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Y. Kozawa and S. Sato Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 399

Demonstration and selection of a single-transversehigher-order-mode beam with radial polarization

Yuichi Kozawa* and Shunichi Sato

Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Katahira 2-1-1, Aoba-ku,Sendai 980-8577, Japan

*Corresponding author: kozawa@tagen.tohoku.ac.jp

Received December 8, 2009; accepted January 10, 2010;posted January 12, 2010 (Doc. ID 121156); published February 10, 2010

Single-transverse higher-order radially polarized beams with multiple rings are selectively generated from aNd:YAG laser cavity by using a reflectivity-modified and polarization-selective photonic crystal mirror. As pre-dicted by theoretical analysis [A. A. Tovar, J. Opt. Soc. Am. A 15, 2705 (1998)], the intensity distribution andpropagation characteristics of the higher-order radially polarized beams are expressed by a Laguerre–Gaussian formula. © 2010 Optical Society of America

OCIS codes: 140.3410, 230.5440, 260.5430.

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. INTRODUCTIONradially polarized laser beam has the potential to pro-

uce a smaller focal spot than a linearly or circularly po-arized Gaussian beam under high-numerical-aperture fo-using conditions [1]. In addition, the attractive featuresf many applications using a radially polarized beamave been investigated theoretically and experimentally2], and various methods to achieve such an inhomoge-eous polarization state of a laser beam have been ex-loited. However, most of the studies on the generation ofradially polarized beam have concerned the so-called

oughnut-shaped beam, which is represented as a funda-ental radially polarized mode (TM01 mode). On the

ther hand, Tovar theoretically deduced the existence ofigher-order transverse modes with radial polarization asresult of the paraxial solution of a vector wave equation

3]. These higher-order radially polarized TM0n modesn=2,3,4, . . . � are predicted to provide a dark focal spot4] and a much smaller focal spot [5,6] at the focus under

tight focusing condition. Recently, a single-transverse-ode operation of a double-ring-shaped TM02 beam was

xperimentally achieved by using a reflectivity-modifiedhotonic crystal mirror (PhCM) [7]. Although the previousesults have opened the possibility of generating much-igher-order radially polarized beams, the experimentalemonstration of purely single-transverse-mode opera-ion has not been achieved because the selection schemef a TM0n mode remains unexplored.

In this paper, several higher-order radially polarizedeams with a multi-ring-shaped intensity distribution (n2, 3, 4, and 5) are experimentally demonstrated from ad:YAG laser cavity by using a reflectivity-modifiedhCM. The transverse-mode order n is readily selected byhanging the position of a lens inserted into the cavity,hich is consistent with the result of the cavity designingased on ABCD matrices assuming the propagation char-cteristics of a radially polarized Laguerre–Guassianode derived by Tovar. Accordingly, it was experimentally

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onfirmed that the radially polarized Laguerre–Gaussianeam (TM0n mode) is one of the eigenmodes of a laseresonator.

. CAVITY DESIGN AND EXPERIMENTALETUPne of the paraxial solutions for the vector Helmholtzquation in cylindrical coordinates was derived by Tovar3] and is represented by the Laguerre–Gaussian form in-olving the complex argument. Applying the conventionalomplex beam parameter used for the propagation of aaussian beam with beam width w, the electric field am-litude of a cylindrically symmetric, radially polarizedM0n mode beam in Tovar’s solution is rewritten as

E�r,z� = E0

w0

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1 �2r2

w2 �exp�−r2

w2��exp�−

ikr2

2R− ikz + i2n�� , �1�

here r is the radius, z is the distance from the beamaist, E0 is the constant, w0 is the minimum beam radiust z=0, R= �z0

2+z2� /z is the radius of curvature of theave front, k is the wave number, 2n� is the Gouy phaseith �=arctan�z /z0�, L is the generalized Laguerre poly-omial of order 1 and degree n−1, and w is the Gaussianeam width defined as w=w0�1+ �z /z0�2 with z0=kw0

