Achromatic N-prism beam expanders: optimal configurations

Rick Trebino

In this paper, we calculate optimal prism configurations for achromatic N-prism beam expanders of a singlematerial; we argue that for moderate to high magnifications, that is, M > [2 - 1(2N-1 -1)]N, the up-up ... up-down configuration is generally optimal, in the sense that it maximizes the transmission for givenmagnification. We also derive exact expressions for the incidence and apex angles that optimize a nonachro-matic N-prism beam expander of arbitrary materials. The use of simple three-prism (up-up-down) andfour-prism (up-up-up-down) single-material achromatic beam expanders is suggested for applications re-quiring compactness, achromaticity, and temperature stability.

1. Introduction

Well known for over a century, 1 the prism beam ex-pander has only recently found application as a laser-related optical device. In the past few years, re-searchers have employed the prism beam expander(PBE) to obtain I-D beam expansion in situations re-quiring compactness, alignment simplicity, low cost,and/or a minimum of aberrations. In particular, thePBE has found application in wavemeters2 and opti-cal-fiber-diameter measurement schemes3 4 and as ageneral-purpose laboratory tool. The most importantapplication of the PBE, however, remains that of beampre-expansion prior to diffraction by a grating.5-2 2

Inside a high-gain laser cavity, this pre-expansion sig-nificantly improves laser linewidth at a small cost inlaser efficiency. More importantly, the 1-D nature ofthe expansion maintains alignment simplicity-a tre-mendous advantage over spherical lenses, cylindricallenses, or mirror telescopes, in which element spacingsare critical or in which unnecessary 2-D expansion oc-curs. As a result, pulsed dye lasers- 2 2 and occasionallyCO2 lasers2 3 contain PBEs.

In many applications it is good practice to employ anachromatic PBE. While the dispersion of a nona-chromatic PBE can be made to add to that of the grat-ing and hence to further improve spectral resolution, the

The author is with Stanford University, Edward L. Ginzton Lab-oratory, Stanford, California 94305.

Received 18 October 1984.0003-6935/85/081130-09$02.00/0.© 1985 Optical Society of America.

temperature-induced wavelength drift associated withthe use of a nonachromatic PBE/grating combinationin a laser cavity is often undesirable. 2 4 In single-axial-mode pulsed dye lasers, such temperature-de-pendent effects are unacceptable,1 1 4 24 and as a result,commercial devices necessarily employ either an ach-romatic PBE25 or temperature stabilization.26 (Fejeret al. 3 4 were also obliged to avoid temperature effectsand hence required approximate achromaticity in thePBE in their optical-fiber-diameter measurement de-vice.) Finally, achromaticity allows extremely accuratesine-bar wavelength tuning of a laser employing a Lit-trow grating.

The simplest achromatic PBE consists of two prismsin a compensating up-down configuration so that thedispersion of the second prism precisely cancels that ofthe first. This configuration has been studied numer-ically by Barr,2 7 who found that the use of very differentprism apex angels can achieve achromaticity in such adevice. Two-prism PBEs achieving moderate to largemagnifications can be quite lossy, however, and as aresult, greater numbers of prisms are often employedso that each prism incidence angle can be reduced(compared with incidence angles required in a two-prism device with comparable magnification) and theoverall PBE transmission improved.1""14 A four-prismachromatic beam expander, involving a compensatingpair of compensating pairs (CPCP), i.e., a down-up-up-down configuration (see Fig. 1), of prisms has be-come popular and is used in several commercial pulseddye lasers.2 5 This device commonly achieves a mag-nification of the order of 40 with a single-pass trans-mission of >50%. Intuitively, this configuration is quiteappealing, but is it optimal? In other words, does thisarrangement yield the maximum transmission for givenmagnification for a four-prism achromatic PBE? Andwhat configurations optimize achromatic PBEs withother numbers of prisms? And, in addition, is it pos-

1130 APPLIED OPTICS / Vol. 24, No. 8 / 15 April 1985

Fig. 1. Standard down-up-up-down (compensating pair of com-pensating pairs) achromatic four-prism beam expander. This deviceconsists of two pairs of nearly achromatic two-prism beam expanders,with the second pair inverted with respect to the first. It achievesnearly collinear input and output beams but does not optimize

transmission.

sible to construct achromatic PBEs with odd numbersof prisms?

