIGURE 2.9Gaussian beam spot size in terms of Rayleigh range after focusing by the lens offocal length f for z R1/f =0.4, 1, 2.5. Geometrical optics predictions are shown bygrey curves for z R1/f =0.4 and 2.5. The dashed curve is the gaussian beam resultfor z R1/f = 1.The location d 2 of the focal spot (second waist) from Eq. (2.43), under the sameconditions, is given byd 2f ≈ 1 − f 2 z 2 1 − d 1=1− z2 R2f f 2 1 − d 1≈ 1 , (2.49)fR1where last step follows because z R2, the Rayleigh range for the focused beam, isusually much less than f. This means the second waist is very nearly in the focalplane of the lens.The peak power density at the second waist isI 02 =2Pπw 2 02= 2P(λ/n) 2 Ω 2 , (2.50)where Ω 2 = πθ2 2 is the solid angle into which the second beam waist radiates [Fig.1.4] and θ 2 is the divergence angle for the focused beam. Similarly, the peak intensityat the first waist isI 01 =2P = 2P(λ/n) 2 Ω 1 , (2.51)πw 2 01

Beam-Like Solutions of the Wave Equation 75where Ω 1 = πθ1 2 is the solid angle into which the first waist radiates. The quantityB = I 01 /Ω 1 = = I 02 /Ω 2 is called the brightness (power emitted per unit2P(λ/n) 2area per unit solid angle: W/m 2·sr) of the source. The brightness of a source isan invariant in the sense that linear optics elements (mirrors, lenses etc.) do notchange it.By expressing w 02in terms of w 01, the result for the peak intensity in the secondwaist can also be written asI 02 =2Pf 2 . (2.51*)Ω 1From this expression we see that the power density that can be obtained by focusinga beam of given power is inversely proportional to the solid angle divergence of thebeam being focused. High degree of directionality (smallness of Ω) of laser beamsthus is of crucial importance for obtaining high power densities in the focal spot.By contrast, a thermal source (ordinary lamp) emits in all directions (2π steradian).If it delivers a power P over an ideal lens aperture, leading to the focal spot powerdensity [See Fig. 1.4]I ≈ P f 2 12π . (2.52)Example: Consider a He:Ne laser with P =1mW,λ=633 nm, and spot size w 01 =1mm. Then its divergence angle (n = 1), solid angle and intensity areθ =λ 0.633 × 10−6=πw o π × 10 −3 ≈ 0.2 × 10 −3 radΩ 1 = πθ 2 = π(0.2 × 10 −3 ) 2 ≈ 1.3 × 10 −7 srI 01 = 2Pπw 2 0= 2 × 10−3π × (10 −1 ) 2 W/cm2 = 64 mW/cm 2If this laser is focused by a lens of f =2.5 cm (the human eye), the peak intensitywill beI 02 =2P 2 × 10 −3f 2 =Ω 1 6.25 × 1.3 × 10 −7 ≈ 2.5 × 103 W/cm 2 . (2.53)Thus, direct viewing of even a lower-power laser beam can result in severe retinaldamage. Thermal lamps would have to emit hundreds of thousands of watts tomatch the intensities achievable by focusing even modest power lasers.The large intensities achievable by lasers are a direct consequence of their lowdivergence. While care must be exercised in dealing with laser beams, their usein repairing detached retinas and other surgical procedures has become practicallyroutine.2.2.2 Hermite-Gauss beam solutionsSo far we have discussed only the fundamental Gaussian beam solution. There areother solutions which have more complex spatial structure. In general, the solutionsof paraxial wave equation (2.9) are labeled by two indices. The well-known solutions

76 Laser Physicsseparable in Cartesian coordinates are the Hermite-Gauss solutions given by[E o ] mn (r) =A w 0w H m( √ 2x/w)H n ( √ 2y/w)e −i(m+n+1)ψ+ikρ2 /2q . (2.54)Here we have suppressed the z−dependence of w(z), complex beam parameter q(z)and phase ψ(z) for simplicity of writing. They are independent of the beam indicesand are given by Eqs. (2.11b)-(2.11f). A Hermite-Gauss beam of indices m, n issometimes denoted by HG mn .H m (x) in Eq.(2.54) is a Hermite polynomial of degree m and argument x. Somelow order Hermite polynomials and recursion relations for computing the higherorder ones are listed belowH 0 (x) =1H 1 (x) =2xH 2 (x) =4x 2 − 2H 3 (x) =8x 3 − 12xH m+1 (x) =2xH m (x) − 2mH m−1 (x)dH m (x)dx=2mH m−1 (x)(2.55)Hermite-Gauss beams maintain their form during propagation. Spot size w(z) setsthe length scale over which the beam profile changes significantly in transversedirections, and z Rsets the length scale over which beam profile changes significantlyas the wave propagates. The intensity distribution for the beam with indices m, nis given bywhere I 0 is given byI mn (x, y) =I 0w 2 0w 2 H2 m( √ 2x/w)H 2 n( √ 2y/w)e −2(x2 +y 2 )/w 2 , (2.56)I 0 = 1 2 0n|A| 2 c. (2.57)The total power of mode with indices m, n is ∞w 2 ∞0P = dxdyI mn (x, y, z) =I 0−∞w 2 dxdyHm( √ 2 2x/w)Hn( √ 2 2y/w)e −2(x2 +y 2 )/w 2−∞= 1 ∞ ∞2 I 0w02 dXHm(X)e 2 −X2 · dY Hn(Y 2 )e −Y 2−∞−∞= 1 2 I 0w02 2m √ πm! · 2 n√ πn! The intensity distribution then can be written asI mn = 2Pπw 2 12 m+n m!n! H2 m( √ 2x/w)H 2 n( √ 2y/w)e −2(x2 +y 2 )/w 2 (2.58)It is easy to check that for m =0=n we recover the intensity distribution of thefundamental gaussian mode.

