524 J. Opt. Soc. Am. B/Vol. 17, No. 4 /April 2000 McDonald et al.

Kaleidoscope laser

G. S. McDonald and G. P. Karman

Huygens Laboratory, Leiden University, Box 9504, 2300 RA Leiden, The Netherlands

G. H. C. New

Laser Optics and Spectroscopy Group, Department of Physics, Imperial College, London SW7 2BZ, UK

J. P. Woerdman

Huygens Laboratory, Leiden University, Box 9504, 2300 RA Leiden, The Netherlands

Received June 11, 1999

We report the first calculations of mode patterns of unstable-cavity lasers with truly two-dimensional trans-verse geometries. A detailed account of numerical techniques, incorporating a nonorthogonal beam-propagation method, and results for cavities with a range of transverse symmetries, such as regular polygonaland rhomboid, are presented. In view of the beautiful complexity of the eigenmodes predicted, a novel kalei-doscope laser is proposed. © 2000 Optical Society of America [S0740-3224(00)00803-1]

OCIS codes: 030.4070, 050.1940, 110.1220, 120.2040, 140.3410.

1. INTRODUCTIONHermite–Gaussian modes have formed the basis for theanalysis and understanding of stable-cavity lasers formany decades. These modes tend to be highly paraxialand have low round-trip loss but often do not exploit thefull volume of the gain medium. On the other hand,unstable-cavity lasers utilize diverging light wave frontsfor high-gain extraction, leading to strong periodic aper-turing effects and high-loss modes.1–3 The transversestructure of these modes is determined, to a large extent,by the detailed symmetry of the aperturing element, re-sulting in mode patterns that are radically different thanthose of conventional stable-cavity lasers.

To date, only unstable-cavity modes for one-dimensional (1D) transverse geometries, and for rectan-gular and circular geometries in two dimensions, havebeen studied.3–12 This is because each of these configu-rations can be investigated by use of a 1D transverse cal-culation. We have undertaken, to our knowledge, thefirst investigation of the eigenmode patterns of unstable-cavity lasers with truly two-dimensional (2D) aperturesymmetries, concentrating specifically on resonators withrhombus and N-sided regular polygon aperture shapes.In this paper we present an account of the numericaltechniques that we have developed for the calculation ofthese fully 2D mode profiles. Results for cavities withtriangular, rhomboid, pentagonal, hexagonal, and octago-nal aperture symmetries are reported. The profound,and indeed fractal, complexity of the mode patterns leadsus to christen such devices ‘‘kaleidoscope lasers.’’

2. THEORYThe simple resonator configuration that we model issketched in Fig. 1(a) and consists of a convex mirror (focal

0740-3224/2000/040524-06$15.00 ©

length f ) and a concave mirror (focal length fv) that areseparated by a free space path of length L. For definite-ness we model a confocal unstable resonator (L [ Lc5 fv 2 f ) in which periodic aperturing of the circulat-ing field occurs at the convex mirror. This aperturingmay arise from the shape of the mirror itself or, more con-veniently, be due to an aperture placed directly againstthis mirror. Figure 1(b) shows the aperture shapes thatwe consider, and it defines their relative sizes through theparameter a. The eigenmodes of the cavity are param-etrized by only two variables, the round-trip magnifica-tion M and the equivalent Fresnel number Neq given, forthis particular configuration, by M [ Mc 5 fv /f and Neq5 (M 2 1)a2/2lL. The same results can also be ap-plied to different experimental configurations, albeit withdifferent definitions of M and Neq . In general, the mag-nification is given by3 M 5 m 1 (m2 2 1)1/2 (for a posi-tive branch resonator), where12 m 5 (Mc 1 1/Mc2 d 2/ffv)/2 and a finite degree of nonconfocality has beenintroduced through L 5 Lc 1 d. The equivalent Fresnelnumber is given by3 Neq 5 a2(M2 2 1)/2lBM, where12

B 5 Lc(1 1 Mc)/Mc 1 (2 f 2 d )d/fv for aperturing atthe convex mirror and B 5 Lc(1 1 Mc) 1 (2 fv 1 d )d/fwhen the aperture is at the other end of the resonator.