In this paper, we examine achromatic multiprismPBEs with regard to optimality, which we take to meanmaximal transmission for given magnification. Weshow that, in general, any number of prisms (that is,except for 1) can produce an achromatic device, andinterestingly, we find that the CPCP configuration isnot in general the optimal achromatic four-prism beamexpander. On the contrary, optimality is generallyachieved by the somewhat unintuitive configurationsin which the dispersions of the first N - 1 prisms add,with the dispersion of the last prism subtracting (i.e.,an up-up ... up-down configuration; see Figs. 2-4). Inparticular, we suggest a unique three-prism single-material achromatic beam expander, using an up-up-down configuration, which should be especiallyuseful for pulsed dye lasers in which the cavity lengthmust be kept short. For other applications, up-up ... up-down configurations of higher numbers ofprisms are suggested. We show that, for magnificationsgreater than about [2 - 1/(2 N-1 - I)N (i.e., about 2 Nfor large N), these arrangements always optimize theperformance of achromatic N-prism devices of a singlematerial.

11. Optimal Nonachromatic N-Prism Beam Expanders

We begin by calculating exactly the optimal solutionfor the nonachromatic N-prism beam expander of ar-bitrary materials, the specific value, N = 2, having beensolved approximately by Rdcz et al. 15 for the case ofidentical material for both prisms. The results of thiscalculation will prove necessary for later achromaticPBE calculations.

Consider an N-prism beam expander composed ofdifferent materials with refractive indices,n,n 2, .. , nN. We wish to determine the incidenceangles and apex angles that maximize transmission fora given value of the total magnification, M, withoutconcern for dispersion. Note, that, for now, the con-figuration (i.e., up-down, etc.) is irrelevant. We assumeAR-coated exit faces, but since broadband AR coatingsdo not generally exist for very high incidence angles(>65°), we assume uncoated entrance faces. Thetransmission of each prism will be determined by re-flective losses at the high-incidence-angle entrance face.

Fig. 2; Three-prism up-up-down beam expander. This configu-ration is optimal for achromatic three-prism beam expanders.

Fig. 3. Four-prism up-up-up-down beam expander. This config-uration is optimal for achromatic four-prism beam expanders of asingle material with total magnification > 10 and probably also for

lower magnifications.

Fig. 4. N-prism up-up ... up-down beam expander. This config-uration is optimal for achromatic single-material prism beam ex-panders of moderate to large magnification, specifically, for magni-fications greater than about [2-1/( 2 N-1 -1)]N. In addition, if eachprism magnification is 2, such a device achieves a total magnificationof 2 N, a dispersion of 1/2 N-1 that of a single prism, and a transmission

of 98% per prism.

Let the ith prism have incidence angle O and apex anglecai, with additional angular quantities defined in Fig. 5.We take the refractive index of this prism to be ni =

ni (X,T), where X is the wavelength of the light incidenton the prism and T is the temperature, which we taketo be constant throughout the prisms. Finally, 'yi = I i- ai + vi is the total deviation angle of an incidentbeam due to this prism (i is here defined to be >0).

15 April 1985 / Vol. 24, No. 8 / APPLIED OPTICS 1131

Observe that the numerator of Eq. (8) is constant andequal to 4 Nn1n2 .. nNM, so that, for the purposes ofoptimization, we can ignore it. The problem thus re-duces to minimizing the denominator, or equivalently,its square root. Thus we wish to minimize

L(Mi) = (n1 + M1)(n2 + M2)... (nN + MN),

subject to the constraint

M1M2 ... MN = M

(9)

Fig. 5. Prism geometry.

A simple Lagrange multiplier calculation yields theoptimal magnifications MPt:

MPt M/N(nzin2 ... flN)l/N 1N

The beam magnification due to the ith prism is22

Mi = .coski coSwi

and the total PBE magnification M isN

M= H M.i=1

which, when n1 = n2 = ... = nN, reduce to MoPt = MoPt=... MVt = M1/N. The optimal incidence angles o0Pt

(1) are found to be

09pt = arcsin (M t 2 -12ip(M )2 _ i1n,

2)

Assuming a p-polarized beam, the transmission Ti ofthe ith prism will be15

T = 4ni cospi/cosOi (3)(ni + coski/cosOi) 2

with the overall PBE transmission T given byN

T = Ti. (4)i=1

The reader is cautioned not to confuse the transmissionwith the temperature, both of which are denoted by theletter T.

Previous authors7 13151718,24,27 have approximatedthe exit angle 1ui by the value zero and consequently Miby the simpler result, cosoi/cosOi. We point out here,however, that this simplification can be deduced exactlyfor optimal designs. Clearly nothing is to be gained byallowing ,ui to be nonzero: nonzero values of Ai actmerely to decrease the magnification without changingthe transmission. For nonachromatic PBEs, then, Aui= 0 is exactly optimal, and it is easy to show that theoptimal values of all the system apex angles are

aopt = 0opt i = 1.N, (5)

where oPt will be determined by OPt, which in turn willbe determined by the optimal magnification.

