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2072 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Ito et al.

Generation of hollow scalar and vector beamsusing a spot-defect mirror

Akihiko Ito,1,2 Yuichi Kozawa,1 and Shunichi Sato1,*1Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Katahira 2-1-1, Aoba-ku,

Sendai 980-8577, Japan2Present address: Institute for Materials Research, Tohoku University, Katahira 2-1-1, Aoba-ku,

Sendai 980-8577, Japan*Corresponding author: sato@tagen.tohoku.ac.jp

Received July 15, 2010; revised August 3, 2010; accepted August 3, 2010;posted August 4, 2010 (Doc. ID 131491); published August 26, 2010

Generation of both scalar and vector hollow beams was demonstrated by using a mirror with a low-reflectivityspot to suppress the oscillation of lower order transverse modes. As scalar beams, several hollow Laguerre–Gaussian beams have been observed from a side-pumped Nd:yttrium aluminum garnet laser cavity. The in-tensity profiles were in an excellent agreement with theoretical ones. The phase front variation around theoptical axis was verified to be spiral. Furthermore, both Laguerre–Gaussian and Bessel–Gaussian vectorbeams have been also observed from the identical cavity. In addition to the verification of intensity profiles,polarization pattern measurement confirmed that the beams had revolving polarization distributions along theazimuthal direction as theoretically predicted. © 2010 Optical Society of America

OCIS codes: 140.3410, 140.3430, 260.5430.

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. INTRODUCTIONxpressions for scalar light beams with homogeneous po-

arization distribution propagating in free space can beerived by solving the Helmholtz wave equation. The ex-ressions depend on the coordinates describing the opticalystem. For example, the Hermite–Gaussian beam is aolution for orthogonal coordinates. Additionally,aguerre–Gaussian (LG) and Ince–Gaussian beams arenown to correspond to cylindrical and elliptical coordi-ates, respectively. The lowest order beam of these ex-ressions is identical, a Gaussian beam with a single in-ensity peak on the beam axis. One of the unique featuresf LG beams is optical angular momentum associatedith their spiral phase front around the beam axis [1]. In

ontrast to spin angular momentum (circular polariza-ion), the generation of angular momentum, namely, theG beam, is not straightforward. Typical generationethods are the use of an interferometer [2], a pair of cy-

indrical lenses [3], a computer-generated hologram [4], aiquid crystal spatial light modulator [5], and a spiralhase element [6]. However, these extra-cavity methodsre accompanied by more or less beam degradation, and its not easy to achieve a high beam quality with them. Byontrast, direct generation from a laser cavity is expectedo produce much better quality beams. LG beams haveeen experimentally obtained by the insertion of a circu-ar absorber [7] or a spiral phase plate [8] in a laser cav-ty, pumping with a ring-shaped beam [9,10], transverse

ode selection in a diode-side-pumped bounce laser cav-ty using a thermal lensing effect [11], and mode conver-ion in a polarization maintaining optical fiber [12].

On the other hand, cylindrical vector beams such as ra-ially and azimuthally polarized beams are attractinguch attention because of their unique features origi-

1084-7529/10/092072-6/$15.00 © 2

ated from the cylindrically symmetric electric field,hich is inherently different from conventional scalareams, whose electric field direction is uniform over thehole beam. Several theoretical works on the cylindricalector beam have predicted the existence of some kinds ofeams including Bessel–Gaussian and LG vector beamstudied by Hall [13] and Tovar [14], respectively. Recently,modified Bessel–Gaussian beam has been also reported

15,16]. Experimental demonstrations of a cylindrical vec-or beam have been achieved by many researchers [17].lthough a beam synthesis method based on a superposi-

ion of plural and conventional beams is convenient, di-ect generation from a laser cavity may be much robustven in the generation of vector beams. The generation ofylindrical vector beam, however, is mostly limited to ei-her radially or azimuthally polarized beam known as theowest order transverse mode of cylindrical vector beams.

Now, notice a similarity between the scalar and vectoreams mentioned above. Namely, the scalar and vectoreams have singularities on the beam axis of phase andolarization, respectively, resulting in the intensity nulln the beam axis. Thus, the direct generation of these hol-ow beams from the identical cavity is expected if cylin-rical symmetry in a laser cavity and the suppression ofower order transverse modes are ensured. Recently, weave demonstrated the generation of a single higher orderransverse mode beam with radial polarization by using aavity mirror, which has polarization selectivity as well asing-shaped reflectivity modulation to suppress the oscil-ation of other transverse modes [18,19]. This reflectivity

odulation of a cavity mirror is a promising method forhe selection of a specific transverse mode because eachransverse mode has its own intensity pattern.

