theory of the unstable bessel resonator

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Hernández-Aranda et al. Vol. 22, No. 9 /September 2005 /J. Opt. Soc. Am. A 1909

Theory of the unstable Bessel resonator

Raúl I. Hernández-Aranda

Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey, México 64849

Sabino Chávez-Cerda

Instituto Nacional de Astrofísica, Optica y Electrónica, Apartado Postal 51/216, Puebla, México 72000

Julio C. Gutiérrez-Vega

Photonics and Mathematical Optics Group, Technológico de Monterrey, Monterrey, México 64849

Received January 5, 2005; revised manuscript received March 2, 2005; accepted March 7, 2005

A rigorous analysis of the unstable Bessel resonator with convex output coupler is presented. The Huygens–Fresnel self-consistency equation is solved to extract the first eigenmodes and eigenvalues of the cavity, takinginto account the finite apertures of the mirrors. Attention was directed to the dependence of the output trans-verse profiles; the losses; and the modal-frequency changes on the curvature of the output coupler, the cavitylength, and the angle of the axicon. Our analysis revealed that while the stable Bessel resonator retains aGaussian radial modulation on the Bessel rings, the unstable configuration exhibits a more uniform amplitudemodulation that produces output profiles more similar to ideal Bessel beams. The unstable cavity also pos-sesses higher-mode discrimination in favor of the fundamental mode than does the stable configuration.© 2005 Optical Society of America

OCIS codes: 140.3300, 140.3410, 140.4780.

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. INTRODUCTIONn recent years, several ways to produce Bessel beamsave been demonstrated; for instance, passive optical sys-ems fed by laser light using a ring aperture and a posi-ive lens,1,2 refractive or diffractive axicons,3,4 holographicethods,5,6 Fabry–Perot interferometers,7 or diffractive

hase elements.8 Active schemes to produce Bessel-typeodes in laser resonators have also been proposed: for ex-

mple, arrangements based on annular intracavitylements,9 output mirrors with annular apertures to pro-uce conical fields,10 graded-phase mirrors,11 and diode-umped Nd:YAG lasers.12

Axicon-based resonators supporting Bessel beams wereroposed independently by Rogel-Salazar et al.13 andhilo et al.14 in 2001. This configuration has the advan-

age that it does not require intracavity optics or specialhapes of the active medium. In 2003, Gutiérrez-Vega etl.15 continued the exploration of the axicon-based reso-ator properties, extending the analysis to concave–pherical mirrors and developing a formal geometric andave analysis of the performance of the bare cavity. Thexicon-based resonator with a convex output coupler op-rating in an unstable regime was proposed by Tsangarist al.16 in 2003. The scope of this initial work was re-tricted to the calculation of the output shape of the domi-ant Bessel mode using the classical iterative Fox–Lilgorithm.17,18

In this paper we present a rigorous analysis of thexicon-based unstable Bessel resonator (UBR). The con-guration we consider presents two important modifica-ions with respect to the conventional stable Bessel reso-ator (SBR) studied in Ref. 15, namely, the output mirror

1084-7529/05/091909-9/$15.00 © 2

s now convex spherical, and the aperture ratio betweenhe output coupler and the conical mirror has been re-uced from unity to one half. The Huygens–Fresnel inte-ral self-consistency equation for the cavity is efficientlyolved using a matrix technique. This method has twoain advantages: It extracts the lowest N modes and

heir eigenvalues at one time, and its accuracy is deter-ined by the order N of the matrix. In our analysis we

ccount for the effects of the finite aperture size of theavity mirrors. Additionally, the Fox–Li algorithm is em-loyed to describe the three-dimensional intracavity fieldistribution of the dominant mode and also to confirm theesults obtained with the Huygens–Fresnel integral ap-roach.We study the effects of varying the curvature of the out-

ut mirror and the cavity length on the output transverserofile, the loss, and the resonance frequency shift of theower modes. The analysis reveals that both the spherical

irror and the cavity length can be modified to minimizehe losses due to diffraction and to adjust the radialodulation of the Bessel rings of the output field. As seen

n our results, the transition between the SBR and theBR is characterized by a continuous change in the asso-

iated diffraction losses and frequency shifts. Higher-rder-mode crossings in the eigenvalue spectrum of theesonator are also present for several values of the radiusf curvature of the output coupler.