2 /2.ote that the complex amplitude of a radially polarizedM0n mode in Eq. (1) is identical to that of a linearly po-

arized Laguerre–Gaussian (LGp1, p=n−1) beam exceptor an azimuthal phase (vortex) term, which produces anrbital angular momentum [8]. Accordingly, Eq. (1) indi-ates that the intensity ��E�2� of a radially polarizedaguerre–Gaussian beam can be treated in the sameanner as a linearly polarized Gaussian beam, while the

olarization vectors are different. Figure 1 shows theoret-

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400 J. Opt. Soc. Am. A/Vol. 27, No. 3 /March 2010 Y. Kozawa and S. Sato

cal intensity profiles of TM0n modes (n=1, 2, 3, and 4)long the radial direction. The horizontal axis is in unitsf the Gaussian beam width w. As shown in Fig. 1, the ra-ius of the circular node varies with the mode orders.onsequently, if the cavity mirror operating with radialolarization possesses an annular narrow-ring regionith low reflectivity, a specific transverse mode whose ra-ius of the circular node coincides with the radius of thennular ring will selectively oscillate to satisfy the mini-um reflection loss in the laser cavity.For generating a radially polarized laser beam (TM01ode), a concentrically patterned PhCM fabricated by an

utocloning technique has been used as a polarization-elective output coupler [9]. Figure 2(a) shows a sche-atic of a PhCM used in this study, which is the same asas used for generating a TM02 mode beam [7]. A concen-

rically patterned region was designed so as to reflect ra-ially polarized light with a reflectivity of 90%, whereashe reflectivity for azimuthal polarization, which is or-hogonal to radial polarization, is below 20%. In addition,he PhCM has a narrow annular region with a diameter

ig. 1. (Color online) Intensity profiles of TM0n modes (n=1, 2, 3aussian beam width w.

ig. 2. (Color online) (a) Schematic of a substrate pattern foronfiguration.

f 550 �m, where the pattern is fabricated in the radialirection. The average pitch of the radial pattern is theame as that of the concentric pattern. Therefore, the re-ectivity of the annulus for radial polarization is regardedo be 20% or less.

The Nd:YAG laser cavity consisting of a highly reflec-ive concave mirror and the PhCM as an output coupler isepicted in Fig. 2(b). A Nd:YAG rod (diameter, 2 mm;ength, 63 mm) was installed in a laser diode side-umped module (Cutting Edge Optronics, RB20-0.33C2).he radius of curvature of the rear mirror was 500 mm.he cavity length was 200 mm. A plano-convex lens withfocal length of 153.8 mm was inserted into the cavity to

hange the beam diameter on the PhCM. The lens wasounted on a mechanical stage to adjust the distance d

etween the lens and the PhCM. A variable aperture waslaced between the lens and the Nd:YAG rod to suppressnwanted higher-order modes.The Gaussian beam propagation inside a geometrically

table laser cavity including any optical elements isvaluated by means of ABCD matrices [10]. By consider-

) along the radial direction. The horizontal axis is in units of the

ectivity-modified PhCM. The units are micrometers. (b) Cavity

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Y. Kozawa and S. Sato Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 401

ng a round-trip ABCD matrix of the cavity in Fig. 2(b),he beam diameter 2w at each element position is calcu-ated as a function of the distance between the lens andhe PhCM, as shown in Fig. 3(a). The cavity is geometri-ally stable when the distance is between 17.8 and14.1 mm. In this calculation, the effect of the thermalensing in a Nd:YAG rod was ignored because the result-nt beam diameter inside the Nd:YAG rod is sufficientlymall in the experiment, and a relatively low-powerumping module with a maximum electric power of100 W was used. As shown in Fig. 3(a), the beam diam-

ter on the PhCM varies widely with change of the posi-ion of the lens. From Eq. (1) and Fig. 3(a), we can calcu-ate the distance d for which the position of the circularode of a TM0n mode coincides with that of the annulus ofhe PhCM �550 �m�. Figure 3(b) summarizes the resultsor different mode orders and node orders (numberedrom inner to outer node). Therefore, the selective oscilla-ion of a specific TM0n mode beam will be made possibley changing the position of the lens so as to move the di-meter of a circular node of the TM0n mode closer to thennulus (with a diameter of 550 �m) on the PhCM. Inhis cavity, for example, the expected values of the d, athich the first node of TM0n modes (n=2, 3, 4, and 5) co-

ncide with the annulus, are 44.5, 31.3, 25.7, and2.9 mm, respectively.

axis across the center of the beam (dotted curve) with the result, and (d) TM mode beams.

ig. 3. (Color online) (a) Calculated Gaussian beam diameterw at each element as a function of the distance d between theens and the PhCM. (b) Node order of a TM0n mode that coincidesith the annulus of the PhCM as a function of the distance d.

ig. 4. (Color online) Measured intensity profiles along the horizontalf the curve-fitting (solid curve) for generated (a) TM , (b) TM , (c) TM