In this paper, we demonstrate the generation of scalar

010 Optical Society of America

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Ito et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A 2073

LG) and vector (LG and Bessel–Gaussian) beams withntensity null on the beam axis from a diode-pumpedd:yttrium aluminum garnet (YAG) laser cavity, whose

ear mirror has a spot defect with low reflectivity to sup-ress the oscillation of lower order modes. This method isery simple and can produce even higher order transverseodes by simply adjusting the cavity length and the

umping strength.

. THEORY AND EXPERIMENTAL SETUPere we summarize the expressions for some scalar and

ector beams in the cylindrical coordinate system. Thelectric field amplitude of the scalar LG mode beam of de-ree zero and order m �LG0m� is expressed by

ELG�s� �r,�,z� = E0

�0

��z�� �2r

��z���m�

exp�− i��m� + 1���z��

�exp− r2� 1

�2�z�+

ik

2R�z��exp�− im��,

�1�

here r is the radius, � is the azimuthal angle, z is theistance from the beam waist, E0 is a constant, �0 is theinimum beam radius at z=0, R= �z2+z0

2� /z is the radiusf curvature of the wave front, k is the wave number,�m�+1���z� is the Gouy phase shift with ��z�arctan�z /z0�, and ��z� is the Gaussian beam width de-ned as ��z�=�0�1+ �z /z0�2, with z0=k�0

2 /2. In this case,he order of the Laguerre polynomial is unity. Note thathe number m is a so-called topological charge corre-ponding to an orbital angular momentum. It is knownrom this equation that every beam has an intensity nulln the beam axis for the order m larger than zero. Accord-ngly, the spot defect with low reflectivity made on the

irror center is expected to suppress the oscillation of theowest order mode—a Gaussian beam, whose intensity

aximum is on the beam axis—if the center of the spot isoincident with the beam axis. Equation (1) also implieshat the intensity distribution of the LG0m mode beamm�0� is a single ring and the dark area near the beamxis expands with increasing the order m. This feature ofhe LG0m mode beams suggests that the oscillation with apecific order m is possible by changing a beam diametern the spot to suppress lower modes or, in other words, byroperly adjusting the laser cavity length because theeam spot diameter on a cavity mirror depends on theavity length.

The general formula of the electric field amplitude andolarization for the vector LG beam has been derived byovar [14]. A simple expression applying the conventionalomplex beam parameter used for a propagating Gauss-an beam has been derived for a cylindrically symmetricadially polarized TM0n mode beam [19]. This beam withmultiple ring structure and pure radial polarization haseen experimentally demonstrated by using a reflectivity-odified photonic crystal mirror mentioned earlier [19].nother expression corresponding to the Laguerre poly-omial of degree zero and order q=m±1 can be written as

ELG�v� �r,�,z� = E0

�0

��z�� �2r

��z���m±1�

�exp�− ikz + i��m ± 1� + 1���z��

�exp− r2� 1

�2�z�+

ik

2R�z��� cos�m��i� � sin�m��ir

±sin�m��i� + cos�m��ir , �2�

here i� and ir are the unit vectors of electric field for azi-uthal and radial directions, respectively. Note that theaguerre polynomial is not appeared in Eq. (2) because itsalue for degree zero is unity. Equation (2) implies thathe intensity pattern for �q��0 is a single ring with differ-nt diameter and width depending on the order of the La-uerre polynomial. Equations (1) and (2) indicate that in-ensity patterns of both beams are the same. By contrast,he polarization pattern is complicated as seen in the lastarentheses of the right-hand side of Eq. (2). Some ex-mples for q�1 are shown in Fig. 1. In the upper case cor-esponding to the upper part of the last parentheses ofq. (2), an azimuthally polarized (TE01 mode) beam iseen for q=1 and m=0. For q=1 and m=2, the beam isimilar to a so-called hybrid mode known as a propaga-ion mode in an optical fiber. The linear polarization re-olves clockwise along the azimuthal direction. It is gen-rally recognized that the polarization revolves q timesuring one revolution. Beams corresponding to the lowerase are shown in the right-hand side of Fig. 1. Note thathe polarization direction is orthogonal to those of the up-er case.The general expression for the vector Bessel–Gaussian

eams has been derived by Hall [13]. A simple expressionan be written as

EBG�v� �r,�,z� = E0�

�0

��z�exp�− ikz + i��z��

�exp− r2� 1

�2�z�+

ik

2R�z��Q�z�T�r,�,z�,

�3�

ig. 1. Spatial polarization patterns of the LG vector beamsith the Laguerre polynomial of degree zero and order q=m±1.