. RESONATOR CONFIGURATIONhe configuration of the UBR consists of a reflective coni-al mirror with characteristic angle � and a convex–
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1910 J. Opt. Soc. Am. A/Vol. 22, No. 9 /September 2005 Hernández-Aranda et al.

pherical output mirror separated a distance L as de-icted in Fig. 1(a). For the sake of generality, we willssume that the transverse radii of the conical mirror �a1�nd the output mirror �a2� are independent and thus canake different values. The conical mirror transforms anncident plane wave into a converging conical wave. Therajectory of a horizontal input ray, except a ray thatrosses through the center of the conical mirror, is alwayshanged at a constant angle 2�0 toward the optical axissee Fig. 1(a)]. In practice, the reflective conical mirroran also be constructed with a refractive axicon with in-ex of refraction n and wedge angle � backed by a per-ectly reflecting plane mirror, as shown in Fig. 1(b). Inhis case the conical angle �0 is related to the axicon pa-ameters by �0=arcsin�n sin ��−��n−1��, where themall-angle approximation sin �� has been used. Inontrast to the procedure in unstable resonators with twopherical mirrors, the laser output in the UBR is taken asdiffraction-coupled beam passing through (rather than

round) the output mirror.The cavity length is chosen to be

L =a1

2 tan �0�

a1

2�0�1�

ig. 1. Design of the resonator with (a) reflective and (b) refrac-ive axicon. The convex–spherical mirror is placed at a distance Lrom the axicon. (c) Lens-guide-equivalent resonator and self-eproducibility condition for ray trajectories after one and twoound trips. RP stands for reference plane.

o satisfy the requirement that the UBR reduces to thedeal SBR13 when the spherical output mirror becomes aat mirror. The radius of curvature R of the output mirror

s then a free parameter that can be used to modify theiffraction properties of the cavity.The case R�0 corresponds to a concave output coupler,

or which the resonator is geometrically stable and theeld at the output plane can be approximated by an lthrder Bessel–Gauss beam,15

u�r� = Jl�ktr�exp�−r2

w2�exp�i�l� + ��, �2�

here the waist w is given by w2=w02�1+ �L /zR�2� with

0= �2zR /k�1/2, zR= �L�R−L��1/2 is the Rayleigh range ofhe equivalent Gaussian beam, �=kr2 /2R is the phase ofhe spherical wave front at the output mirror, and ktk sin �0 is the transverse wave number. In the limit of aat output coupler �R→��, Bessel–Gauss modes reduce toessel beams Jl�kt��exp�il��. In real resonators the finitextent of the mirrors slightly modifies the transversehape of the ideal Bessel modes such that, even in thease of a plane output mirror, the resonating modes areodulated by a bell-shaped envelope.Output fields with specific transverse features can be

enerated by properly choosing the geometric parametersf the UBR in Fig. 1. As we will confirm in the remainderf the paper, under the paraxial regime and neglectinghe finite size of the apertures, the output field may be ap-roximated by the product of an ideal Bessel beam andhe field produced by a half-symmetric unstable resonatorn which one mirror is planar (axicon plane) and the others convex (output plane). When considering the finite ex-ent of the mirrors, the output function seems to be modu-ated by a near-top-hat function that is a good approxima-ion of the ideal situation of having a constant amplitudeodulation. As occurs in the SBR, the radial separation of

he Bessel rings in the UBR will depend only on the angle0 of the conical mirror.

A first useful picture, even if not fully detailed, of theode properties of the UBR can be obtained from a purely

eometric analysis. The equivalent lens-guide system ofhe UBR is shown in Fig. 1(c). The conical mirror is rep-esented by a double refractive axicon whose ABCD rayransfer matrix is written as

�r2

�2 = � 1 0

− 2�0/r1 1�r1

�1 , �3�

here �r1 ,�1� and �r2 ,�2� are the position and slope of thenput and output rays, respectively.15 Note that the ra-ius r is explicitly part of the ABCD matrix for the doublexicon and, eventually, of the self-consistency condition.ith the reference planes placed just before the double

xicon, the ABCD matrix for the complete cavity is giveny

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A B

C D = �1 L

0 1� 1 0

2/R 1�1 L

0 1� 1 0

− 2�0/r1 1 �4a�

=1 +2L

R+ �−

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R�

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We are interested in stable trajectories inside the cav-ty. The mathematical problem is that of finding theigenvectors �r ,�� of the self-reproducibility equation

�A B

C D�r

�� = ± �r

�� , �5�

here the plus and minus signs correspond to trajectorieshat are self-reproducible after one and two round trips,espectively [see Fig. 1(c)]. Nonzero solutions are possiblenly if

det�A � 1 B

C D � 1 = 0. �6�

Taking advantage of the fact that AD−BC=1, we canee that Eq. (6) vanishes only if A+D= ±2. By replacing And D from Eq. (4b) into conditions A+D= ±2 and solvingor r, we get the eigenvalues corresponding to self-eproducing trajectories after one and two round trips, re-pectively:

rone = �L + R��0, rtwo = L�0 = a1/2. �7�

As expected, the eigenangles � associated with each rre found to be �one=�two=�0. Note that rtwo depends onlyn the aperture of the axicon, so it is always present inhe cavity, but the existence of one round-trip stable tra-ectory is not always guaranteed because it depends onhe radius of curvature of the output mirror.