We now observe that, with ui = 0, the optimal indi-vidual prism magnification is exactly given by

cosojM costj (6)

The individual prism transmissions can now be writtenin terms of Mi without approximation:

Ti = (+M4niM (7)(ni + M,) 2

so that the total transmission is

T = 4n1Ml 4n2M24nNMN

(n, + M1)2 (n2 + M2)2 (nN + MN)2

so that the optimal apex angles aPt are [using Eq. (5)and Snell's law]

Thus, for example, a PBE made entirely with materialof refractive index 1.52 and achieving a magnificationof 3 per prism should employ apex angles of 39.5° andincidence angles of 75.10. The optimal transmissionis easily obtained by substitution into Eq. (8) and, forthe case of a single-material PBE, is M[4n/(n +M1IN)2]N. For the above example, the transmissionwill be 89% per prism. The transmission dependssomewhat sensitively on the incidence angles, but theapex angles do not enter sensitively at all, with thetransmission independent of these angles, and themagnification varying by <10% despite as much as a 200variation in one of the apex angles. Thus, a very nearlyoptimal device can be achieved with apex angles quitedifferent from aoPt.

11. Dispersion in Prism Beam Expanders:Discussion

The dispersion of the ith prism is9

O'yi sinai

dni cosgi cosvi

Dispersions of the form 3yi/d9X or a-Yi/dT are easilyobtained from Eq. (14) via the chain rule. We also de-fine the orientation of the ith prism to be Ei, with ei =+1 indicating an up orientation, and Ei = -1 indicatinga down orientation. Here, up and down refer towhether the ith prism bends the beam clockwise orcounterclockwise, respectively, rather than to theprism's actual orientation in space (see Figs. 1-4). Wenow define y (N) to be the total angular deviation of theN-prism PBE:

NT(N) = (y

i-1

1132 APPLIED OPTICS / Vol. 24, No. 8 / 15April 1985

(10)

(12)

aqt = arcsin [n2(M9 t)2- 1 (13)

(14)

(15)

i = 1, N, (11)

The total dispersion will be ay (N)/ax, where x = nX orT:

E ex (16)ax- i=l Ox

At a given wavelength and temperature, a PBE isconsidered achromatic to first order if a-y (N)/,9X = 0 andthermally stable to first order if dy (N)/aT = 0. In fact,second-order variation of these quantities can cause aPBE, designed to be achromatic to first order at onewavelength, to have a nonzero value of ay(N)/aX atother wavelengths. This is an important point becausePBEs in pulsed dye lasers can expect to see radiationfrom 300 nm to 1 um in wavelength and beyond. In-deed, Barr27 has shown that two-prism PBEs designedto have ay( 2)/aX = 0 at a given wavelength have varyingamounts of dispersion at other wavelengths. A trulyachromatic PBE would have zero-valued higher-orderderivatives also: aky (N)/aXk = 0. Similarly, a trulythermally stable PBE would have zero-valued higher-order derivatives: ak ,y (N)/aTk = 0. (Alternatively, adesigner could require first-order achromaticity orfirst-order thermal stability at two or more distinctoperating points, simultaneously, which would also havethe effect of minimizing the magnitudes of the higher-order derivatives.) The complexity of the problemmakes it extremely difficult if not impossible to designa perfectly flat, zero-valued y (N) vs X or T curve, soperfect achromaticity or thermal stability is unlikely.Most applications, however, have not required extremeachromaticity or thermal stability, and first-order cal-culations have generally been sufficient.27 For exam-ple, consider a single-axial-mode (0.001-A linewidth)laser employing a Littrow grating and first-order-ach-romatic PBE designed for the wavelength A0 and tem-perature To. Now suppose that the laser operates ata wavelength quite distant from 0. Assuming aworst-case scenario, will the laser's wavelength nowexhibit significant thermal drift? The grating disper-sion will be -3 X 103 A/rad; the prism material (takeBK-7, for example) will have Idn/dTI 3 X 10-6; andBarr27 shows that, for first-order-achromatic two-prismdesigns, the rate of change of dispersion with respect ton, that is, 02 y(2 )/an2 , varies from 0.8 to 2 for rea-sonable prism parameters. Taking a rather largechange in the prism refractive index (due to the varia-tion in wavelength) of 0.015, resulting in ad)y(2)/an <0.03 at the new wavelength, we find that y/aTI 3 X10-4 A/C. Thus, a (quite large) 3C temperaturechange is now required to cause a wavelength drift ofone linewidth. We conclude that first-order achro-maticity is adequate in this case. That at least first-order achromaticity is required here is clear: the dis-persion of a single beam-expanding prism, ay(l)/an, isof order unity, resulting in a value of j aX/aT I about 30times larger than that above. We will thus restrict ourattention to first-order achromaticity or thermal sta-bility, although it is probably a good idea to calculatethe appropriate second-order quantity for a desireddesign. Such higher-order calculations are beyond thescope of this paper, however. In any case, we will con-

