OPTICAL ACTIVITIES IN INDUSTRY

Prism for in-line beam expansion in one dimension

Karl A. Stetson

A four-sided prism is described that expands or compresses a beam in one dimension without altering thedirection of the beam or displacing its centerline.

Optical Activities in Industry is re-ported by Stephen D. Fantone, OptikosCorporation, 286 Cardinal MedeirosAvenue, Cambridge, Massachusetts02141. Stephen welcomes letters,news, and comments for this column,which should be sent to him at theabove address.

Stephen D. Fantone

1. Introduction

An anamorphic beam expander has been described byFantone1 that makes use of a simple three-sidedprism for expanding or compressing a beam in onedimension. The primary difficulty with that elementis that it displaces the centerline of the beam, asshown in Fig. 1. Although the beam can be broughtback to its original axis by a tilted etalon or a secondprism, there are applications in which a single ele-ment is desired that will expand or compress thebeam without displacing it. Figure 2 illustrates aprism with four sides, A, B, C, and D, which providesfour internal reflections so that the beam can exit inline with the incoming beam. The beam enters sideAand exits side B, or vice versa, and sides B and D areparallel so that only two angles need be specified.As might be expected, the design of this prism issomewhat more complex than the simple three-sidedprism and requires not only the correct grinding oftwo angles but also proper sizing of the prism and

The author is with Karl Stetson Associates, 2060 South Street,Coventry, Connecticut 06238.Received 28 November 1994; revised manuscript received 27

January 1995.0003-6935@95@224634-03$06.00@0.

r 1995 Optical Society of America.

4634 APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995

silvering of a segment of side B. In this paper isprovided the design equations and their derivation forthis type of prism beam expander.

2. Angle Specifications

Because this prism has two adjustable angles, thebeam expansion is not solely dependent on the indexof refraction but can be adjusted by the prism angleU1between sides A and B, which in turn is equal to theangle of incidence relative to the surface normal of thenarrow beam entering the prism at side A. Theexpansion can be calculated with the help of Fig. 3:

cos U1 5 h1@a, cos U2 5 h2@a. 112

The magnificationm is

m 5 h2@h1 5 cos U2@cos U1. 122

Fig. 1. Three-sided beam-expanding prism. The beam is ex-panded by refraction and reflected so as to exit in the samedirection but with a displacement of its center.

We can express U2 in terms of the angle U1 usingSnell’s law and Eq. 122 to obtain

U1 5 sin2131m2 2 12@1m2 2 1@n2241@2. 132

The second prism angle F can be calculated withFig. 4. If we assume that the beam enters side A, itmust reflect from side C so that after the beam reflectsfrom sides D and A it exits the prism in the samedirection it was going when it approached side A.Figure 4 shows that the ray travels from side B to sideC at an angleU1 2 U2 to the horizontal andmust leaveside C at an angle 2U1 2 180 to the horizontal. If wetake a positive angle to be counterclockwise from thehorizontal to the ray in question, the angle of side Cfrom the horizontal is the average of the two rayangles from the horizontal. Therefore

F 2 90 5 12U1 2 180 1 U 1 2 U22@2, 142

F 5 13U1 2 U22@2. 152

3. Dimensional Specifications

The dimensions of the prism are determined by therequired clear aperture and the requirement thatthe central, undeviated ray should be centered in theclear aperture. It is obvious from Fig. 2 thatthe silvered segment of side B should extend to butnot beyond the point where a perpendicular line fromthe corner between D and A intersects B. Thisresults in the maximum clear aperture for the prism.Given the angles of the prism and the dimensions

Fig. 2. Four-sided beam-expanding prism. Four internal reflec-tions are used to permit the beam to exit the prism on its originalaxis. Sides B and D are parallel for convenience of fabrication,and side B is slivered for a section of its length.

Fig. 3. Refractive beam expansion. The beam expansion causedby refraction is determined by the ratio of the cosines of U1 and U2,which are related by Snell’s law.

indicated in Fig. 5, we can write

h1 5 a tan190 2 U12 5 a@tan U1, h3 5 b@tan U1, 162

h4 5 b tan1U1 2 U22,

h5 5 h4 2 h1 5 b tan1U1 2 U222 a@tan U1, 172

h2 5 h5 cos1U1 2 U22. 182

If h2 5 h1, then

b@a 5 31 1 1@cos1U1 2 U224@tan U1 tan1U1 2 U22. 192

So, given the clear aperture ap, we have

a 5 1⁄2ap tan U1,

b 5 a311 1@cos1U1 2 U224@tanU1 tan1U1 2 U22, 1102

T 5 a 1 b, 1112

which specifies the thickness of the prism.To specify the base of the prism x, refer to Fig. 6:

f 5 a tan1180 2 2U12 5 2a tan 2U1,

g1 5 1c 2 T2tan 2U1, 1122

g2 5 c tan1U1 2 U22, f 1 g1 5 h4 1 g2, 1132

Fig. 4. Diagram for determination of the prism angles. U1 isdetermined by magnification requirements, and F is determinedby the requirement that the original propagation direction bepreserved.