In Fig. 1(c) we show the stable trajectories correspond-ng to one and two round trips. Note that the two round-rip trajectories always intersect the axicon plane at ra-ius r=a2 /2 and the output mirror at its center.

. WAVE-OPTICS ANALYSIS. Canonical Formulation of Unstable Bessel Resonatorshe geometric description provides a useful but limitedpproximation to the mode properties of the UBR. A moreetailed understanding of the mode distribution is ob-ained by solving the self-consistency Huygens–Fresnelntegral equation for a given resonating mode.

In the equivalent lens-guide system shown in Fig. 1(c),he complete round trip inside the resonator can be bro-en into two segments. The first segment corresponds tohe propagation from just before the double axicon �RP1�o the output mirror �RP2� and then to the original start-ng plane �RP3�.

The Huygens–Fresnel integral for the propagation ofode ulp with azimuthal mode index l=0, ±1, ±2, . . . andradial mode index p=1,2,3, . . . through a paraxial cy-

indrical system from plane RPm [coordinates �rm ,�] tolane RPn [coordinates �rn ,�] is given by18

ulp�rn� =�0

am

Kl�rm,rn�Tm�rm�ulp�rm�drm, �8�

here Tm�rm� is the transmittance function of the opticallement or aperture located at plane RPm, and the kernell�rm ,rn� is defined in terms of the ABCD elements for

he propagation segment between the planes m and n,amely,

Kl�rm,rn� = �− i�l+1� k

B�rmJl� k

Brmrn�exp� ik

2B�Arm

2 + Drn2� ,

�9�

ith Jl�·� being the lth-order Bessel function. In our case,he ABCD elements correspond to the free-space propaga-ion through a distance L, namely, A=D=1 and B=L. Thepper limit in Eq. (8) corresponds to the radius of the ap-rture at RPm.

The transmittance functions of the double axicon andhe convex output mirror are given by

T1�r1� = exp�− i2k�0r1�, r1 a1, �10�

T2�r2� = exp�− ikr22/R�, r2 a2, �11�

here a time dependence exp�−i�t� has been assumednd irrelevant constant phase shifts have been ignored.ote that the double axicon exhibits a linear phase radialariation instead of the quadratic variation of the spheri-al lens. For generality, in the following analysis the radiif the double axicon and the output mirror can assumeifferent numerical values.The round-trip propagation for the UBR is described by

he following pair of coupled integral equations19,20:

ulp�r2� =�0

a1

Kl�r1,r2�T1�r1�ulp�r1�dr1, �12�

ulp�r3� =�0

a2

Kl�r2,r3�T2�r2�ulp�r2�dr2. �13�

y direct substitution of Eq. (12) into Eq. (13), we find theound-trip integral from RP1 to RP3,

ulp�r3� =�0

a1

Hl13�r1,r3�ulp�r1�dr1, �14�

here we have used the fact that the radial coordinates inoth planes are numerically the same, as illustrated inig. 1(c).The kernel in Eq. (14) includes information about both

ropagation kernels in Eqs. (12) and (13) and the trans-ittance functions T1 and T2. We have explicitly

Hl13�r1,r3� =�

0

a2

Hl23�r2,r3�Hl

12�r1,r2�dr2, �15�

ith Hl12�r1 ,r2�=Kl�r1 ,r2�T1�r1�, Hl

23�r2 ,r3�=Kl�r2 ,r3�T2�r2�, where the superscripts �m ,n� mean that the ker-els are evaluated from RP to RP .
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Finally, at the reference plane each eigenmode ulp inhe cavity satisfies the self-consistency integral equation

lpulp�r3� =�0

a1

Hl13�r1,r3�ulp�r1�dr1, �16�

here the complex eigenvalue

lp = lp exp�i�lp� �17�

efines the fractional power loss per transit

�lp = 1 − lp 2 �18�

nd the phase shift �lp suffered by the mode in addition tohe longitudinal phase shift kL. The resonant conditionequires that the total phase shift � along the axis of theavity be an entire multiple of � radians, thus

� = kL + �lp = q�, �19�

here q is the number of half-wavelengths of the axialtanding-wave pattern.

