Irradiance redistribution of collimated radiation using a

simple intensity redistributor

Dubravko Risovi6

The transformation from an annular beam intensity distribution to a cylindrical one is considered. A simple

refractive optical radiation intensity redistributor was designed and fabricated, and its performance is

discussed. Experimental results show that the redistributor is a useful optical element for transformations of

collimated annular beam irradiance. Key words: Irradiance redistribution, annular beam, laser intensity

redistribution.

1. Introduction

Optical systems that modify the irradiance distribu-tion of a collimated beam of radiation are useful pri-marily in applications involving lasers, but the need forirradiance redistribution emerges also in some devicesusing conventional light sources (e.g., solar simula-tors). Uniformity of the flux distribution deliveredfrom a source with a Gaussian intensity distribution toa working plane is a frequent requirement in manyapplications involving lasers. This problem has beendealt with in various ways: using refractive optics1 orreflectors, 2 aspheric lenses, 3 fly's-eye lenses,4 or holo-grams.5 Another similar problem is the intensity re-distribution of a highly nonuniform annular beamemerging from unstable laser resonators or some con-ventional sources provided with elliptic or parabolicmirrors. A solution of this problem was attemptedwith various beam integrators6 which overlap differentparts of the original beam, providing considerable uni-formity of intensity distribution, but only in one planeat a certain predetermined distance from the source.Another method is the use of specially designed mir-rors7 which are rather difficult to fabricate.

In this paper we describe an approach to opticalbeam shaping that is simple and is capable of trans-forming a highly nonuniform annular intensity distri-bution into a relatively collimated beam with quasi-Gaussian intensity distribution.

The author is with Rudjer Boskovi6 Institute, Department of

Laser & Atomic Research & Development, P.O. Box 1016, 41001Zagreb, Yugoslavia.

Received 28 November 1989.

0003-6935/90/182675-04$02.00/0.© 1990 Optical Society of America.

II. Optical Design of the Intensity Redistributor

The concept of the beam intensity redistributor isbased on beam shifting toward the beam axis. This isaccomplished by a simple specially designed opticalelement. Consider a cylindrical optical element withbases that are not flat but conical and with apexespointing in the same axial direction. The cross sectionof this element is given in Fig. 1.

A meridional ray falling on this element will be dis-placed laterally by

D = L F(nO), (1)

where L is the length of the cylinder and F is given by

sinO - ( 1 - sin2O )1/2]Io n2 - sin 2o) (2)

Here 0 is the angle between the conical surface and aplane perpendicular to the axis and n is the index ofrefraction of the material.

Therefore, by choosing the index of refraction of theoptical material, the sharpness of the redistributor'sconical bases, and its length we can achieve practicallyany desired lateral displacement of an incoming ray.For a chosen material (given n) and certain 0, thedisplacement is proportional only to the length of theelement. Dependence of the lateral displacement onangle 0 is more complicated, as illustrated in Fig. 2,where the dependence of normalized lateral displace-ment DIL on 0 is presented for two indices of refrac-tion, namely, that of BK7 and Plexiglass. It is seenthat, at first linear, this relationship saturates and thatincreasing the redistributor angle over 500 contributesvery little to the lateral displacement.

This is valid only for the rays that impinge on theredistributor at a distance R > Ro from the optical axis,that is, equal to or greater than a certain critical radiusR0, characteristic for the given redistributor. The

20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2675

Fig. 3. Transmission of an annular beam through the redistributor.basic parameters of the light intensityredistributor.

0.30 -

0.25 B

0.20 -_

0.15 -

0.10 -

0.05 -

n nn . . . . ... Fig. 4. Light intensity redistributor.. . . . . . . . . . . . . . . . . . I , , - -0.00 20.00 40.00 60.00

Redistributor angle (deg)

Fig. 2. Dependence of the normalized lateral displacement on an-gle 0 for the optical glass (BK7) and Plexiglass.

critical or displacement radius Ro of the redistributoris determined by

Ro = L F(n,0) (3)

and is equal to the displacement produced by thatredistributor.