Upper” and “lower” in the figure correspond to the upper andower parts of the last parentheses of Eq. (1), respectively.

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2074 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Ito et al.

here E0� is a constant and Q�z�=exp�i�2z / �2k�1− iz /z0���,ith a constant �. T�r ,� ,z� has two families of

ransverse-electric field solutions Te and transverse-agnetic field solutions Tm, respectively, expressed by

Te�r,�,z� = �Jm−1�u� − Jm+1�u���− sin�m��

cos�m�� �i�

+ �Jm−1�u� + Jm+1�u���cos�m��

sin�m���ir, �4�

Tm�r,�,z� = − �Jm−1�u� + Jm+1�u���cos�m��

sin�m���i�

+ �Jm−1�u� − Jm+1�u���− sin�m��

cos�m�� �ir, �5�

here Jm�u� is the Bessel function of the first kind of or-er m with u=�r / �1− iz /z0�. Here we assumed that theime dependence of the electric field is exp�i�t� to be con-istent with Eqs. (1) and (2). For m=0, Eqs. (4) and (5)epresent the beams with pure azimuthal and radial po-arizations, respectively. For m�1, the polarization varia-ion along the azimuthal direction is a combination of sinend cosine functions and is similar to that of the LG vec-or beam. However, the spatial amplitude pattern is notimple now. Figure 2 shows examples of the intensity pat-ern calculated using Eq. (4). The intensity patterns areike petals of a flower for large m. Note that the ratio ofadial and azimuthal components is not equal due to un-qual amplitudes as expressed in Eq. (4). It is emphasizedhat all the beams shown in Figs. 1 and 2 have intensityull on the beam axis.Figure 3 shows a schematic of a laser cavity used for

he generation of both scalar and vector beams. A Nd:YAGod is side-pumped by three diode lasers. The gain distri-ution of the rod is well uniform. The laser cavity consistsf two mirrors, an output coupler, and a rear mirror. Theear mirror has a small spot defect with low reflectivityear the beam axis to suppress the oscillation of lower or-er transverse mode beams, which have finite intensitiesear the beam axis, such as a Gaussian beam. In fact, all

ig. 2. (Color online) Examples of calculated total intensity andolarization patterns of the Bessel–Gaussian vector beam.

he beams experimentally observed were hollow. In addi-ion, because this suppression of lower transverse modesas no effect on polarization, the spot-defect mirror isalid for both scalar and vector beams. In the experiment,wo mirrors with spot diameters of 0.25 and 0.5 mm aresed. The spot was made by finely grinding the surface ofconventional dielectric multi-layer coating mirror. The

eflectivity of an output coupler is 90%. Although both theear mirror and the output coupler are flat, the cavity istable owing to a thermal lensing effect of the Nd:YAGod.

. RESULTS AND DISCUSSIONhen the laser cavity was carefully aligned to ensure a

ylindrical symmetry of the optical system including mir-ors and rod and pumping, the beams experimentally ob-erved had a doughnut-shaped intensity pattern corre-ponding to a LG mode as shown in Fig. 4. In the top row,he total intensity distributions observed are shown.rom the left- to the right-hand side, the beam diameterf each ring increases with increasing the order m. Thentensity patterns along the azimuthal direction are

ostly uniform. In the middle row, the intensity profilesbserved are plotted by dotted curves. Solid curves corre-pond to theoretical profiles of a LG mode of the order m,hich is indicated in the bottom row. Note that the pro-les observed are in excellent agreement with theoreticalurves. Especially, the intensity on the beam axis is al-ost zero indicating a sufficient suppression of lower or-

er LG modes owing to the low-reflectivity spot on theear mirror located near the beam axis.