In general, the kernel in Eq. (16) is symmetric but notermitian; therefore the eigenvalues lp are complex and

he existence of a complete set of eigenfunctions cannot beuaranteed in advance. The eigenfields are not power or-hogonal in the usual sense; rather, they areiorthogonal.21

. Numerical Considerationse have employed two methods to solve Eq. (16). The first

ne is a matrix method that consists of converting theound-trip integral equation into a matrix eigenvaluequation.22,23 This method makes use of a Gaussian–egendre quadrature rule and has some important ad-antages: It extracts the lowest N modes and eigenvaluest the same time, its accuracy is determined by the size Nf the matrix,15 and it can handle the case of closelypaced eigenvalues that is a difficult task for iterativeethods. The second method is the classical Fox–Li itera-

ive scheme.17 This method extracts the dominant or theowest-loss eigenmodes, but, as already mentioned, it isnefficient near eigenvalue degeneracies. The Fox–Li

ethod has the advantage that for cylindrical symme-ries, the kernels in Eqs. (12) and (13) resemble a Hankelransform and, as a consequence, it is possible to use fastlgorithms for propagating each radial eigenmode and ei-envalue in the resonator.24

Typically the value of the radial coordinate is boundedy a maximum value where the input and output fieldsre small enough to be neglected; otherwise it is boundedy the radius a1 of the double axicon. Let r�r1 ,r2 , . . . ,rN� and w= �w1 ,w2 , . . . ,wN� be the abscissasnd weight factors for Gauss–Legendre quadrature in theange �0,a1�. If ulp= �u1 ,u2 , . . . ,uN� is a column vectorepresenting the eigenfield evaluated at radius r, thenq. (16) takes the matrix form

ulp = �H � W� � ulp, �20�

here H is a N�N matrix with elements Hm,nH�rm ,rn� and W is a diagonal matrix with elements

w ,w , . . . ,w �.
1 2 N
The matrix product H�W corresponds to the propaga-ion matrix from plane 1 to plane 3. Since we are breakingp the round-trip propagation by performing two plane-o-plane propagations, this product can be seen as theroduct of two propagation matrices P12 and P23 corre-ponding to the equivalent matrix representation for theropagation integrals of Eqs. (12) and (13) respectively.quation (20) can be rewritten as

ulp = �P23 � P12� � ulp, �21�

here Pm,n=Hm,n�W. The eigenvalues and eigenvectorsf Eq. (21) can be easily extracted using the well-knownatrix eigenvalue algorithms.

. Resonating Modeshe physical parameters for the resonator used in our cal-ulations correspond to typical values of a gas-discharge,ast axial flow cw CO2 laser resonator: aperture size of thexicon a1=10 mm, n=2.4 (corresponding to commerciallyvailable zinc selenide axicons), axicon wedge angle �0.5°, and wavelength �=10.6 �m. The radius of the out-ut mirror is chosen to be a2=5 mm.From approximation (1) the values for the cavity length

nd apex angle are calculated, yielding L=40.92 cm and0=12.22�10−3 rad. A 200-point Gauss–Legendreuadrature was used to solve Eq. (21).To make appropriate comparisons, we consider first the

undamental mode �l ,p�= �0,1� of the resonator withlane output mirror (i.e., R→�). Figure 2 shows the am-litude and phase of the transverse profiles at both thexicon plane �RP1=RP3� and output mirror �RP2�. Theigenfield and the eigenvalue at the axicon plane were de-ermined by using the matrix method described in Sub-ection 3.B. As stated in Section 2, the field at the outputirror corresponds to an ideal zeroth-order Bessel beam

0�kt�� modulated by a bell-shaped amplitude function.he modulation of the output beam is evident from theomparison with the theoretical Bessel beam shown in

ig. 2. Transverse profiles of the magnitude and phase of theundamental Bessel mode at (a), (b) the axicon plane and (c), (d)he output plane for a resonator with output flat mirror and a25 mm.

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ig. 2(c). It can easily be seen how the amplitude of theutput beam decreases near the edge of the output mirror.or the range �0,3 mm� the plots are very similar and al-ost overlap such that differences are negligible. In Fig.