For rectangular and circular aperturing, semianalyti-cal techniques are useful for determining an approxima-tion to the mode profile. One of the more commonly usedmethods is the virtual source technique6 in which the cav-ity is unfolded into a sequence of effective apertures.The modal pattern is then constructed by taking aweighted sum of edge waves from each aperture and anundiffracted plane-wave component. At the heart ofsuch methods is knowledge of the diffraction patternsthat result from particular aperture shapes. However, itis quite surprising that there is almost no published workdescribing Fresnel diffraction at apertures with nonor-

2000 Optical Society of America

McDonald et al. Vol. 17, No. 4 /April 2000 /J. Opt. Soc. Am. B 525

thogonal edges. A notable exception to this is Newton’stwo-knife experiment,13 which has only recently attractedsome deserved attention.14 Newton reported that hyper-bolic fringes appear in the diffraction patterns from thinwedge apertures as one moves toward the far-field do-main. Nevertheless, the complexity of the apertureshapes considered here, and the absence of appropriateanalytical work, render the generalization of existingsemianalytical techniques quite involved; we thereforeadopt a fully numerical approach.

In the numerical scheme used, the intracavity field isinitialized with a plane wave and then evolved by circu-lation around the resonator by repeated application of around-trip ABCD matrix and aperturing. This mappingis iterated until the transverse profile converges, withsufficient accuracy, to the lowest-loss mode. Resolutionof the aperture edges and the resulting diffraction pat-terns demands relatively dense discretization of thetransverse plane. To minimize memory and run-time re-quirements of the simulations, we use a transverse basisthat most closely matches the symmetry of the apertureshape. Figure 2 shows the geometry of the transversecomputational axes in the most general case. For eachtransverse point, orthogonal and nonorthogonal coordi-nates are defined, (x1 , y1) and ( p1 , q1), respectively.Denoting the angle between the p –q axes by F, we seethat these sets of coordinates are related through p15 x1 2 y1 /tan F and q1 5 y1 /sin F.

To evolve the field profile over each cavity round trip,we find it is advantageous to transform the actual confo-cal cavity to an equivalent lens guide, consisting of a se-quence of concave lenses and apertures [Fig. 1(c)]. Re-quiring that this lens guide has the same value of Neq asthe original cavity, we find the focal length of each lens tobe fe 5 2L(M 1 1)/(M 2 1)2. The mode profile imme-

Fig. 1. Bare cavity configuration of the kaleidoscope laser. (a)Schematic picture of the unstable cavity, consisting of a convexmirror (focal length f ) and a concave mirror (focal length fv)separated by a distance L. The circulating field is apertured oneach round trip by one of the mirrors or, equivalently, by an ap-erture placed directly against that mirror. (b) The transverseshapes of the apertures considered are plotted. (c) In the mod-eling we transform the above system to the equivalent lens guidesketched, which is a series of diverging lenses (focal length fe anda distance Le apart) and apertures adjacent to each of theselenses.

diately before the concave lens in the equivalent guideUe1( p, q) is related to the associated profile U( p, q) ap-proaching the convex mirror through

Ue1~ p, q ! 5 U~ p, q !

3 exp@2ipb~ p2 1 q2 1 2pq cos F!/a2#,

(1)

where b 5 (a2/lL)(M 2 1)/(M 1 1). The defocusingand aperturing actions of the lens then transform thefield to Ue2( p, q), where

Ue2~ p, q ! 5 Ue1~ p, q ! exp@2ipb~M 2 1 !

3 ~ p2 1 q2 1 2pq cos F!/a2# (2)

inside the aperture and is zero otherwise. Free spacepropagation of Ue2( p, q) over a distance Le 5 L(11 M)/M then yields Ue3( p, q) and completes the schemefor a single cavity transit.

The propagation step is implemented in the spatial fre-quency domain and thus involves a pair of fast Fouriertransforms (FFT’s). Although there exists a computa-tionally faster scheme in which a cavity transit can bewritten in terms of only one FFT,3 we find that this latterscheme imposes an awkward constraint on the spatialgrid. In 1D transverse calculations the effect of this con-straint can be minimized by use of a very large number oftransverse points, but in the 2D case this is not possibleand the constraint ultimately limits the accuracy of thesolution. We also find that the double FFT method per-mits a straightforward accommodation of a nonorthogo-nal transverse basis.

In terms of the Cartesian basis x –y and the longitudi-nal coordinate z, propagation of the field involves the so-lution of

]U~x, y !

]z5

i

2k S ]2

]x2 1]2

]y2DU~x, y !, (3)

where k is the free space propagation constant. In thenonorthogonal basis the field evolution is governed by

Fig. 2. Orthogonal (x –y) and nonorthogonal ( p –q) axes in thetransverse plane. An arbitrary point in this space, which is la-beled 1, has two pairs of coordinates, (x1 , y1) and ( p1 , q1).

