JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Asymmetric focusing of a laser beam in TEM01 doughnut mode in anonlinear dielectric*

M. S. Sodha and V. P. NayyartDepartment of Physics, Indian Institute of Technology, New Delhi-110029, India

V. K. TripathiSchool of Radar Studies, Indian Institute of Technology, New Delhi-110029, India

(Received 9 July 1973)

We have studied the self-focusing/defocusing of a laser beam in a nonlinear dielectric when the laser isoperating in the TEM01 doughnut mode. The cylindrical asymmetry of the irradiance distribution increasesinside the medium; the power gets concentrated around the points of maximum irradiance. The maximumirradiance in different directions (in a plane transverse to the direction of propagation) occurs at differentvalues of r, the distance from the z axis. The polar representation of maximum irradiance in the transverseplane turns out to be a figure of eight.

Index Headings: Nonlinear optics; Fibers; Focus.

The problem of self-focusing of an intense laser beamin a nonlinear medium has been studied by manyworkers.1 -3 These investigations assume the irradiancedistribution of the incident beam to be gaussian in thetransverse plane, which implies that the laser (as istrue in a majority of cases) operates in the TEMoomode. However, in certain situations higher-ordermodes (e.g., the TEMoi doughnut mode) also becomeimportant. Many lasers have been operated withmirrors having annular output-coupling apertures.This type of configuration tends to suppress the lowest-order mode (TEMoo) and encourages operation in thehigher-order modes, such as the TEM 01 doughnut mode.Since this mode can be focused to a spot that is notmuch larger than that of the lowest-order mode, suchoperation may be entirely satisfactory for many uses ofthe laser.

In this communication, we have investigated theself-focusing/defocusing of a laser beam operating inthe TEM01 doughnut mode. The irradiance distribution(z= 0) of the incident beam with respect to a cylindricalcoordinate system is'

EE*== Eo2 - cos20 exp(-r 2 /r 02 ),

r02 (1)

would diminish. However, focusing should occur aroundthe points of irradiance maxima. Thus accentuation ofthe doughnut-shape irradiance of the beam is expectedas a result of self-focusing of the beam.

To show these effects, we have solved the waveequation in the WKB approximation and have derivedequations for the beam-width parameter in the twolimiting cases

(i) x (= r cos0)<<ro, y (=r sino)<<ro

and

(ii) |ro-x|<<ro, y<<ro;

the first case corresponds to the usual paraxial-rayapproximation.

SOLUTION OF THE WAVE EQUATION

We consider the propagation of a laser beam in the zdirection in a nonlinear medium described by Eq. (2).To simplify the analysis of the problem (arising fromthe cylindrical asymmetry of the beam), we employ acartesian coordinate system. In this system, the irradi-ance distribution of the incident beam can be expressedas

x 2

EE* = Eo2- exp( - x 2/r02) exp(-y 2 /r 0

2 ).where the direction of propagation has been assumedto be along the z axis and r refers to a cylindrical co-ordinate system. The beam has an irradiance minimumat the axis (r=O) and also at 0=7r/2 . The irradiance ofthe beam is maximum at the points r=ro, 0=0, r. Thenonlinear dielectric constant of the medium has adependence on irradiance of the form

E= Eo+E2 EE*. (2)

Recalling the focusing properties of nonlinearity, wecan intuitively say that defocusing of the beam wouldoccur on the axis; hence the irradiance of paraxial rays

(3)

The behavior of the electric vector inside the mediumis governed by

EC02

V2E= - E,C2

(4)

where the time dependence of E is taken as exp(iwt)and a term of order V2 lne<<k2 has been neglected.

Following the technique of Akhmanov et al., 2 weexpress the electric vector as

E = Ao exp[-ik(z+S)], (5)

941

VOLUME 64, NUMBE;R 7 JULY 1974

SODHA, NAYYAR, AND TRIPATHI

FIG. 1. Variations of fl, f2 , gl, and g2 as functions of distanceof propagation for Rd

2/Rn

2= 10.0.

where Ao is the real amplitude and S is an additionaleikonal. A0 and S in the WKB approximation aregoverned by'

as (a S 2 aS\ 2 e2A o2 1 /a2 A 0 a2A 02-+ -)+ - = +----+- ),(6)az ax ay EO k 2A 0 ax

2 ay

2

aA 2 as aA 02 as aAo2 /a 2S a2

s\-+- +- +AO2( +-- =0. (7)az ax ax ay ay ax2 ay2/

General solution of Eqs. (6) and (7) is very difficult;hence we solve them in only two limiting cases.

