1312 PROCEEDINGS OF THE IEEE, VOL. 54, NO. 10, OCTOBER, 1966 Laser Beams and Resonators

H. KOGELNIK AND T. LI

Abstract-This paper is a review of the theory of laser beams and 2. PARAXIALRAY ANALYSIS resonators. It is meant to be tutorial in nature and usefulin scope. No attempt is made to be exhaustive in the treatment. Rather, emphasisis A study of the passage of paraxial rays through optical placed on formulations and derivations which leadto basic understand- resonators, transmission lines, and similar structures can ing and on results which bear practical significance. reveal many important properties of these systems. One such “geometrical” property is the stability of the struc- 1. INTRODUCTION ture [6], another is the loss of unstable resonators [12]. HE COHERENT radiation generated by lasers or The propagation of paraxial rays through various optical masers operating in the optical or infrared wave- structures can bedescribed by ray transfer matrices. Tlength regions usually appears as a beam whose Knowledge of these matrices is particularly useful as they transverse extent is large compared to the wavelength. also describe the propagation of Gaussian beams through The resonant properties of such a beam in the resonator these structures; this will be discussed in Section 3. The structure, its propagation characteristics in free space,and present section describes briefly someray concepts which its interaction behavior with various optical elements and are useful in understanding laser beams and resonators, deviceshave been studied extensively in recentyears. and lists the ray matrices of several optical systems of This paper is a review of the theory of laser beams and interest. A more detailed treatment of ray propagation resonators. Emphasis is placed on formulations and can be found in textbooks [13] and in the literature on derivations whichlead to basic understanding and on laser resonators [ 141. results which are of practical value. Historically, the subject of laser resonators had its origin when Dicke [ 11, Prokhorov [2], and Schawlow and Townes [3] independently proposed to use the Fabry- Perot interferometer as a laser resonator. The modes in such a structure, as determined by diffraction effects, were first calculated by Fox and Li [4]. Boyd and Gordon [5], and Boyd and Kogelnik [6] developed a theory for resonators with spherical mirrors and approximated the modes by wave beams. The concept of electromagnetic L..h,-J It-h, ..A wave beams was also introduced by Goubau and Schwe- I I I I I p--P;Z;;p--l I ring [7], who investigated the properties of sequences of INPUT OUTPUT PLANE PLANE lenses for the guided transmission of electromagnetic Fig. 1. Reference planes of an optical system. waves. Another treatment of wave beams was given by A typical ray path is indicated. Pierce [8]. The behavior of Gaussian laser beams as they interact with various optical structures has been analyzed by Goubau [9], Kogelnik [lo], [l 1 1, and others. 2.1 Ray Transfer Matrix The present paper summarizes the various theories and A paraxial ray in a given cross section (z=const) of an is dividedinto three parts. The first part treats the passage optical system is characterized by its distance x from the of paraxial rays through optical structures and is based optic (z) axis and by its angle or slope x’ with respect to on geometrical optics. The second part is an analysis of that axis. A typical ray path through an optical structure laser beams and resonators, taking into account the wave is shown in Fig. 1.The slope x’ of paraxial rays is assumed nature of the beams but ignoring diffraction effects due to be small. The ray path through a given structure de- to the finite size of the apertures. The third part treats the pends on the optical properties of the structure and on the resonator modes, taking into account aperture diffrac- input conditions, i.e., the position x1and the slope x; of tion effects. Whenever applicable, useful results are pre- the ray in the input plane of the system. For paraxial rays sented in the forms of formulas, tables, charts, and the corresponding output quantities xzand xi are linearly graphs. dependent on the input quantities. This is conveniently written in the matrix form Manuscript received July 12, 1966. H. Kogelnik is with Bell Telephone Laboratories, Inc., Murray Hill, N. J. T. Li is with Bell Telephone Laboratories, Inc., Holmdel, N. J. KOGELNIK AND LI: LASERBEAMS AND RESONATORS 1313

TABLE I 1 f = -- RAY TRANSFERMATRICES OF SIX ELEMENTARYOPTICAL STRUCTURES C OPTICAL SYSTEM RAYTRANSFER MATRIX D-1 hl = ___ (3) C 1 d A-1 hz = ___ C 0 I where hl and hz are the distances of the principal planes from the input and output planes as shown in Fig. 1. In Table I there are listed the ray transfer matrices of