Beam-Like Solutions of the Wave Equation 77Intensity distributions of some low order Hermite gaussian beams areI 00 (x, y, z) = 2P +y 2 )/w 2πw 2 e−2(x2I 10 (x, y, z) = 2P 1 8x 2+y 2 )/w 2πw 2 2 w 2 e−2(x2I 01 (x, y, z) = 2P 1 8y 2+y 2 )/w 2πw 2 2 w 2 e−2(x2 (2.59)I 11 (x, y, z) = 2P 1 8x 2 8y 2+y 2 )/w 2πw 2 4 w 2 w 2 e−2(x2 √ I 20 (x, y, z) = 2P 1 2xπw 2 8 H2 2 e −2(x2 +y 2 )/w 2wIn terms of scaled variables X = √ 2x/w, Y = √ 2y/w, we obtain slightly morecompact expressionI mn (X, Y )=Pπ2 m+n m!n! H2 m(X)H 2 n(Y )e −(X2 +Y 2) . (2.60)for the intensity. Expressed in terms of X and Y , the intensity distribution is forminvariant. Note that the power normalization condition is nowI mn (X, Y )dXdY = P.2.2.3 Astigmatic Hermite-Gauss BeamsIt is possible for gaussian beams to have different spot sizes in the two transversedimensions. The fundamental beam will then have an elliptical cross-section. Forthis reason such a beam is called an elliptical beam. They are good models for thelight emitted by semiconductor lasers. Elliptical beams will converge or diverge atdifferent rates along the principal axes of the ellipse. Such beams will therefore beastigmatic. If a elliptic beam is passed through a lens, beam waists after the lens

78 Laser Physicsdo not, in general, lie in the same plane. Fundamental elliptical beam has the form„w0x w 0yE o (r) =Aw x (z)w y (z) eik1q x (z) = 1R x (z) + i 2kwx(z)21q y (z) = 1R y (z) + i 2kwy(z)2 2(z −wx 2 = wox2 zx )1+⎡wy 2 = woy2 ⎣1+R x (z) =(z − Z x )⎡kw 2 0xx 22qx(z) + y22qy(z) 2 ⎤ 22(z − z y )⎦kw0y2 kw21+ 0x2(z − Z x ) 2 2⎤«−iψ(z)kw2R y (z) =(z − Z y ) ⎣1+0y⎦2(z − Z y )ψ(z) = 1 2(z − Zx )2 tan−1 kw0x2 + 1 2(z − Z y )2 tan−1 kw0y2(2.61)Beam waist occurs at z = Z x in the x − z plane and at z = Z y for the y − z plane.These two planes in general do not coincide. The beam in general has ellipticalprofile. Semiconductor lasers emit this type of beams. Such beam can be convertedinto symmetric Hermite-Gauss beams by using prisms or cylindrical lenses. Forw 0x = w 0y = w 0 ( which also requires Z x = Z y = Z, we recover the fundamentalcircularly symmetric gaussian beam with waist at Z.2.2.4 Laguerre-Gauss BeamsParaxial wave equation (2.9) admits beam solutions that reflect other symmetries.For example, in the presence of circular cylindrical symmetry about the z-axis, Eq.(2.9) admits Laguerre-Gaussian beam solutions2(p!)[E o ] p (r) =Aπ(p + )!w 0w√2ρw 2ρe iϕ L ||2kρ2−i(2p++1)ψ+ipw 2 e 2q . (2.62)where L |p (x) is the associated Laguerre polynomial and w(z) and R(z) are independentof the mode indices. Some low order associated Laguerre polynomials and

Beam-Like Solutions of the Wave Equation 79recursion relations for computing the higher order polynomials are ( >0)L 0(u) =1L 1(u) =−u + +1L 2(u) = 1 u 2 − 2( + 2)u +( + 1)( + 2) 2(2.63)(p + 1)L ||p+1 (u) =(2p + +1− u)L|| p (u) − (p + )L ||p−1 (u)u dL|| p (u)= pL ||p (u) − (p + )L ||p−1du(u)We see that the lowest order solution (p =0=) coincides with the fundamentalGaussian beam solution for the HG family. For p = 0 and = 1 we obtain theintensityw02 2ρ 2/wI 01 (ρ) =I 2 0w 2 w 2 e−2ρ2 (2.64)This has a dark center and is sometimes called the donut mode. Note that the“donut” shaped intensity distribution sometimes seen in lasers is most often a mixtureof HG 01 and HG 10 modes.Let us calculate the total power of the beam. With I o = 1 2 0cn|A| 2 ,wehaveP = ∞0ρdρ 2π0= I o2(p!)π( + p)!dϕ I(ρ, z) w0w ∞22π 4π(p!) wo 2 w2= I oπ( + p)! w 40ρdρ ∞0√2ρw 2 2L ||p (2ρ 2 /w 2 ) e−2ρ 2 /w 2 2du u L ||p (u) e−u (p!)= I o w 2 ( + p)!o = I o wo 2 (2.65)( + p)! (p!)Hence we can write the intensity of LG p beam asI p = 2Pπw 2 p!( + p)!√2ρw 2L ||p (2ρ 2 /w 2 ) 2e−2ρ 2 /w 2 (2.66)There are also the so-called Bessel beam or non-diffracting beam solutions [3]. Inpractice, symmetries other than the rectangular symmetry are difficult to realize.The presence of Brewster surfaces and other asymmetric optical elements in laserresonators naturally leads to beams with Cartesian symmetry. For this reason onlythe Hermite-Gauss solutions are usually considered.