A. Focusing in the Paraxial-Ray Approximation(Around the Point x = 0, y = 0)

In this case, the solutions for S and A o2 are of the form

andRn = ro(eo/e2E0 2 ) -.

The solution of Eq. (11) for an initially plane-wavefront (i.e., dfi, 2/dz=0 at z=O) is

f2= (1+72 ) . (12)

By use of this expression in Eq. (10), f' can be obtainednumerically. To show the behavior of fI and f2 with -a plot of f1 and f2 with - is given in Fig. 1 for feasibleparameters.

B. Focusing Around the Points of Irradiance Maxima(x = xo, y = yo); Geometridal-Optics

Approximation

If we take x=ro+x 1 , then Eqs. (6) and (7) become

as /S \2 aS\2 e2A2-+ a +-as ax11 \ay / eo

1 + a 2Ao a2A,\+_ + = 0,

k2A ox 12 ay2

A 02 as aA 0

2 as aA02 / a2S a2S\+-* +-* +AO2 +- =0.

az ax1 ax1 ay ay ax12

Iy2 I

(13)

(14)

Solution of Eqs. (13) and (14) for S in the limitof I x1i <<ro is

XI2 y2

S= -Yl(z) + 2(Z) +±I'(Z)2 2

(15)

which, when used in Eq. (13), gives the irradiancedistribution

x2 y2

S= ,031(Z)-+±f2(Z)-+4(4z)2 2

andx2

A o2= Eo2 exp(-x 2 /rO 2 f12) exp(-y 2 /r 02 f2

2 ),f2fw

3ro2

where1 dfi

I=--dfj dz

1 df 232= d

f2 dz

and fi and f2 in the approximationy2<<rO2 f2

2 are governed by

d 2 f, Rd 2 1 1

dq 2 Rn

2 f12f2 fi3

d2 f 2 1

dq 2 f23

whereRd = kro2, q = z/Rd

E02 /xI\ 2

(8) A02 =-(1+-)expE- ()+x1/rogi) 2]

glg2 rog/

Xexp(-y 2/r02g 22 ), (16)

(9)

of x2 <<rO2 f12,

where1 dg1

gi dz

1 dg272=--.g2 dz

(17)

To obtain equations for g, and g2, we employ theapproximations xj<<rogI and y<<rog2; the equationsobtained are

d2g1

d-q2(10)

(11)

2Rd2 1

2 2Rn2 gl~g2(18)

(19)d2g2 Rd2 1

dv2 Rn2 glg2

To portray self-focusing around the points of irradiancemaxima, we have solved the coupled equations nu-merically and show the results in Fig. 1.

942 Vol. 64

fN

ASYMMETRIC FOCUSING OF LASER BEAM TEM 0 1 MODE

DISCUSSION

The axial part of the beam suffers strong divergencedue to diffraction and also due to the nonlinearity ofthe medium. The power concentrates around theirradiance maxima, because of the dependence of thedielectric constant on irradiance. We see from Fig. 1that f1 increases much faster with n than f2; similarlygl decreases much faster with j than g2. Thus thetransfer of power from the central portion of the beam(as it propagates) to the portion of the beam having

a- 12

bI

Cogr 2 6

7.0 - to

b

0B0

0 - 3 /2

FIG. 2. Polar representation of irradiance IE 1max in a planetransverse to z axis, the direction of propagation for (a) =O and(b) 7=0.39. If a straight line is drawn from the origin to anypoint on the curve, its length is proportional to the irradiance inthe direction 0, where 0 is the angle that the line makes with the0=0 axis. The curve b is incomplete because the approximations,involved in the analysis, break down for the range of 0 where thecurve is incomplete.

maximum irradiance is more pronounced in the xdirection than in the y direction. The maximum irradi-ance in different directions (in a plane transverse to thedirection of propagation) occurs at different values of r,the distance from the z axis. The polar representation ofmaximum irradiance in the transverse plane turns outto be a figure eight; the length of the line from the originto any point on the curve in Fig. 2 is a measure of themaximum irradiance in the direction 0, where 6 is theangle made by the line with 0=0 axis (i.e., x axis).

This analysis is valid when the WKB approximationis meaningful, i.e., when Rn2>>X2 (slowly convergingbeams).

REFERENCES

*Work partially supported by NSF, USA.tOn visit from Physics Department, Punjabi University,

Patiala, India.IS. A. Akhmanov, A. P. Sukhorukov, and R. V. Khoklov, Zh.

Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys.-JETP 23,1025 (1966)].

2S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov,Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys.-Usp. 10, 609(1968)].

'E. Garmire, R. Y. Chiao, and C. H. Townes, Phys. Rev.Lett. 16, 347 (1966).

'H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).

943July 1974