If1 six elementary optical structures. The matrix of No. 1 I 0 describes the ray transfer over a distance d. No. 2 de- scribes the transfer of rays through a thin lens of focal I -- 1 lengthf. Here the input and output planes are immediately f to the left and right of the lens. No. 3 is a combination 12 of the fist two. It governs rays passing first over a dis- tance d and then through a thin lens. If the sequence is reversed the diagonal elements are interchanged. The matrix of No. 4 describes the rays passing through two 1 d structures of the No. 3 type. It is obtained by matrix multiplication. The ray transfer matrix for a lenslike -- d r '-T medium of length d is given in No. 5. In this medium the I 2 refractive index varies quadratically with the distance r from the optic axis.

n = no - +n2r2. (4) An index variation of this kind can occur in laser crystals and in gas lenses. The matrix of a dielectric material of index n and length d is given in No. 6. The matrix is I I +zd 1-1 d dz dl +Ld,d, referred to the surrounding medium of index 1 and is fl f2 fl f, f, f2f2 flf2 computed by means of Snell's law. Comparison with No. 1 shows that for paraxial rays the effective distance is shortened by the optically denser material, while, as is well known, the "optical distance" is lengthened.

2.2 Periodic Sequences Light rays that bounce back and forth between the spherical mirrors of a laser resonator experience a periodic focusing action. The effect on the rays is the same as in a periodic sequence of lenses [15] which can be used as an optical transmission line. A periodic sequence of identical I d/n optical systems is schematicallyindicated in Fig. 2. A singleelement of the sequenceis characterized by its 0 1 ABCD matrix. The ray transfer through n consecutive

1 2 elements of the sequence is described by the nth power of this matrix. This can be evaluated by means of Sylves- ter's theorem where the slopes are measured positive as indicated in the A B* 1 figure. The ABCD matrix is called the ray transfer matrix. -- Its determinant is generally unity IC Dl - sin@

AD - BC = 1. (2) (5) B sin n@ The matrix elements are related tb the focal length f of A sin nO - sin(n - l)@ the system and to the location of the principal planes by C sin& D sin nO - sin(n - l)@ 1314 PROCEEDINGS OF THE IEEE OCTOBER

where cos 0 = +(A + 0). Periodic sequences can be classified as either stable or unstable. Sequences are stable when the trace (A+D) obeys the inequality -1 < $(A + 0)< 1. (7) Inspection of (5) shows that rays passing through a stable sequence are periodicallyrefocused. For unstable sys- tems, the trigonometric functions in that equation be- come hyperbolic functions, which indicates that the rays f2 f2 become more and more dispersed the further they pass Rt =2fj , R~=2f2 through the sequence. Fig. 3. Spherical-mirrorresonator and the equivalent sequence of lenses.

I I

Fig. 2. Periodic sequence of identical systems, each characterized by its ABCD matrix.

2.3 Stability of Laser Resonators A laser resonator with spherical mirrors of unequal curvature is a typical example of a periodic sequence that can be either stable or unstable [6].In Fig. 3 such a resonator isshown together with its dual, whichis a sequence of lenses. The ray paths through the two struc- tures are the same, except that the ray pattern is folded in the resonator and unfolded in the lens sequence. The focal lengths fl and f2of the lenses are the same as the focal lengths of the mirrors, i.e., they are determined by the Fig. 4. Stabilitydiagram. Unstable resonator systems lie in shaded regions. radii of curvature R1 and R2 of the mirrors cfi= R,/2, f2=R2/2). The lens spacings are the same as the mirror spacing d. One can choose, as an element of the peri- 3. WAVEANALYSIS OF BEAMSAND RESONATORS odic sequence, a spacing followed byone lens plus another In this section the wave nature of laser beams is taken spacing followed by the second lens. The ABCD matrix into account, but diffraction effects due to the finite size of such an element is given in No. 4 of Table I. From this of apertures are neglected. The latter will be discussed in one can obtain the trace, and write the stability condition Section 4. The results derived here are applicable to (7) in the form optical systems with“large apertures,” i.e., with apertures that intercept only a negligible portion of the beam power. A theory of light beams or “beam waves” of this kind was o< 1-- ( first given by Boyd and Gordon [5] and by Goubau and Schwering [7]. The present discussion followsan analysis To show graphically which type of resonator is stable given in [ 1 11. and which is unstable, it is useful to plot a stability dia- gram on which each resonator type is represented by a 3.1 Approximate Solution of the Wave Equation point. This is shown in Fig. 4 where the parameters d/Rl Laser beams are similar in many respects to plane and d/R2 are drawn as the coordinate axes; unstable waves; however, their intensity distributions are not uni- systems are represented by points in the shaded areas. form, but are concentrated near the axis of propagation Various resonator types, as characterized by the relative and their phase fronts are slightly curved. A field com- positions of the centers of curvature of the mirrors, are ponent or potential u of the coherent light satisfies the indicated in the appropriate regions of the diagram. Also scalar wave equation entered as alternate coordinate axes are the parameters gl V2u k2u = 0 (9) and g2 which play an important role in the diffraction + theory of resonators (see Section 4). where k = 2r/X is the propagation constant in the medium. 1966 KOGELNIK AND LI: LASER BEAMS AND RESONATORS 1315