(b) we see that the phase of the field is almost linearith slope k�0; this fact confirms that the field at the axi-

on behaves as a conical wave.Let us now consider the case of the UBR. We replace

he plane output mirror by a convex–spherical mirrorith radius of curvature R=−50L. The amplitude andhase of the fundamental mode at both the axicon planeRP1� and just before the output mirror �RP2� are depictedn Fig. 3. Note that the radial variation of the output fieldshown in Fig. 3(c)] still resembles closely a Bessel beamhose rings preserve the same separation as is the case

or R→� shown in Fig. 2. This result is in agreement

ig. 3. Transverse profiles of the magnitude and phase of theundamental Bessel mode at (a) (b) the axicon plane and (c) (d)he output plane for a UBR with R=−50L and a2=5 mm.

ig. 4. Transverse profiles of the magnitude and phase of theecond-order Bessel mode at (a), (b) the axicon plane and (c), (d)he output plane for a UBR with R=−50L and a2=5 mm.

ith the fact that the ring separation is determined byhe characteristic angle of the conical mirror only, namely,e have kt=k sin �0�k�0; thus the separation between

onsecutive Bessel fringes is �0 /2�. As expected, thehase just before the axicon plane behaves linearly ac-ording to k�0r.

Higher-order solutions ulp and lp for Eq. (16) can alsoe obtained. When the output coupler is plane, the idealransverse field is given by Eq. (2). Figure 4 shows theagnitude and phase of the eigenfield corresponding to

he second-order �l ,p�= �2,1� for a radius of curvature R−50L at the axicon and output planes.

. Intracavity Field Distributionhe classical Fox and Li iteration method was imple-ented to determine numerically the passive three-

imensional field structure of the cavity modes. For thisurpose, the diffractive field calculations are based on thengular spectrum of the plane-waves representation uti-izing the fast-Fourier-transform algorithm. The trans-erse field is sampled in the reference plane over a grid of12�512 points. Typically around 150 round trips are re-uired for the process to converge, starting from an arbi-rary field distribution. The three-dimensional intracavityeld distribution was obtained by calculating the field at00 transverse planes evenly spaced through the unfoldedavity.

The intracavity field distribution of the dominantowest-order mode in the UBR with convex output mirrorR=−50L� is presented in Fig. 5. Forward propagationoes from the axicon plane at z=0 to the output mirrorlane at z /L=1. Reverse propagation goes from the out-ut mirror at z /L=1 to the axicon plane at z /L=2. Thexicon and the output mirror extend in transverse dimen-ion from −1 to 1 and from −0.5 to 0.5 in normalized units/a1, respectively. The corresponding transverse fields athe axicon and output mirror planes were already calcu-ated through eigenequation (16) and depicted in Fig. 3.

The results shown in Fig. 5 clearly illustrate the conicalature of the field within the cavity. At the middle planef the axicon, the field is approximately a plane wave; af-er crossing the axicon, the diffraction pattern of the coni-al wave exhibits a bright-line focus surrounded by a se-ies of cylindrical concentric sidelobes with gradually

ig. 5. Passive three-dimensional intracavity field distributionn the UBR with R=−50L. The eigenfield is first forward propa-ated from the axicon plane to the output plane and later re-urned backward to the axicon plane.

dsuwltam

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bbr

l�f

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iminishing intensity. We performed a number of Fox–Liimulations for a variety of initial conditions, includingniform plane waves, Gaussian profiles with differentidths, and random noisy transverse patterns. Regard-

ess of the initial condition, the field always converged tohe dominant mode of the cavity with the expected profilend radial frequency characteristics imposed by the geo-etrical parameters.

. ANALYSIS OF THE DIFFRACTIVEOSSES AND FREQUENCY SHIFTSrom a practical point of view, it is of great interest totudy the effect of varying the resonator parameters. Theelevant output characteristics are the transverse fieldrofile, the diffractive loss, and the resonant frequencyhift.

. Effect of Varying the Radius of Curvature of theutput Couplerhe effect of varying the radius of curvature of the outputirror on the field distribution at the axicon and output

lanes is shown in Fig. 6. For R=−200L there exist fewifferences between the fundamental eigenmode of theesonator and the ideal Bessel beam shown in Fig. 2. As Recreases, the transverse field at the axicon plane tendso concentrate around the vertex of the conical mirror,nd the Bessel rings are modulated by a radial amplitudeactor. Note that the ring separation remains constant un-er the variation of R. The output radial distribution ofhe J2 Bessel mode for several radii of curvature is de-icted in Fig. 7. One can see that the calculated eigen-elds are multiringed and that as R increases the pattern

ig. 6. Transverse profiles of the magnitude of the fundamentalessel mode at the axion and output planes of the UBR for sev-ral ratios R /L.

ecomes more similar to a theoretical second-order Besseleam. Note that higher-order modes experience the sameadial modulation as the fundamental mode.