tinue to use the technically incorrect term "achromatic"as a shorthand for the longer "achromatic to first orderat a given wavelength and temperature," hoping thatthis usage does not cause undue confusion.

For a PBE constructed from a single material of re-fractive index n, our interest will lie only in ay(N)/anbecause an/aT or an/aX will factor out of all terms of Eq.(16), and it is these remaining terms that we must setequal to zero. For a single-material PBE, then, ach-romaticity will be solely a geometrical consideration,with material considerations unimportant, except fora single parameter, the refractive index. Achromaticityfor such a device, then, will not depend on possiblyerror-prone measurements of the material dispersionand will not be hurt by variations in this quantity fromprism to prism. In addition, for such a device, achro-maticity (i.e., wavelength-independent operation) willbe equivalent to high temperature stability. This is thecase because ay(N)/aX and y(N)/aT are both propor-tional to ay(N)/an; the material dispersion (an/aX) orthermal derivative (dn/dT) factors out. For this rea-son, single-material achromatic PBEs are generallypreferred. 2 8

Duarte and Piper20 show that

aEi

d ) (N) N ax= E

ax i= Ni Ml

j=i+i

(17)

More specifically, for two-, three-, and four-prism beamexpanders,

de(2) 1 O-y' OY2-2=-El - + E2 -ax M 2 ax ax

ax =3M=M-l + e(92 + 3 373Ox M2M x M3 Ox Ox

Oy(4) 1 aY i 1 e Y2 1 eY 3 974ax M EMM 1 + x E2-+-f3-+ 4 -Ox M 2M 3M 4 Ox M 3M 4 Ox M 4 Ox Ox

(18)

(19)

(20)

Thus, the contribution of each individual prism to thetotal PBE dispersion is reduced by the total magnifi-cation experienced by the beam after that prism. Thisis why two identical prisms with identical incidenceangles cannot produce an achromatic [i.e., aqy(2)/ax =

0] PBE: different apex angles,2 7 incidence angles, ormaterials for each prism are required-a situationsomewhat analogous to the design of an achromaticdoublet lens.

Equations (17)-(20) also illustrate a somewhat phil-osophical point about the achromatic PBE: that itsconstruction from adjacent pairs of inverted, compen-sating prisms is neither necessary nor desired to attainachromaticity. To see this, note that only the finalprism's dispersion contribution remains undiminishedby any magnification-dependent factor. It is thuspossible-indeed, probable-that, in high-magnifica-tion devices, the single dispersion term due to the finalprism may dominate in Eqs. (17)-(20). As a result, allother prisms will have to be arranged to compensate for

15April 1985 / Vol. 24, No.8 / APPLIED OPTICS 1133

the dispersion of this single prism; otherwise severedesign compromises will have to be made-reducingperformance. The next section illustrates this point.

IV. Example

Consider an N-prism beam expander constructedfrom a very large number of prisms of the same material.Suppose that we desire an achromatic expander, andfinally, suppose that we desire a total magnification ofM = 2 N. We proceed now to construct an exactly op-timal device for the case N - .

We have shown that to maximize the transmission ofa single-material PBE, all prism magnifications shouldbe equal, so we begin by setting Mi = 2 for all i. Thisrelation determines all the prism incidence angles, andin conjunction with the use of normal exit angles fromthe prisms, determines the prism apex angles. So far,we have merely constructed the maximal-transmissionPBE of magnification 2 N, paying no attention to theconfiguration or the device dispersion. It is likely thatthe above parameters cannot produce an achromaticPBE. In this example, however, the achromaticityconstraint will prove satisfiable by appropriate choiceof the various Ei, that is, by appropriate choice of con-figuration. And clearly, since our choices of incidenceand apex angles optimize the transmission of a PBEwithout regard to dispersion, this set of angles in con-junction with the achromatizing configuration will op-timize the achromatic PBE of N-prisms with a magni-fication of 2 N as N Co.