For light traveling in the z direction one writes u = $b, Y, 2) exP(-M ( 10) I‘ where $ is a slowlyvarying complex function which represents the differencesbetween a laser beam and a plane wave,namely: a nonuniform intensity distribu- tion, expansion of the beam with distanceof propagation, curvature of the phase front, and other differences dis- cussed below. By inserting (10) into (9) one obtains I r Fig. 5. Amplitude distribution of the fundamental beam.

When (17) is insertedin (12) the physical meaningof these where it has been assumed that $ varies so slowly with z two parameters becomes clear. One sees that R(z) is the that its second derivative &/az2 can be neglected. radius of curvature of the wavefront that intersects the The differential equation (1 1) for $ has a form similar axis at z, and ~(z)is a measure of the decrease of the to the time dependent Schriidinger equation. It is easy to field amplitude E with the distance from the axis. This see that decrease is Gaussian in form, as indicated in Fig. 5, and w is the distance at which the amplitude is l/e times that on the axis. Note that the intensity distribution is Gaus- sian in every beam cross section, and that the width of is a solution of (1 l), where that Gaussian intensityprofile changes along the axis. The parameter w is often called the beam radius or “spot T2 = x2 y2. ( 13) + size,” and 2w, the beam diameter. The parameterP(z) represents a complex phase shift which The Gaussian beam contracts to a minimum diameter is associated with the propagation of the light beam, and 2w0 at the beam waist where the phase front is plane. If q(z) is a complexbeam parameter whichdescribes the one measures z from this waist, the expansion laws for Gaussian variation in beam intensity with the distance r the beamassume a simple form. The complexbeam from the optic axis, as well as the curvature of the phase parameter at the waist is purely imaginary front which is spherical near the axis. After insertion of . rno2 (12) into (1 1) and comparing terms of equal powers in r qo =.I- one obtains the relations x q’ = 1 ( 14) and a distance z away from the waist the parameter is and

p’= --j (15) After combining (19) and (17) one equates the real and (2 imaginary parts to obtain where the prime indicates differentiation with respectto z. The integration of (14) yields q2 = q1 + z (16) and which relates the beam parameter q2 in one plane (output plane) to the parameter q1 in a second plane (input plane) R(z) = z 1 (32]. separated from the first by a distance z. [ + 3.2 Propagation Lawsfor the Fundamental Mode Figure 6 shows the expansion of the beam according to A coherent light beam with a Gaussian intensity pro- (20). The beam contour ufz)is a hyperbola with asymp- file as obtained above is not the only solution of (1 l), totes inclined to the axis at an angle but is perhaps the most important one. This beam is often x called the “fundamental mode” as compared to thehigher e=-. (22) order modes to be discussed later. Because of its impor- rno tance it is discussed here in greater detail. This is the far-field diffraction angle of the fundamental For convenience one introduces two real beam param- mode. eters R and w related to the complex parameter q by Dividing (21) by (20),one obtains the useful relation 1316 PROCEEDINGS OF THE IEEE OCTOBER

ilar properties, and they are discussed in this section. These solutions form a complete and orthogonal set of functions and are called the “modes of propagation.” Every arbitrary distribution of monochromatic light can be expanded in terms of these modes. Because of space limitations the derivation of these modes can only be sketched here. a) Modes in Cartesian Coordinates: For a system with a rectangular (x, y, z) geometry one can try a solution for (1 1) of the form