We consider now the loss behavior corresponding to theower-order modes resonating within the UBR. The loss01 for the fundamental mode is depicted in Fig. 8(b) as a

unction of the normalized radius R /L for two different

ig. 7. Transverse profiles of the magnitude of the second-orderessel mode at the axion and output planes of the UBR for sev-ral ratios R /L.

ig. 8. (a) Diffractive losses �=1− 2 in terms of the normal-zed radius R /L for the fundamental mode with different wedgengles. Comparison of results with the matrix and Fox–Li meth-ds. (b) Comparison of the results for different aperture sizes ofhe output mirror.

vvcl�F

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lR=

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f

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alues of the axicon wedge angle � and for two differentalues of the output mirror radius a2. These plots wereomputed by finding the eigenvalues from Eq. (16) for aarge number of radii of curvature in the range −200

R /L�−30 and were also corroborated by using theox–Li algorithm [� symbols Fig. 8(a)].There are some conclusions that can be inferred from

he curves in Fig. 8(a): In general, the mode loss increasess the output mirror becomes more and more convex, lossncreases as the aperture diameter of the output mirror2 moves away from the value a1 /2, and loss increases ashe axicon wedge angle decreases. It has been found thathe lowest-loss curve occurs when the value of the outputperture is a2�a1 /2.It is expected that the resonant frequency of a particu-

ar resonating mode in the UBR will change if the value ofchanges. The resonant frequency can be written as �

� +��, where � is the resonance frequency of the reso-

Table 1. Diffractive Los

R=−50L

Mode Loss Mode

0,1 0.11346 0,11,1 0.16585 1,10,2 0.16826 2,10,3 0.21633 0,21,2 0.24841 1,22,1 0.25777 3,11,3 0.28333 0,35,1 0.28878 2,26,1 0.29033 4,14,1 0.29376 1,3

ig. 9. Transition between the stable and the unstable regionszed phase shifts �� /� are depicted as a function of the normalizb) the phase curves for the considered modes in (a) are contained

� �

ator with the plane output mirror and �� is the fre-uency shift introduced by the variation of the radius ofurvature. Let us now define the relative phase shift ex-erienced by a resonating mode inside the cavity as ��R�−��, with ��R� being the phase angle of the corre-ponding eigenvalue in terms of R and �� the angle forhe case when R→�, i.e., plane output mirror.

The frequency shift �� in terms of the phase shift �� isound to be

�� = − �0��/��, �22�

here �0=c /2L is the fundamental beat frequency, i.e.,he frequency spacing between successive longitudinalesonances. Figure 9(b) illustrates the normalized phasehift �� /� for the first four azimuthal modes l�0,1,2,3� and p=1. It is readily evident from these

or the First Ten Modes

R=50L

Loss Mode Loss

0.0178 0,1 0.002070.0302 1,1 0.003760.0593 2,1 0.008660.0633 0,2 0.017450.1072 3,1 0.020570.1132 1,2 0.031460.1583 4,1 0.047220.1720 2,2 0.061030.1984 0,3 0.077250.2216 5,1 0.09664

Bessel resonator. (a) Diffractive losses �=1− 2 and (b) normal-ius R /L. Mode crossings are present for higher-order modes. Inn the region between the phase curves for modes (0,1) and (0,4).

ses f

R→�

of theed radwithi

cp

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urves that the resonant frequency increases as the out-ut coupler becomes more convex.

. Transition between the Unstable and Stable Besselesonators

n Fig. 9 we show the eigenvalue spectrum of the UBRnd the SBR as a function of the normalized radius of cur-ature R /L. Numerical values are included in Table 1 forhe first ten modes sorted in ascending order by loss forhe UBR �R=−50L�, the SBR �R=50L�, and the resonatorith a plane output mirror �R→��. The plots in Fig. 9

learly illustrate that the transition between the UBRnd the SBR is characterized by a continuous change inhe associated diffraction losses and frequency shifts.hese results were determined from matrix equations

21) taking equal diameters for the axicon and the outputirror. The loss curves of all modes exhibit a monotoni-

ally increasing behavior as the output mirror becomesore convex. This result is expected from the fact that in

he SBR the field is constantly refocused by the concaveirror and the axicon, whereas in the UBR only the axi-

on forces the field to propagate toward the optical axis.ote that mode crossing points are present for higherodes. In Fig. 10 we compare the output field profiles of

he SBR and the UBR for the J0 and the J2 Bessel modes.hereas the SBR retains a Gaussian radial modulation

n the Bessel rings, the UBR exhibits a more uniformodulation.