We proceed to determine this optimizing configura-tion by first observing that all Oi are equal, and all cai areequal. In addition, the exit angles vi, which are func-tions of Oi, ai, and n, are also equal. Equation (14) thusimplies that all prism dispersions will be equal. Fac-toring out the individual prism dispersions, we find thatthe total PBE dispersion for this example is

___(N fN-1 N-2+ EN-3 El (1-cc EN +-+t-+ 8+***+ 2N1 (21)On 2 4 8 2 N-1

Remembering that Ei = +1 depending on the prismorientation, we see that Eq. (21) sums to values between-2 and +2, depending on the Ei. In the limit, N - ,the series equals 0 exactly when el = E2 = 3 = ... = EN-1

=- EN, i.e., for the up-up-up ... up-down (or, equiv-alently, the down-down-down ... down-up) configu-ration (see Fig. 4 for N = 10). Other arrangements willnot achieve achromaticity and will, as a result, requirecompromises, such as different magnifications (andhence less transmission), or different apex angles, whichwill result in less magnification and/or transmission.The above arrangement thus optimizes the achromaticN-prism beam expander with magnification 2 N in thelimit of very large N.

This example appears somewhat contrived, but infact it is actually quite representative of the problemsencountered in the construction of achromatic PBEs.29

Suppose, for example, that the PBE is to contain onlya small number of prisms. In this case, the dispersionof the final prism will exceed that of all the precedingprisms combined. Of all possible configurations, the

up-up ... up-down configuration, using i = 0Qpt andai = a9Pt, will most closely approach achromaticity, andwill thus require the fewest design compromises, hencemaximizing the transmission. Suppose, instead, thata greater magnification is required. Again, the up-up ... up-down configuration will most closely ap-proach achromaticity with identical magnifications andapex angles and hence will require the fewest designcompromises.

On the other hand, the example also illustrates sit-uations in which the above configuration will not beoptimal. Low-magnification expanders, requiring in-dividual magnifications somewhat less than 2, will re-quire additional prisms oriented in the same directionas the final prism, as the total dispersion contributionof the first N - 1 prisms will now exceed that of a similarfinal prism. In addition, the use of different materialsfor the various prisms could also mandate a differentoptimal achromatic configuration. Most useful ach-romatic PBEs will, however, consist of prisms of a singlematerial to eliminate thermal and chromatic effectssimultaneously. In addition, useful devices will gen-erally demand a magnification per prism of -2 orgreater. Thus the up-up ... up-down arrangement willusually prove optimal in practice. The next sectiontreats the optimal-configuration question more quan-titatively.

V. Optimal Configurations for Achromatic N-PrismBeam Expanders of a Single Material

Analytical optimization of an achromatic N-prismbeam expander requires the maximization of thetransmission subject to two constraints: that the totalmagnification M is equal to a given value and that thetotal dispersion is zero. At its simplest, this problemis quite difficult, and Barr,27 who took on the problemof the two-prism achromatic beam expander, did sonumerically. The many-prism problem is furthercomplicated by the large number of configurations thatare possible even for intermediate numbers of prisms;for example, there are thirty-one possible nonredundantsix-prism devices to consider. In this work, our goal isto ascertain the configuration appropriate to a givenmagnification for an N-prism achromatic device. Weapproach this problem by asking when the optimal so-lutions for nonachromatic PBEs will yield achromaticPBEs simply by proper choice of configuration. We willfind that corresponding to each possible configurationwill be a characteristic magnification for which thatconfiguration yields an achromatic PBE with trans-mission equaling that of the optimal nonachromaticPBE. These values thus clearly determine the optimalachromatic PBE of the characteristic magnification. Asa result, we will have obtained exact solutions to theproblem of optimizing a single-material N-prism ach-romatic beam expander, but only for a few values of themagnification. Because the functions involved arecontinuous in the regions of interest, however, a con-figuration that is optimal at a given magnification willbe the optimal configuration in a neighborhood ofmagnifications about the given magnification. The few

1134 APPLIED OPTICS / Vol. 24, No. 8 / 15 April 1985

optimal solutions derived here will then approximatelydetermine the optimal configuration as a function ofmagnification, since devices requiring a magnificationnear one of the characteristic magnifications will nec-essarily employ the same configuration to achieve op-timality.