Fig. 5. Diagram for determination of the thickness 1fromB to D2 interms of the clear aperture required. The ratio of a to b is set sothat the undeviated ray is centered in the clear aperture.

1 August 1995 @ Vol. 34, No. 22 @ APPLIED OPTICS 4635

from which we derive

c 5 231T 1 a2tan 2U1 1 b tan1U1 2 U224

@3tan1U1 2 U22 2 tan 2U14. 1142

The base x is equal to

x 5 h3 1 h4 1 g2 1 c@tan F, 1152

from which

x 5 b31@tan U1 1 tan1U1 2 U224

1 c3tan1U1 2 U22 1 1@tan F4. 1162

Two other important angles are the angles to thesurface normals for rays reflecting from sides C andD. If these angles are less than what is required fortotal reflection, the side in question must be silvered.These are

UD 5 1802 2U1, UC 5 U2 2 U1 1 13U1 2 U22@2. 1172

4. Design Summary

The design equations for the four-sided beam-expand-ing prism are summarized as follows:Given that m is the expansion, n is the refractive

index, and ap is the clear aperture,

U1 5 sin2131m2 2 12@1m2 2 1@n2241@2,

U2 5 sin2131sin U1@n4 1Snell’s law2,

F 5 13U1 2 U22@2,

a 5 1⁄2ap tan U1,

b 5 a31 1 1@cos1U1 2 U224@tan U1 tan1U1 2 U22,

T 5 a 1 b,

c 5 231T 1 a2tan 2U1 1 b tan1U1 2 U224

Fig. 6. Diagram for determination of the length of the prismbase x.

4636 APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995

@3tan1U1 2 U22 2 tan 2U14,

x 5 b31@tan U1 1 tan1U1 2 U224

1 c3tan1U1 2 U22 1 1@tan F4.

A short computer program has been written to acceptm, n, and ap as inputs and provide U1, F, T, and x asoutputs. In Table 1 are listed the angles and dimen-sions of prisms with magnification factors from 1.5 to2.5 for a refractive index of 1.519 1BK7 at 530 nm2 anda clear aperture of 5 mm. Included in the table arethe angles to the surface normal of the internalreflections at faces D and C. It can be seen that formagnifications greater than 2.2, face D must besilvered to prevent escape of the beam; otherwise thereflections are beyond the critical angle for totalreflection. It is clear that prisms of this sort requireantireflection coating on side A for reflection losses tobe reduced, especially for light polarized normal tothe plane of the prism. Examples of this prism havebeen fabricated and function as predicted by thedesign equations.

5. Conclusions

This design for a beam-expanding prism offers advan-tages over the simplier three-sided design in that itdoes not deviate the centerline of the beam andpermits free selection of refractive index and there-fore a wider choice of magnifications. In applica-tions in which one prism is to be followed by another,with one or both rotated for two-dimensional controlof the beam profile, this design may have advantagesin that the beam center may be kept from beingtranslated as the different magnifications are ob-tained. The disadvantage of this design is its higherfabrication cost.

Reference1. S. D. Fantone, ‘‘Anamorphic prism: a new type,’’ Appl. Opt. 30,

5008–5009 119912.

Table 1. Angles and Dimensions of Prisms a

Magnifi-cation

U1

1deg2F

1deg2Thickness

1mm2Base1mm2

DAngle1deg2

CAngle1deg2

1.5000 56.0491 67.5239 16.0289 26.0968 67.9019 44.57421.6000 58.9240 71.2249 15.6134 23.0728 62.1520 46.62311.7000 61.2979 74.3116 15.3829 20.9183 57.4042 48.28421.8000 63.3004 76.9382 15.2747 19.3066 53.3993 49.66251.9000 65.0178 79.2086 15.2523 18.0566 49.9643 50.82712.0000 66.5107 81.1957 15.2922 17.0594 46.9785 51.82572.1000 67.8229 82.9531 15.3791 16.2459 44.3543 52.69262.2000 68.9869 84.5208 15.5023 15.5699 42.0262 53.45302.3000 70.0277 85.9297 15.6542 14.9996 39.9445 54.12582.4000 70.9649 87.2040 15.8292 14.5123 38.0703 54.72572.5000 71.8137 88.3632 16.0231 14.0911 36.3727 55.2642

aRefractive index 5 1.519, clear aperture 5 5.0 mm.