. Effect of Changing the Cavity Lengthhe relationship in approximation (1) between the cavityarameters shown in Fig. 1 seems to be restrictive. Totudy the effect of varying the cavity length, let us firstefine the length factor �=L /L0, where L0 is the un-hanged cavity length in approximation (1) and L is theurrent length. The fundamental mode patterns at the

ig. 10. Comparison between the transverse modes for the SBRnd the UBR.

ig. 11. Transverse field patterns at the output and axiconlanes corresponding to the length factors �=0.8, 1, and 1.2 foronvex output mirror with R=−50L and �=0.5.

xicon and the output mirror planes are depicted in Fig.1 for �= �0.8,1.0,1.2�. Note that the radial separation ofhe Bessel rings remains practically constant, whicheans that the transverse component of the propagation

ector is not affected by a change in the cavity length. Theain difference between the profiles shown in Fig. 10 is

elated to diffraction losses. We can see that the modes for=0.8 and 1.2 present larger losses than the mode for �1; therefore it is expected that the amplitudes of theirutput profiles decrease faster.

The curve for the losses is depicted in Fig. 12(a) for therst two azimuthal modes �l=0,1� and R=−50L. As in thease reported for the SBR,15 it is remarkable that theinimum for the loss curves does not occur at the value of=1 as expected, but at a slightly different value of �. For

he fundamental mode (0, 1) it occurs at ��0.958,hereas for mode (1, 1) it is �1.035. Note that mode (1, 1)ecomes the fundamental mode of the resonator as theavity length increases; this mode crossing due to thearying cavity length also occurs between other higher-rder modes.

Figure 12(b) depicts the relative phase-shift behavioror l=0 and 1. We redefined the relative phase shift nown terms of the length L as ��=��L�−�L0

, where ��L� ishe phase angle of the eigenvalue as a function of thearying length and �L0

is the angle for the cavity lengthiven by approximation (1). It can be readily seen that theelative phase-shift behavior is almost linear and that itecreases for an increasing �.

. CONCLUSIONSdetailed analysis of the resonating modes in the axicon-

ased unstable Bessel resonator with a spherical–convexutput mirror has been presented. We studied the modeehavior under the variation of the radius of curvature ofhe output coupler, the axicon angle, and the cavityength, taking into account the finite aperture size of the

ig. 12. Loss and resonant frequency shift behavior as a func-ion of the length factor � for the first two angular modes �l ,p��0,1� and (1,1), with R=−50L and �=0.5.

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irrors composing the cavity. The most important conclu-ions are summarized as follows:

• UBRs support higher-order Bessel-like modes thatatisfy a biorthogonal relation rather than an orthogonalelation.

• The eigenvalue matrix method, based on the dis-retization of the Huygens–Fresnel self-consistency equa-ion, is particularly useful for extracting the first N eigen-elds and eigenvectors (i.e., losses and frequency shifts) ofhe resonating modes at the output coupler and the axi-on plane of the UBR. The Fox–Li algorithm was used toxtract the dominant mode of the cavity, obtaining an ex-ellent agreement with the matrix method.

• UBRs possess higher transverse-mode discrimina-ion in favor of the fundamental mode than SBRs.

• While the SBR retains a Gaussian radial modula-ion on the Bessel rings, the UBR exhibits a more uniformmplitude modulation that produces output profiles moreimilar to ideal Bessel beams. It now seems clear that forlaser system characterized by a least moderate gain perass (50% per pass), the best practical cavity for obtainingearly ideal Bessel beams will be a UBR with R=−50L.

• Given the light wavelength, the radial separation ofhe Bessel rings in the UBR depend only on the charac-eristic angle of the conical mirror.

• The transition between the UBR and the SBR isharacterized by a continuous change in the associatediffraction losses and frequency shifts.

• In general, the mode loss increases as the outputirror becomes more convex, and the loss increases as the

xicon wedge angle decreases. The lowest-loss curve oc-urs when the value of the aperture radius of the outputoupler is a2�a1 /2.

• The mode �l ,p�= �0,1� exhibits the lowest-loss curveor practically all values of the radius of curvature of theutput mirror; mode crossing points are, however, presentor higher modes.