/PHASE FRONT Fig. 6. Contour of a Gaussian beam. which can be usedto express woand z in termsof w and R :

wo2 = w2 1[ 1 + (;?‘I where g is a function of x and z, and h is a function of y and z. For real g and h this postulates mode beams whose intensity patterns scale according to the width 2w(z) of a z = R/ [l +($)‘I. Gaussian beam. After inserting this trial solution into (1 1) one arrives at differential equations for g and h of the To calculate the complex phase shift a distance z away form from the waist, one inserts (19) into (15) to get d2Hm dHm -- 2x -+ 2mH, = 0. (31) P’=---j j dx2 _- (26) dx Q + j(WO2/V This is the differential equation for the Hermite poly- Integration of (26) yields the result nomial Hm(x)of order m. Equation (1 1) is satisfied if

~P(z)= ln[l - ~(XZ/W~Z)] = Indl + (XZ/WO~)~- j arc tan(hz/xwo2). (27) where and n are the (transverse) mode numbers. Note The real part of P represents a phase shift differenceCP be- m tween the Gaussian beam and an ideal plane wave, while that the same pattern scaling parameter w(z) applies to modes of all orders. the imaginary part produces an amplitude factor WO/W which givesthe expected intensity decreaseon the axis due Some Hermite polynomials of low order are to the expansion of the beam. With these results for the H,(x) = 1 fundamental Gaussian beam, (10) can be written in the H,(x) = x fonn H,(x) = 4x2 - 2 wo U(T, 2) = - H~(z)= 8x3 - 122. (33) W Expression (28) can be used as a mathematical descrip- tion of higher order light beams,if one inserts the product g.h as a factor on the right-hand side. The intensity pat- where tern in a cross section of a higher order beam is, thus, de- scribed bythe product of Hermite and Gaussianfunctions. 9 = arc tan(Xz/mo2). (29) Photographs of such mode patterns are shown in Fig. 7. They were produced as modes of oscillation in a gas laser It will be seen in Section 3.5 that Gaussian beams of this oscillator [16]. Note that the number of zeros in a mode kind are produced by many lasers that oscillate in the pattern is equal to the correspondingmode number, and fundamental mode. that thearea occupied bya mode increases withthe mode 3.3 Higher Order Modes number. The parameter R(z) in (28) is the same for all modes, In the preceding section only one solution of (1 1) was implying that the phase-front curvature is the same and discussed,i.e., a light beam with the property that its changes in the same way for modes of all orders. The intensity profile in every beam cross section is given by phase shifta, however, isa function of the mode numbers. the same function, namely, a Gaussian. The width of this One obtains Gaussian distribution changes as the beam propagates along its axis. There are other solutions of (1 1)with sim- @(m,n; z) = (m + n + 1) arc tan(k/mo2).(34) 1966 KOGELNIK ANDBEAMS LI: LASER AND RESONATORS 1317

As in the case of beams with a rectangular geometry, the beam parameters w(z) and R(z) are the same for all cylin- drical modes. The phase shift is, again, dependent on the mode numbers and is given by

@(p,I; z) = (2p + I+ 1) arc tan(Az/mo2). (39)

3.4 Beam Transformation by a Lens A lens can be used to focus a laser beamto a small spot, or to produce a beam of suitable diameter and phase- front curvature for injection into a given optical structure. An ideal lens leaves the transverse field distribution of a beam mode unchanged, i.e., an incoming fundamental TEM- TEMx, Gaussian beam will emerge from the lens as a funda- mental beam, and a higher order mode remains a mode of the same order after passing through the lens. However, a lens does change the beam parameters R(z) and 42). As these two parameters are the same for modes of all orders, the following discussion isvalid for all orders; the relationship between the parameters of an incoming beam (labeled here with the index 1) and the parameters of the corresponding outgoing beam (index 2) is studiedin detail. An ideal thin lens of focal lengthftransforms an incom- ing spherical .wave with a radius R1 immediately to the left of the lens into a spherical wave with the radius R2 immediately to the right of it, where Fig. 7. Mode patterns of a gas laser oscil- lator (rectangular symmetry). 111 -=---. (40) R2 R1 f This means that the phase velocity increases withincreas- Figure 8 illustrates this situation. The radius of curvature ing mode number. In resonators this leads to differences is taken to be positive if the wavefrontis convex as in the resonant frequencies of the various modes of oscil- viewed from z= The lens transforms the phase fronts lation. co . of laser beamsin eactly the same way as those of spherical b) Modes in Cylindrical Coordinates:For a system with waves. As the diameter of a beam is the same immediately a cylindrical (r, 4, z) geometry one uses a trial solution to the left and to the right of a thin lens, the q-parameters for (1 1) of the form of the incoming and outgoing beams are related by 111 -=---> (41) Q2 91 f After some calculation one finds where the q's are measured at the lens. If q1 and q2 are measured at distances dl and d2from the lens as indicated g = (1/2;)'..: (2 2) in Fig. 9, the relation between them becomes where is a generalized Laguerre polynomial, and (1 - d2/.f)q1+ (dl + dz - dde/f) 15,' p (22 = . (42) and 1 are the radial and angular mode numbers. LpL(x) -(q1/!f) + (1 - dl/f) obeys the differential equation This formula is derived using (16)and (41). d2L,' dL,' More complicated optical structures, such as gas lenses, X- + (2 + 1 - 2) -+ pL,' = 0. (37) combinations of lenses, or thick lenses, can be thought of dx2 dx dx2 as composed of a series of thin lenses at various spacings. Some polynomials of low order are Repeated application of (16) and (41) is, therefore, suffi- cient to calculate the effect of complicated structures on Lo'(x) = 1 the propagation of laser beams. If the ABCD matrix for the transfer of paraxial rays through the structure is Ll'(2) = 1 - 2 I + known, the q parameter of the output beam can be cal- Ls'(2) $(I?+ 1)(Z + 2) - (1 + 2)~+ 4~~. (38) culated from 1318 PROCEEDINGSOCTOBER OF THE IEEE