Our approach then, remembering the results of Sec.II, is to assume equal magnifications and dispersions foreach prism in the expander, with each individual prismmagnification defined to be m and with the total mag-nification given by M = MN. Factoring out the indi-vidual prism dispersion from each term in Eq. (17),setting the total dispersion a9y (N)/an equal to zero, andmultiplying through by mN , we have the conditionfor simultaneous optimality and achromaticity:

NZ_ eimi-i = 0,

i=l(22)

a simple polynomial in the variable m. We will refer toEq. (22) as the characteristic polynomial of theE,,E2, .. . ,EN configuration; there is a one-to-one corre-spondence between PBE configurations and polyno-mials with unity-magnitude coefficients. The root ofa characteristic polynomial represents the individualprism magnification for which the E1, 2,... ,EN config-uration is optimal.

To eliminate redundancy in the analysis that follows,we now define EN to be -1 so that the final prism pointsdownward. There are thus 2N-1 possible nonredund-ant configurations for an N-prism beam expander, andrejecting the all-down configuration, which cannot bemade achromatic, 2 N-1 - 1 possible configurationsremain. There are thus three possible three-prism andseven possible four-prism nonredundant achromaticprism beam expanders. Roots of the characteristicpolynomials were obtained using the SOLVE routineof an HP-15C calculator. Table I lists these roots forall three-, four-, five-, and six-prism beam-expanderconfigurations. Imaginary roots and real roots that areless than one have been neglected, as they have nophysical value here.

Observe that the up-up ... up-down configurationalways yields the highest magnification, and that thismagnification approaches 2 N for large N, reminiscentof the example in Sec. IV. Configurations of the formdown-up-up ... up-down yield the next highest mag-nification, since the first prism's dispersion is reducedby the greatest number of magnification factors [seeEqs. (17)-(29)], so that its orientation matters the least.Note that many configurations have no physically in-teresting roots and hence are generally not usefulchoices for achromatic PBEs. In some cases, two con-figurations yield the same root, meaning that either chnbe employed to achieve the appropriate magnificationoptimally. These degeneracies occur because theseconfigurations actually consist of pairs of smaller con-figurations. Specifically, the two six-prism expandersobtaining M = 17.94 actually contain two achromaticthree-prism (up-up-down) beam expanders (one isinverted). The same holds true for the M = 4.24 pairof six-prism beam expanders: nonachromatic three-

Table I. Exact Optimal Achromatic PBE Solutions

Number ofprisms (N) Configuration m M = mN T

3 uud 1.62 4.42 1.00

uuud 1.84 11.44 0.96duud

4 udud 1.00 1.00 0.85uudd)

uuuud 1.93 26.61 0.92duuud 1.72 15.15 0.98

5 uduud 1.51 7.93 1.00uudud 1.29 3.58 0.97uuudd 1.18 2.28 0.93

uuuuud 1.97 57.73 0.89duuuud 1.88 44.60 0.93uduuud 1.79 33.16 0.95uuduud

6 dduuudJ 1.62 17.94 0.99uuudud 1.41 7.78 0.99uuuuddduduud 11.27 4.24 0.96several 1.00 1.00 0.78

Note. Roots of the PBE characteristic polynomials and the re-sultant magnifications of exactly optimal achromatic PBEs of a singlematerial. Here, m is the individual prism magnification, while Misthe total magnification of the PBE. For each magnification shownhere, the corresponding configuration is proved to be optimal (seetext). Total transmissions T are listed assuming n = 1.5. Note thatthe up-up ... up-down configuration, whose root m approaches 2 asN , always yields the largest magnification. Use of individualprism magnifications less than the material refractive index n iswasteful; such low-magnification solutions, which require incidenceangles below Brewster's angle, are included here only for complete-ness. (In this table, u = up and d = down.)

prism beam expanders appear as the even and oddprisms in these devices.

For magnifications other than those derived here,approximate optimal solutions should be obtainable byemploying the configuration suggested by Table I, i.e.,that of the listed magnification nearest to the desiredmagnification. Thus, a six-prism beam expander witha magnification of 42 (which is close to 44.60) shouldemploy a down-up-up-up-up-down configuration forgreatest transmission. We do not as yet know theprecise angles to use for such a device, but the solutionlisted here serves as a good starting point toward findingthis solution or approximating it empirically. Table Ithus represents an approximate correspondence be-tween the desired magnification and the optimal con-figuration for that magnification. This correspondenceis illustrated in Fig. 6, which plots the optimal config-uration for an achromatic PBE vs the desired magnifi-cation. Figure 7 illustrates (estimated) optimaltransmission vs magnification curves for achromaticPBEs of various configurations with the exact optimaltransmission vs magnification for a nonachromatic PBEalso shown for reference.