526 J. Opt. Soc. Am. B/Vol. 17, No. 4 /April 2000 McDonald et al.

]U~ p, q !

]z5

i

2k sin2 F

3 S ]2

]p2 1]2

]q2 2 2 cos F]

]p

]

]q DU~ p, q !.

(4)

To employ FFT’s for the solution of Eq. (4), we define aspectral function U(K1 , K2) such that

U~ p, q ! 5 E2`

` E2`

`

U~K1 , K2!

3 exp@i~K1p 1 K2q !#dK1dK2 , (5)

whereby free space propagation is effected by15

Ue3~K1 , K2!

5 Ue2~K1 , K2!

3 expF2iLe

2k sin2 F~K1

2 1 K22 2 2 cos F K1K2!G .

(6)

Once a particular mode is established, the field profilessatisfy the self-consistency condition Ue3( p, q)5 gUe1( p, q), where g is the (complex) mode eigenvalue.Equation (1) can then be inverted to determine the modeprofile U( p, q) in the actual cavity.

3. RESULTSIn recent experiments,9–11,16,17 unstable cavities with lowNeq values and linear magnifications between 1 and 2have been studied. We have concentrated our investiga-tions of eigenmode patterns in this regime. To facilitatea comparison between transverse modes with differentaperture shapes, we present results for a single value ofmagnification,17 M 5 1.3.

First, we checked that the full 2D code gave resultsconsistent with those of 1D codes for apertures withsquare and circular shapes. Second, since the transversegrid cannot sample individual edges of all polygon aper-tures in exactly the same fashion, checking the overallsymmetry of each modal pattern proved to be a good testof whether we were using a sufficient number of trans-verse points. For M 5 1.3 we found that 256 3 256transverse points were required for Neq up to 2.0 and thata grid density of 512 3 512 points was needed for highervalues (up to Neq ' 4). Higher-order corrections in thediscretizations, which weight individual sampling pointsalong the edges of each aperture, have also been em-ployed, and the results obtained were checked againstthose presented here. Finally, the value of the Peter-mann K factor (the so-called excess quantum-noise factor)for any particular eigenmode pattern depends very sensi-tively on the detail of its transverse structure, since the Kfactor is the inverse of an almost vanishing modal overlapintegral.7 We used this quantity as a diagnostic to testthe accuracy of 2D patterns using corresponding resultsfrom 1D calculations and to monitor the convergence ofthe field profiles. In common with studies of the 1D

problem,12 we find that the number of cavity transits forconvergence depends on the precise value of Neq consid-ered and whether there is a nearby mode crossing (a pointat which two eigenmodes of the cavity become degener-ate).

Most graphical packages require data defined on a Car-tesian basis, so it is convenient to transform the modalprofiles U( p, q) back to a rectilinear representationU(x, y). To undertake this transformation using con-ventional interpolation techniques would be rather in-volved and may introduce inaccuracies. As an alterna-tive, we use the Fourier transform shift property andperform the required transformation in the frequency do-main. For each q, we define a function U(K) such that

U~ p ! 5 E2`

`

U~K !exp~iKp !dK, (7)

whereby x 5 p 1 q cos F yields

U~x ! 5 E2`

`

U~K !exp~iKq cos F!exp~iKp !dK. (8)

Thus in practice, the reconstruction of U(x, y) involvesscanning through q and, on each line of data, performingan FFT, applying a phase ramp given by Kq cos F, andthen using another FFT.

Column 1 of Fig. 3 shows results for triangular aper-turing. The transverse intensity profiles of the modeshave increasing complexity as Neq is increased from 0.5 to4.0 (moving downward in the column). The values of Neqused were chosen to give an overview of the distinct pat-terns that arise as the aperture size is slowly increased.The choice of Neq values for other aperture shapes ismade on the same basis. For Neq 5 0.5, a single spotwith overall triangular symmetry is evident. Closer in-spection of this pattern reveals the presence of smoothcurves connecting the bright regions that arise from theaperture corners. Since this modal pattern may be con-sidered to be a superposition of aperture diffraction pat-terns that extend into the far field, we believe that thesesmooth contours are related to a generalization of New-ton’s result in which each aperture of the unfolded systemgives rise to a combination of three wedge-type diffractionprofiles.