These beams were observed by changing the cavityength, the pumping strength, and the spot diameter. Theaussian beam width �0 on the cavity mirror is deter-ined by a cavity length and a focal length of a thermal

ens under our experimental condition. For a long cavityength and weak pumping (large focal length of a thermalens of a Nd:YAG rod), the width �0 on the cavity mirrorecomes large. When �0 is relatively large compared tohe spot diameter, a lower order LG beam was observed.y contrast, a higher order beam was observed for small0. For example, the LG01 and LG08 mode beams were ob-erved for the estimated values 2�0 of 1.21 and 0.80 mm,espectively, with the spot of 0.25 mm in diameter. Theeams shown in Fig. 4 were quite stable against mechani-al and thermal noises occurring in a common laboratory.ere we consider the polarization of the beam. Once a

ingle LG mode beam oscillates in a laser cavity, the po-arization of the beam will be determined. However, theolarization observed was random. Since the cavityength in the experiment ranges from 175 to 750 mm,

ig. 3. A schematic diagram of an experimental setup. A laserain medium is a Nd:YAG rod. The cavity consists of two mirrors,n output coupler, and a rear mirror. The rear mirror has a smallpot with low reflectivity.

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Ito et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A 2075

here exist many longitudinal modes. More importantly,he spot-defect mirror used in this experiment as well as aechanical aperture has a selectivity of modes with the

ame intensity patterns but with sufficiently different po-arization properties. This fact means that all the beamsith the same intensity patterns can simultaneously os-

illate in a laser cavity. Therefore, the random polariza-ion observed may be a natural result.

A single transverse mode operation with linear polar-zation was easily achieved by inserting a Brewster platento a laser cavity as normally demonstrated for many la-ers. We put a transparent thin glass plate at a Brewsterngle in the laser cavity. In the top row of Fig. 5, the totalntensity patterns for m=1, 2, and 3 are shown. The pat-erns are similar to those in Fig. 4 as expected. The polar-zation was verified to be linear by observing a transmit-ed intensity pattern through a linear polarizer. Asentioned earlier, a LG0m mode beam has the orbital an-

ular momentum of m. This can be verified by an inter-erence pattern between the whole LG beam and a refer-nce beam. Results are shown in the middle row of Fig. 5.ere the LG beam was divided into two beams and a

mall fraction of one beam was used as a reference beamith the spherical phase front. In the case of m=1, the in-

erference pattern is a single spiral indicating that thiseam has the angular momentum of unity. For m=2 and, the number of spirals are two and three, respectively,s expected. The rotational direction of the spiral pattern

ig. 4. (Color online) Intensity distributions of scalar LG beamshe middle row, the measured intensity profiles are plotted by deam of the order of m, which are indicated in the bottom row.

ig. 5. (Color online) Observed intensity distributions of lin-arly polarized LG beams for m=1, 2, and 3. In the top row, theotal intensity distributions are shown. In the middle row, the in-erference patterns of the LG beam with a spherical referenceeam are depicted.

ndicates the direction of the spiral phase front of the LGeam or, in other words, the sign of order m. In Fig. 5, theotational direction for m=1 is opposite to those for m2 and 3. Although this rotational direction is not control-

able at present, it is stable and reproducible once the la-er oscillation is established and the cavity alignment isestored.

A mechanical aperture has been generally used to ob-ain a Gaussian mode beam or lower transverse modeeams. In this technique, the oscillation of higher trans-erse mode beams is suppressed based on the fact thatigher order transverse mode beams are spatially ex-anded than lower mode beams and receive a higher cav-ty loss due to the aperture. By contrast, the spot-defect

irror with low reflectivity used in this experiment sup-resses the oscillation of lower transverse mode beams.hus, the mirror with a spot defect acts as a reversed ap-rture that blocks the inner portion of a beam. It is foundhat a mirror with a metal sphere adhered on the mirrorurface produces similar results to the spot.

By removing the Brewster plate and slightly titling onef the cavity mirrors, the polarization pattern changed ashown in Fig. 6. On the top row, the total intensity pat-ern and polarization patterns after the passage through

linear polarizer are shown. The total and polarization

ed. In the top row, the total intensity distributions are shown. Inurves. Solid curves correspond to theoretical profiles of the LG

ig. 6. (Color online) Intensity distributions of observed vectorG beams. On the left column, total intensity distributions arehown. Other four images in each row correspond to intensityistributions after passing through a linear polarizer. The direc-ion of the linear polarizer is indicated by an arrow.