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402 J. Opt. Soc. Am. A/Vol. 27, No. 3 /March 2010 Y. Kozawa and S. Sato

. RESULTS AND DISCUSSIONhe laser beam generated in this cavity without an aper-ure was almost multi-transverse mode with the compli-ated intensity distributions at any positions of the lens.he polarization of generated beams was radial for allavity configurations due to the polarization selectivity ofhe PhCM. However, by adjusting the diameter of theariable aperture inserted, a double-ring-shaped beamas observed when the lens was placed at d=44 mm. Fig-re 4(a) shows the horizontal profile (dotted curve) acrosshe center of the generated beam whose intensity distri-ution is depicted in the inset. The solid curve representshe result of the curve fitting to the theoretical profile of aM02 mode by using Eq. (1). The beam propagation factor2 was measured to be 4.1, which is close to the theoret-

cal value �M2=4� of a TM02 mode beam. These results in-icate that the generated beam was a single-transverseM02 mode. The maximum output power was 683 mWith a drive current of 23 A of the pumping laser diode.Then the inserted lens was moved at d=31 mm, and a

riple-ring-shaped beam was generated by adjusting theiameter of the aperture as shown in Fig. 4(b). The inten-ity profile across the beam center is well fitted to a TM03ode. The beam propagation factor was 6.1, which agreesell with the theoretical value �M2=6� of the TM03 mode.ower-order-mode beams such as TM01 and TM02 beamsere not observed even if the aperture diameter was fur-

her decreased. Therefore, the laser cavity was operateds a single-transverse TM03 mode in this cavity configu-ation. The output power in a TM03 mode operationeached 780 mW. On the other hand, a TM04 mode beam,hich is characterized by a fourfold ring pattern, wasenerated at d=25.5 mm as shown in Fig. 4(c). The beamropagation factor was 8.2, while the theoretical value is. Furthermore, a TM05 mode beam with a fivefold ringattern was also generated at d=24 mm as shown in Fig.(d).The polarization state of the generated beams was veri-

ed by observing the intensity distribution after passage

ig. 5. (Color online) Polarization distributions of the generatedistributions. (b)–(d) and (f)–(h) Intensity patterns after passage

hrough a linear polarizer as shown in Fig. 5. Figures 5(a)nd 5(e) show the total intensity distributions (without ainear polarizer) for TM03 and TM04 beams, respectively.igures 5(b)–5(d), 5(f), and 5(g) indicate the correspond-

ng intensity distributions after passage through a linearolarizer. Each arrow denotes the direction of the polariz-r’s axis. The six-arc patterns along the polarizer’s axishown in Figs. 5(b)–5(d) imply that the generated beam isolarized in the radial direction. Likewise, the eight-arcatterns suggest that Fig. 5(e) is a radially polarizedM04 mode.Thus, the transverse mode of the generated radially po-

arized beams was selected by simply changing the posi-ion of the inserted lens with the help of the variable ap-rture. Each transverse mode was generated at thearticular lens position that is in good agreement withhe theoretical value based on the propagation character-stics of a radially polarized Laguere–Gaussian beam. As

result, the higher-order modes predicted by Tovar werexperimentally verified for the first time to our bestnowledge by selectively generating a single-transverseode directly from a laser cavity. These single-transverseigher-order TM0n mode beams are quite suitable for pro-ucing a much smaller focal spot [5,6] under a tight focus-ng condition, which is very attractive for many applica-ions such as super-resolution microscopy or materialano-processing.

. CONCLUSIONn summary, we demonstrated the generation of higher-rder radially polarized beams with Laguere–Gaussianistributions from a Nd:YAG laser cavity by using aeflectivity-modified photonic crystal mirror. The selectiveeneration of a single transverse mode was achieved byimply changing the position of the inserted lens. The in-ensity distribution and propagation characteristics of theigher-order radially polarized beam were coincidentith the Laguerre–Gaussian formula.

(a)–(d)] and TM04 mode [(e)–(h)] beams. (a) and (e) Total intensitygh a linear polarizer. Each arrow indicates the polarizer’s axis.

TM03 [

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Y. Kozawa and S. Sato Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 403

CKNOWLEDGMENTShis research was supported in part by the Japan Society

or the Promotion of Science (JSPS) and by the Japan So-iety and Technology Agency (JST), Core Research forvolutional Science and Technology (CREST). The au-

hors thank Photonic Lattice, Inc. for fabricating thehCM.

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