From this analysis we conclude that any achromaticPBE with a desired magnification per prism approxi-mately equal to or greater than the root of the charac-teristic polynomial of the up-up ... up-down configu-ration (always the largest root) should employ that

15 April 1985 / Vol. 24, No. 8 / APPLIED OPTICS 1135

NUMBEROF

PRISMS

6

5

4

3

OPTIMAL CONFIGURATIONS

duduud uuduud duuuudseveral uuuudd uuudud dduuud uduuud uuuuud02222222 ::222:O22: 41-2 2222:2 22:5Y-2222 is :

uuudd uudud uduud duuud uuuud2a 1*i:W2 - vBvv 2 .

duudududuudd uuud

uud22*gWa2BB<BX 2BB'X2>

1 2 5 10 20 50 100ACHROMATIC PBE MAGNIFICATION

Fig. 6. Optimal prism configurations vs magnification for single-material three-, four-, five-, and six-prism achromatic beam ex-panders. Exact solutions derived herein are shown as dark dots,which are extended to indicate the estimated range of magnificationsfor which each configuration is optimal. The up-up ... up-downconfiguration appears optimal for large magnifications for all

PBEs.

100% [-

z0U)U)

U)2Hl

OPTIMALNON -ACHROMATIC

DEVICE ,

ACHROMATIC CONFIGURATIONS

MAGNIFICATION

Fig. 7. Estimated optimal transmission vs magnification for severalconfigurations for an N-prism single-material achromatic beam ex-pander. Each configuration is optimal at some characteristic mag-nification and also in a neighborhood about that point but becomesnonoptimal near another configuration's characteristic magnification.(The horizontal axis can be interpreted as total magnification ormagnification per prism). The dashed line illustrates the optimaltransmission for a nonachromatic single-material PBE (see Sec. II),which attains maximal (100%) transmission for Brewster angle inci-dence, when the total magnification is nN. Note that at the charac-teristic magnifications, ml, .i n , M5 , optimal nonachromatic trans-mission and optimal achromatic transmission (using the appropriate

configuration) are equal.

configuration to obtain optimal performance. It is easyto show that this root is approximately given by 2 -1/( 2 N-1 - 1). Since this root always corresponds to amagnification per prism of less than 2 and practicalprism beam expanders employ larger magnifications,we conclude that in all practical situations the up-up ... up-down configuration will be the optimal con-figuration. Precisely how much better this configura-tion will be than the down-up-up ... up-down ar-

rangement (the next best configuration) is not clearfrom our analysis, although we estimate that for mag-nifications near the up-up ... up-down root, flippingthe first prism will not decrease transmission signifi-cantly. This conclusion follows from the insensitivityof the magnification and transmission to small varia-tions in the exit angle of the prism, which translates toinsensitivity to apex angle. Thus, by varying prismapex angles, a nonoptimal configuration will probablyachieve a transmission near to that of an optimal con-figuration for magnifications not too far from thenonoptimal configuration's characteristic magnification.For magnifications far from a configuration's charac-teristic magnification, the prism apex angles will nec-essarily deviate significantly from apt, prism exit angleswill deviate significantly from zero, and demagnificationat the exit face will become important. Higher inci-dence angles will be required to compensate for thisdemagnification in order to achieve the desired mag-nification, and as a result, the transmission will decreasesignificantly. Thus, for large magnifications, the up-up ... up-down configuration will not only be optimal,but will be significantly more transmissive than otherconfigurations. A quantitative resolution of this issuewill appear in a future publication dealing with thespecial cases N = 3 and 4.

The up-up ... up-down configuration may proveinconvenient for some applications due to the beamsteering involved (see Figs. 2-4), but for other applica-tions, such as preexpansion before a diffraction grating,this effect is generally not a problem since beam steeringnecessarily occurs anyway at the grating. The up-up-down three-prism and up-up-up-down four-prismachromatic PBEs may be of particular value for thisapplication inside lasers, and in particular, in pulsed dyelasers. Replacement of current down-up-up-downdevices with higher-transmission up-up-up-down de-vices (see Fig. 8) could improve dye laser efficiency.Alternatively, replacement with simpler up-up-downthree-prism devices (see Fig. 9) will decrease dye lasercost and complexity. If the optimal achromatic up-up-down PBE proves less efficient than down-up-up-down devices of the same magnification, the in-crease in laser efficiency due to the shortening of thedye-laser cavity length by one prism may offset this loss.Finally, the decrease in the amount of glass in the cavityis desirable.