If the rays enter the redistributor at a distance fromthe optical axis that is less than R0, they will be refract-ed and deflected from their original path and, there-fore, lost from the beam, because for these rays theredistributor acts as a divergent lens. The rays thatenter the redistributor at a still lesser distance fromthe axis will experience total internal reflection at theexit surface.

Consider now a collimated monochromatic annularbeam of radiation falling on the redistributor charac-terized by L, n, and 0, or simply with D. All the rays inthe beam will experience a lateral displacement of anamountD toward the axis (Fig. 3). If the redistributoris designed so that D r, where r denotes the innerradius of the annulus, the output beam will have nocentral dark area.

Since the sections of annulus with greater radii areeffectively contracted to the lesser radii, the intensityincreases toward the axis, and if the redistributor werenot to act as a divergent lens for rays entering thedisplacement radius about the axis, and if the apexwere ideally sharp, the intensity would increase toward

infinity at the axis and eventually destroy the tip of theexit conical surface. Since the apexes of the conicalsurfaces are not ideal (because of the fabrication pro-cess they have a small but finite radius), the intensityat the axis is finite.

The radial intensity distribution in the outgoingbeam is given by

I(r) = Ii(r + D)(1 +-); r > O, (4)

where r is the radial distance from the optical axis andIi is the input intensity distribution.

111. Experimental Results

The principles of beam shaping were verified experi-mentally using an acrylic (Plexiglass) redistributorand a He-Ne laser with a modified annular beam crosssection. The redistributor was machined from acrylicwith n = 1.491, 30°, and L = 30 mm, resulting indisplacement D = 5 mm. To minimize the overalllength, conical surfaces were segmented and shifted ina way similar to the method used for Fresnel lenses(Fig. 4). This introduces certain small inhomogenei-ties in the output beam, but these can be disregardedfor the present purpose. To obtain an annular beamthe output from a He-Ne laser was expanded and thentransmitted through an annular aperture that blockedout the central part of the beam. We denote the radialposition of the maximum intensity in the annulus (ra-dius of maximal intensity) by rM, the outer radius ofthe annulus by r, and the inner radius of the annulus

2676 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990

.0

aJ

C)

.SK)

-

-5

K)

tID

N

V

02

Fig. 1. Outline and the

25.00- Input intensity

- Theoretical output

0 Measured output (10 cm)

20.00

uW 15.000)

Q)10.00

5.00

0.00 .. . ..-. . .-30.00 -20.00 -10.00 0.00 1 0.00 20.00

Distance from the axis (mm)

Fig. 5. Calculated and measured intensity distributions for case 1:rM > D > rl. The measurements were made at the entrance plane ofthe redistributor and in the plane that is at a distance of 10 cm from

the redistributor.

Fig. 6. Case 1. (a) The cross section of the input beam and input

intensity distribution along the axis shown as recorded with a Hama-

matsu C158 measuring video system with a C1000 vidicon camera.

(b) The cross section of the output beam and output intensitydistribution along the axis shown.

(the radius of the dark area) by rj. The radial distribu-tion of intensity was measured with a PIN photodiodeoperated in the photovoltaic mode and mounted on anX-Y positioner to permit scanning across the beam.The dimensions of the annular beam varied while theredistributor's parameters remained the same. Theinput beam intensity distribution was measured be-fore the redistributor, and the output beam intensitydistribution was measured at a short distance from theredistributor. Some of the results are presented in

- Input intensity

- Theoretical outputo Mesured output (locm)

+ Measured output (20cm)

>,100.00

00

0

-0.00 -20.00 -10.0.Distance from the axis (mm)

Fig. 7. Calculated and measured intensity distributions for case 2:

rM _ D. The redistributor's displacement is optimal. The mea-surements were made at the entrance plane of the redistributor andin the planes that are at distances of 10 and 20 cm from the redistrib-

utor.