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2076 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Ito et al.

atterns were doughnut-shaped and two-lobe, respec-ively, indicating that the beam generated was azimuth-lly polarized. A radially polarized beam was also ob-erved by slightly and differently changing the cavitylignment. The polarization state was reproducible if theavity alignment is restored. When the cavity length washanged, other doughnut-shaped beams with larger inneriameters were observed as shown in the lower rows ofig. 6. In these cases, the polarization pattern had a largeumber of lobes. For example, in the second row, the pat-ern had four lobes. The pattern also rotated with rotat-ng the linear polarizer. The rotation angle of the polar-zation pattern, however, was just a half of the rotationngle of the polarizer. The number of lobes and the rota-ion angle of the polarization pattern can be summarizeds follows. Namely, the number of lobe is 2q and the ro-ation angle of the polarization pattern is � /q for the ro-ation angle � of the polarizer. This observation is in ex-ct agreement with the beams depicted in the left-handide of the second column of Fig. 1. Note that the polar-zation pattern shown in Fig. 6 was stable once the laserscillation was established.

The mechanism of the single transverse mode opera-ion of the cylindrical vector beam is not clear now. How-ver, the contribution of a cavity mirror with a low-eflectivity spot is dominant for selecting a single higherrder transverse mode, which is doughnut-shaped andas a LG intensity pattern. In addition, the introductionf polarization anisotropy for reflectivity induced by slightilting of a cavity mirror may be of great help in the se-ection of polarization state.

The first and the third rows of Fig. 7 show typical in-ensity patterns obtained from the same laser cavityhen the pump strength was simply increased. As shown

n the left-hand side of the top row, the total intensity pat-ern was like a flower with 12 petals as seen in the bottomow of Fig. 2. Measured horizontal and vertical polariza-

ig. 7. (Color online) Observed and calculated intensity pat-erns of vector Bessel–Gaussian beams of m=6 and 7. On theeft-hand column, the total intensity distributions are shown.ther two images in each row are the intensity distributions af-

er passing through a linear polarizer, namely, horizontal andertical polarization components. The direction of the polarizer isndicated by an arrow.

ion components shown in Fig. 7 were slightly differentrom the total intensity pattern. These patterns were alsoifferent from those of vector LG beams shown in Fig. 6.he second row of Fig. 7 shows calculated patterns of the

otal intensity and polarization components for a vectoressel–Gaussian beam for m=6 in Eq. (4). These patternsere very close to each other. Accordingly, the beam ob-

erved in the experiment was identified as a vectoressel–Gaussian beam. By changing the cavity length, aector Bessel–Gaussian beam corresponding to m=7 wasbserved. Measured and calculated patterns are depictedn the third and bottom rows of Fig. 7, respectively.

Equation (2) indicates that the composition ratio be-ween radial and azimuthal polarizations is equal. Byontrast, as shown in Fig. 2, the higher order transverseodes of vector Bessel–Gaussian beam have a flower-like

attern but with a different polarization composition ratios indicated from Eqs. (4) and (5). When the pumpingtrength was increased, thermal birefringence in additiono thermal lensing effect has became significant and effec-ive to discriminate between radial and azimuthal polar-zation components resulting in different optical path andain in a Nd:YAG rod and consequently selective oscilla-ion with one polarization state [20]. In the same way, theeneration of a vector Bessel–Gaussian beam can be at-ributed to the effective discrimination of the gain be-ween radial and azimuthal polarizations due to the ther-al birefringence of a Nd:YAG rod.

. CONCLUSIONn conclusion, we demonstrated the generation of hollowcalar [Laguerre–Gaussian (LG) beam] and vector (LGnd Bessel–Gaussian beams) beams from a Nd:YAG laseravity, whose rear mirror has a spot with low reflectivityo suppress the oscillation of lower order transverseodes. The intensity patterns observed were in excellent

greement with theoretical curves. A single transverseode operation with linear polarization is achieved by in-

erting a thin glass plate at a Brewster angle for scalarG beams. Vector LG and Bessel–Gaussian beams wereenerated from the identical laser cavity by slightly tilt-ng a cavity mirror and increasing the pumping, respec-ively. As mentioned above, the technique used in this ex-eriment is very simple but effective for the singleransverse mode operation of hollow beams. The point iso suppress lower transverse modes such as the funda-ental transverse mode (Gaussian beam) by a spot-defectirror, which acts as a reversed aperture.

CKNOWLEDGMENTShis work was supported in part by Japan Society for theromotion of Science, and Japan Science and Technologygency (JST), Core Research for Evolutional Science andechnology (CREST).

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