• The mode (1,1) becomes the fundamental mode ofhe resonator as the cavity length increases beyond �1.15imes the unchanged cavity length defined by approxima-ion (1). Mode crossing points due to the varying cavityength also occur between other higher-order modes.

• The cavity loss for the fundamental mode is mini-ized when the cavity length is �96% of the value pre-

icted by geometrical optics.• The reduction of the aperture size of the output con-

ex mirror reduces edge-diffraction effects and preventsignificant energy spillage across the vertex of the axiconirror.

The analysis presented in this paper consolidates pre-ious works on the production of Bessel and Bessel–auss beams in laser resonators.13–16

CKNOWLEDGMENThe authors acknowledge financial support from Conacyt-éxico grant 42808 and from the Tecnológico de Monter-

ey Research Chair in Optics grant CAT-007.

Corresponding author Julio C. Gutiérrez-Vega can beeached by e-mail: at [email protected].

EFERENCES1. J. Durnin, “Exact solutions for nondiffracting beams. I. The

scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).2. J. Durnin, J. J. Micely, Jr., and J. H. Eberly, “Diffraction-

free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).3. G. Indebetouw, “Nondiffracting optical fields: some

remarks on their analysis and synthesis,” J. Opt. Soc. Am.A 6, 150–152 (1989).

4. G. Scott and N. McArdle, “Efficient generation of nearlydiffraction-free beams using an axicon,” Opt. Eng.(Bellingham) 31, 2640–2643 (1992).

5. J. Turunen, A. Vasara, and A. T. Friberg, “Holographicgeneration of diffraction-free beams,” Appl. Opt. 27,3959–3962 (1988).

6. A. Vasara, J. Turunen, and A. T. Friberg, “Realization ofgeneral nondiffracting beams with computer-generatedholograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).

7. Z. L. Horváth, M. Erdélyi, G. Szabó, Zs. Bor, F. K. Tittel,and J. R. Cavallaro, “Generation of nearly nondiffractingBessel beams with a Fabry–Perot interferometer,” J. Opt.Soc. Am. A 14, 3009–3013 (1997).

8. W.-X. Cong, N.-X. Chen, and B.-Y. Gu, “Generation ofnondiffracting beams by diffractive phase elements,” J.Opt. Soc. Am. A 15, 2362–2364 (1998).

9. J. Durnin and J. H. Eberly, “Diffraction free arrangement,”U.S. patent 4,887,885 (December 19, 1989).

0. K. Uehara and H. Kikuchi, “Generation of nearlydiffraction-free laser beams,” Appl. Phys. B 48, 125–129(1989).

1. P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).

2. A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J.Simonen, P. Pääkkönen, and J. Turunen, “Bessel–Gaussoutput beam from a diode-pumped NdYAG laser,” Opt.Commun. 238, 335–340 (2004).

3. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda,“Bessel–Gauss beam optical resonator,” Opt. Commun.190, 117–122 (2001).

4. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description andexperiment,” J. Opt. Soc. Am. A 18, 1986–1992 (2001).

5. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S.Chávez-Cerda, “Bessel–Gauss resonator with sphericaloutput mirror: geometrical and wave-optics analysis,” J.Opt. Soc. Am. A 20, 2113–2122 (2003).

6. C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar,“Unstable Bessel beam resonator,” Opt. Commun. 223,233–238 (2003).

7. A. G. Fox and T. Li, “Resonant modes in a maserinterferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

8. A. E. Siegman, Lasers (University Science, 1986).9. B. Lissak and S. Ruschin, “Transverse pattern

modifications in a stable apertured laser resonator,” Appl.Opt. 29, 767–771 (1990).

0. B. Duszcyk, M. P. Newell, and S. J. Sugden, “Numericalmethods for solving the eigenvalue problem for a positivebranch confocal unstable resonator,” Appl. Math. Comput.140, 427–443 (2003).

1. A. E. Siegman, “Unstable optical resonators,” Appl. Opt.13, 353–367 (1974).

2. W. D. Murphy and M. L. Bernabe, “Numerical proceduresfor solving nonsymmetric eigenvalue problems associatedwith optical resonators,” Appl. Opt. 17, 2358–2365 (1978).

3. W. P. Latham, Jr., and G. C. Dente, “Matrix methods forbare resonator eigenvalue analysis,” Appl. Opt. 19,1618–1621 (1980).

4. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computationof quasi-discrete Hankel transforms of integer order forpropagating optical wave fields,” J. Opt. Soc. Am. A 21,53–58 (2004).

theory of the unstable bessel resonator

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