Fig. 8. Case 2. (a) The cross section of the input beam and input

intensity distribution along the axis shown as recorded with a Hama-

matsu C158 measuring video system with a C1000 vidicon camera.

(b) The cross section of the output beam and output intensitydistribution along the axis shown.

Figs. 5-9, where the measured and calculated intensitydistributions are shown. The theoretical output in-tensity distribution was calculated according to Eq.(4), starting with the measured input intensity distri-bution and taking into account the photovoltaic modeof the detector operation. Three cases were investi-gated:

(1) Displacement D is smaller than the radius ofmaximum intensity of the input beam but greaterthan the radius of the dark area: rM > D > rI. The

20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2677

1 50.00

150.00

,1 00.00

u)

a) ,

a 50.00 -a)

0.00 - 1r-30.00 -20.00 -10.00 0.00 10.00 20.00

Distance from the axis (mm)Fig. 9. Calculated and measured intensity distributions for case 3:ro > D > rM. The measurements were made at the entrance plane ofthe redistributor and in the planes that are at distances of 10 and 20

cm from the redistributor.

intensity distribution of the output beam still showsan annular structure but without a dark area and witha pronounced central maximum due to intensity am-plification near the axis [see Figs. 5 and 6(a) and(b)].

(2) Displacement D is approximately equal to theradius of maximum intensity of the input beam: D rM. The intensity distribution of the output beam isoptimal: the energy loss from the input beam is small,and the distribution is quasi-Gaussian strongly peakedat the axis. This distribution is preserved over a con-siderable distance along the propagation path, de-pending on the degree of collimation of the input beam[see Figs. 7 and 8(a) and (b)].

(3) Displacement D is greater than the maximumintensity radius rM but smaller than the outer radius ofthe input beam: r > D > r. The redistributor's

displacement is too large for the given input beam.There is no central dark area, and the output beamintensity distribution is similar in shape to the previ-ous case but, due to a much greater loss of energy fromthe input beam, the output intensity is considerablysmaller and approaches that of the input beam (seeFig. 9).

IV. Conclusion

We have experimentally shown that a highly non-uniform intensity distribution in a light beam can beredistributed to obtain a more homogeneous axiallycentered distribution. To achieve this an intensityredistributor, a simple optical element that can beeasily fabricated to suit a given purpose, is needed.The design of the redistributor is simple and its param-eters can be varied in a wide range to achieve thedesired intensity distribution. The best results withan annular beam are achieved when the redistributor'sdisplacement is approximately equal to the radius ofmaximum intensity.

References

1. P. W. Rhodes and D. L. Shealy, "Refractive Optical Systems forIrradiance Redistribution of Collimated Radiation: Their De-sign and Analysis," Appl. Opt. 19, 3545-3553 (1980).

2. T. E. Horton and J. H. McDermit, "Design of a Specular AsphericSurface to Uniformly Radiate a Flat Surface Using a NonuniformCollimated Radiation Source," Trans. ASME J. Heat TransferC94, 453-456 (1972).

3. D. Shafer, "Gaussian to Flat-Top Intensity Distributing Lens,"Opt. Laser Technol. 14, 159-160 (1982).

4. Y. Ozaki and K. Takamoto, "Cylindrical Fly's Eye Lens for Inten-sity Redistribution of an Excimer Laser Beam," Appl. Opt. 28,106-110 (1989).

5. C. Y. Han, Y. Ishii, and K. Murata, "Reshaping Collimated LaserBeams with Gaussian Profile to Uniform Profiles," Appl. Opt. 22,3644-3647 (1983).

6. D. M. Dagenais, J. A. Woodroffe, and I. Itzkan, "Optical BeamShaping of a High Power Laser for Uniform Target Illumina-tion," Appl. Opt. 24, 671-675 (1985).

7. J. W. Ogland, "Mirror System for Uniform Beam Transformationin High-Power Annular Lasers," Appl. Opt. 17,2917-2923 (1978).

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