The second profile in column 1 also displays a patternthat has bright spots from the aperture corners. Thischaracteristic was found for all polygon mode profileswhen Neq ' 1. As Neq is increased to 3.0 (the third tri-angular profile) a different type of eigenmode pattern isfound that is composed of bright sides enclosing a nestedstructure. This profile is in fact similar to the Fresneldiffraction pattern of the closest aperture of the unfoldedsystem. In the final plot of column 1, where Neq 5 4.0,the mode characteristics have changed yet again, and thepattern exhibits bright spots emanating from the aper-ture edges.

Our nonorthogonal computational technique is ideallysuited to the modeling of configurations with rhomboidaperturing. In column 2 of Fig. 3 we show eigenmodeprofiles when the cavity aperture has the shape of a 60°rhombus. For this configuration we show the cases Neq5 0.5, 1.0, 1.5, and 3.0. Making a comparison with themodes for triangular aperturing, we clearly see that the

McDonald et al. Vol. 17, No. 4 /April 2000 /J. Opt. Soc. Am. B 527

same numerical value of Neq leads to much more devel-oped modal structure. In fact, the parameter a is definedsuch that, on the one hand, the rhombus aperture hastwice the area of the corresponding triangular aperturebut, on the other hand, has the same area as the corre-sponding square aperture. This highlights the somewhatarbitrariness in the definition of Neq . We define a suchthat there is consistency with square and circular aper-tures, since in these cases Neq is formally defined. In do-ing so we regard a circle as a regular polygon with an in-finite number of sides.

The equivalent Fresnel number actually has its originin the relative phase of (magnifying and demagnifying)geometrical eigenmodes of the cavity and is distinct fromthe collimated Fresnel number Nc 5 Ma2/Bl. This lat-ter parameter is more closely related to the number of dif-fraction ripples across the opening of the aperture3 and isgenerally larger than Neq . This fact explains the rela-tive complexity of modal patterns for such low values ofNeq . The rhomboid mode profiles for the lowest values ofNeq considered confirm the generality of the bright-corners characteristic, and, with increasing Neq , themode can be seen to cycle through patterns with bright

sides and nested structures. The final frame in column 2displays a mode that is reminiscent of those found forsquare aperturing,11 even though the overall symmetry isrhomboid.

Eigenmode profiles for other types of polygonal aper-turing are shown in columns 3–5 of Fig. 3, where thetransverse symmetry of the cavities are pentagonal, hex-agonal, and octagonal, respectively. In each column asimilar sequence of modal characteristics, as describedearlier, is evident. For low Neq the modes become pro-gressively more circular as the number of aperture sidesis increased. For higher values of Neq , it is interesting tonote that the results are similar to snowflake patterns.This leads one to enquire whether the eigenmodes havean underlying fractal nature. Two of us have recentlyundertaken calculations of the fractal dimension of modeprofiles of 1D and radially symmetric cavities.18 In thesestudies the dimension of the modes was found to lie be-tween 1 and 2, confirming their fractal character. A sim-plified physical picture, involving the repeated magnifica-tion of the modal pattern on subsequent round trips, wasalso presented to support their findings. We note thatthis interpretation is essentially independent of the aper-

Fig. 3. Cross section of the transverse-mode profiles of the kaleidoscope laser. The defining aperture of the cavity has a different shapein each column of this figure. The profiles generally exhibit increasing complexity as the equivalent Fresnel number is increased (mov-ing downward through each column). In column 1 the aperture has the shape of an equilateral triangle (Neq 5 0.5, 1.5, 3.0, and 4.0).Columns 2–5 are for rhomboid (Neq 5 0.5, 1.0, 1.5, and 3.0), pentagonal (Neq 5 0.5, 1.5, 2.0, and 4.0), hexagonal (Neq 5 0.5, 1.5, 3.0, and4.0), and octagonal aperturing (Neq 5 0.5, 1.0, 3.0, and 4.0), respectively. The colors yellow, red, green, blue, and black are used torepresent different (decreasing) levels of light intensity.

528 J. Opt. Soc. Am. B/Vol. 17, No. 4 /April 2000 McDonald et al.

ture shape. On this basis the modes of cavities with po-lygonal and rhomboid symmetries are also expected to befractal.

Since the fractal dimensions of 1D and square geom-etry modes are identical, we suspect that the modal di-mension for each polygonal configuration also lies be-tween 1 and 2. It is therefore not surprising that anyunderlying fractal structure can be visualized in thetransverse plane since this surface represents an embed-ding dimension of 2 (which is larger than the dimension ofthe pattern). To obtain accurate estimates of fractal di-mension, one generally needs many thousands of distinctsampling points. Although this requirement can easilybe met in 1D calculations, the 2D spectral method that weemploy dictates that the modal structure occupies only aportion of the whole transverse grid, and this featuregreatly reduces the number of distinct and useful datapoints that are available for quantitative dimension esti-mates.