trip. If the complex beam parameter is given by ql, im- mediately to the right of a particular lens, the beam parameter q2, immediately to the right of the next lens, can becalculated by meansof (1 6) and (41) as

f Fig. 8. Transformation of wavefronts by a thin lens. Self-consistency requires that ql=q2=q, which leads to a quadratic equation for the beam parameter q at the lenses (or at the mirrors of the resonator): 111 -+-+-=oo. (45) q2 fq fd The roots of this equation are I f I 9, 92 1 1-11 Fig. 9. Distances and parameters for a - - - (+) j - - beam transformed by a thin lens. _- (- (46) P 2f fd 4f2 where only the root that yields a real beamwidth is used, (43) (Note thatone getsa real beamwidth for stable resonators only J This is a generalized form of (42) and has been called the From (46) one obtains immediately the real beam ABCD law [lo]. The matrices of several opticalstructures parameters defined in (17). One seesthat R is equal to the are given in Section 11. The ABCD law follows from the radius of curvature of the mirrors, which means that the analogy between the laws for laser beams and the laws mirror surfaces are coincident with the phase fronts of obeyed by the spherical waves in geometrical optics. The the resonator modes. The width 2w of the fundamental radius of the spherical waves R obeys laws of the same mode is givenby form as (16) and (41) for the complex beam parameter q. A more detailed discussionof this analogy is givenin [ 11 1. 3.5 Laser Resonators (Infinite Aperture) The most commonly used laser resonators are com- To calculate the beam radius wo in the center of the reso- posed of two spherical (or flat) mirrors facing each other. nator where the phase front is plane, one uses (23) with The stability of such “open” resonators has been discussed z=d/2 and gets in Section2 in termsof paraxial rays. To study the modes x -- of laser resonators one has to take account of their wave wo2 = - dd(2R - d). (48) nature, and this is done here by studying wave beams of 2a the kind discussed aboveas they propagate back and forth The beam parameters R and w describe the modes of between the mirrors. As aperture diffraction effects are all orders. But the phase velocities are different for the neglected throughout this section, the present discussion different orders, that the resonant conditions depend applies only to stable resonators with mirror apertures so on the modenumbers. Resonance occurs when the phase that arelarge compared to the spot size of the beams. shift from one mirror to the other is a multiple of a. A mode of a resonator is defined as a self-consistent Using (28) and (34) this condition can be written as field configuragion. If a mode can be represented by a wave beam propagating back and forth between the kd - 2(m + n + 1) arc tan(Xd/2rwo2) = r(q + 1) (49) mirrors, the beam parameters must be the same after one complete return trip of the beam. This condition is used where q is the number of nodes of the axial standing-wave to calculate the mode parameters. As the beam that repre- pattern (the number of half wavelengths is q+l),l and m sents a mode travelsin both directions between the mirrors and n are the rectangular mode numbers defined in Sec- it forms the axial standing-wave pattern that is expected tion 3.3. For the modes of circular geometry one obtains for a resonator mode. a similar condition where (2p+Z+ 1) replaces (m+n+ 1). A laser resonator with mirrors of equal curvature is The fundamental beat frequency yo, i.e., the frequency shown in Fig. 10 together with the equivalent unfolded spacingbetween successive lo