4. KALEIDOSCOPE LASERSUnstable-cavity lasers are usually associated with high-power applications. Here, we have proposed a novel ka-leidoscope laser for which, particularly in the high Neq re-gime, dynamic tuning between modes and real-timeadjustments of the shape and size of the aperture wouldresult in stunning visual complexity. A prototype kalei-doscope laser17 has actually been constructed by our-selves using a miniature He–Xe laser. A simple tech-nique, based on modifying commercial iris diaphragms,has been developed that yields the required (variable-sizeand variable-shape) apertures. Using the most sensitivediagnostic available (the Petermann K factor), we havefound excellent agreement between theoretical predic-tions and experimental results.17 Noting that theFresnel number of a cavity depends on the aspect ratio(and not on the absolute size) of the system, the simplicityand the high-gain characteristics of the kaleidoscope lasergeometry make it particularly suited to further miniatur-ization; relatively inexpensive and compact devices maybe envisaged.

The original kaleidoscope device, invented by Sir DavidBrewster in 1816, essentially consists of just two planemirrors that share a common edge.19 The angle u be-tween these mirrors defines the N-fold polygonal symme-try of the patterns produced through the relation N5 2p/u. A formal connection with the kaleidoscope la-ser can be established by employing Babinet’s principleand decomposing an N-sided polygon aperture of the un-stable cavity into N congruent isosceles triangles. Theinternal angle of each of those component triangles,2p/N, then corresponds to the angle between the mirrorsof Brewster’s kaleidoscope.

This suggests that kaleidoscope mirrors themselvescould be employed as an optical element and could giverise to novel laser designs. They could impose polygonaltransverse boundary conditions and thus replace morecomplex aperture assemblies. The result would be a la-ser whose modes could be easily scanned through an infi-nite number of possible polygonal symmetries (by vari-

ance of the angle u). Alternatively, other fixed-symmetrysystems could be built; gain rods whose longitudinal sur-faces were polished or more exotic lasers that exploit athree-mirror kaleidoscope tube are possible.

5. SUMMARYWe have reported the first calculations, to our knowledge,of unstable-cavity eigenmodes for configurations withtruly 2D transverse geometries. Modal patterns for ap-erturing with triangular, rhomboid, pentagonal, hexago-nal, and octagonal symmetries have been mapped out as afunction of the equivalent Fresnel number of the cavity.In contrast to the lowest-loss modes of stable-cavity la-sers, a rich and beautiful complexity is revealed that sug-gests such systems be termed kaleidoscope lasers.

The use of a nonorthogonal computational grid madean important contribution to the analysis. Although thistechnique 15 has received little attention since it was firstproposed in 1993, it is clearly a very powerful and flexiblemethod with a wide range of potential applications.These include the modeling of tapered optical structuresand, more generally, two-and three-dimensional propaga-tion problems that involve nonorthogonal features andwhere efficient discretization is a necessity. Moreover,there is almost no computational overhead arising fromundertaking propagation calculations in a nonorthogonalframe.

We plan further theoretical and experimental investi-gations of kaleidoscope lasers, including alternative cav-ity designs, the study of more general aperture shapes,and considerations of higher-order modes.12 This willlend more insight into general Fresnel diffraction prob-lems and the character of excess quantum noise, as wellas permit a fuller exploration of the diversity of modalpatterns that are possible.

ACKNOWLEDGMENTSThe authors acknowledge numerous invaluable discus-sions with M. van Eijkelenborg and T.I.J. Albaho, as wellas interaction over many years with A. E. Siegman. Theproject was supported by the Foundation for Fundamen-tal Research of Matter of the Netherlands, ESPRIT con-tract 20029 (ACQUIRE), TMR contract ERB4061PL95-1021 (Micro-Lasers and Cavity QED) and the UKEngineering and Physical Sciences Research Council un-der grant GR/L90583.

Please send correspondence to G. H. C. New, Laser Op-tics and Spectroscopy Group, Department of Physics, Im-perial College, London SW7 2BZ, UK. Telephone:144 20 7594 7791. Fax: 144 20 7823 8376. E-mail:g.new@ic.ac.uk.

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