Another possibility is the use of the achromaticthree-prism up-up-down device in a hybrid PBE/grazing-incidence dye-laser design, following the designof Rdcz et al.,15 who employed a two-prism nonachro-matic beam expander in a grazing-incidence dye-lasercavity in order to decrease the high incidence angle ofthe grating to improve its efficiency from its value of-5% at 890 incidence angle. Their design proved moreefficient than either the simple grazing-incidence orfour-prism Littrow-grating designs25 at most usefullinewidths. Since the diffraction efficiency of a2400-lines/mm holographic grating in the visible re-mains relatively high (70%) for incidence angles ashigh as -700 and then trails off rapidly, a PBE magni-

1136 APPLIED OPTICS / Vol. 24, No. 8 / 15 April 1985

l

PUMP

OUTPUT BEAMWINDOW

H

CELL

PUMPBEAM

ACHROMATIC3-PRISM

BE AMEXPANDER

/LITTROWGRATING

Fig. 8. Four-prism (up-up-up-down) achromatic PBE/Littrow-grating dye laser. The up-up-up-down configuration should be moreefficient than the down-up-up-down configuration for relatively large

magnifications, such as -40.

GRAZING INCIDENCE GRATING

Fig. 10. Three-prism (up-up-down) achromatic PBE/grazing-incidence-grating dye laser. Such a hybrid design is more efficientthan the PBE/Littrow-grating design or the grazing-incidence de-sign.' 5 Use of a three-prism achromatic design (vs the nonachromatictwo-prism design of Ref. 15) should allow thermally stable single-mode operation and should further improve efficiency. (In Figs. 8-10,the pump beam is shown entering from above for diagrammatic

simplicity only.)

PUMP

BE AM ACHROMATIC3-PRISM BEAM

EXPANDER

- M 40 LITTROW

OUTPUTWINDOW

Fig. 9. Three-prism (up-up-down) achromatic PBE/Littrow-gratingdye laser. For a magnification of -40, a transmission of -50% can

be achieved.

fication of -20 in such a hybrid design will continue tofill an ,70° incidence angle, 5-cm grating (maintaininggood linewidth) and, assuming high PBE transmission,will yield very efficient dye-laser operation. A three-prism achromatic up-up-down glass PBE with a mag-nification of 20 will transmit >70% of the light incidenton it3O and hence should prove ideal for such a hybriddye-laser design (see Fig. 10). At present we are oper-ating such a three-prism hybrid dye laser pumped bya frequency-doubled Nd:YAG laser, and with it we areachieving much higher efficiencies than we have witha simple grazing-incidence arrangement. Due to itssignificantly higher efficiency, and the weak toleranceson all prism orientation angles, its alignment is easierthan that of the grazing-incidence dye laser, and thebeam bending due to the prisms does not prove prob-lematic. Our application of this laser required only-1-GHz wavelength stability, so we do not here reportthermal-stability data.3 '

VI. Conclusions

We have obtained an exact solution to the problemof maximizing the transmission of an N-prism nona-chromatic beam expander of arbitrary materials andhave employed this solution to obtain exact optimalsolutions for single-material achromatic N-prism beamexpanders of various magnifications. From these re-sults we have shown that the value of the magnificationdetermines the optimal configuration of an achromaticN-prism beam expander, and we have tabulated theseconfigurations for achromatic three-, four-, five-, andsix-prism single-material beam expanders. We arguedthat for practical achromatic PBEs, the up-up ... up-down configuration always maximizes the transmissionand hence is to be preferred in most applications. Inparticular, three-prism (up-up-down) and four-prism(up-up-up-down) single-material achromatic devicesmay be of great use, especially in pulsed dye lasers.

Our analysis of achromatic single-material PBEs,which was based on finding zeros of polynomials withunity-magnitude coefficients, can be easily generalizedto treat the more general problem of multimaterialPBEs by simply allowing more general coefficients.Our analysis does not, however, indicate exact incidenceand apex angles for general achromatic prism beamexpanders. In addition, it gives no indication preciselyhow superior one configuration is over another. Thesetwo issues will be the subject of a future paper in whichnumerical solutions for three- and four-prism achro-matic beam expanders will be reported.

The author would like to thank Adnah Kostenbauderfor helpful conversations and suggestions and GregMagel for a critical reading of the manuscript. R. L.Byer also contributed very useful information to thisendeavor, for which the author is very grateful. Theauthor also appreciates the financial assistance of theAir Force Office of Scientific Research.

15 April 1985 / Vol. 24, No. 8 / APPLIED OPTICS 1137

ACHROMATIC

4-PRISMBEAM

EXPANDERM-40 OUTPUT

WINDOW CELL

References1. I. W. Jackson, An Elementary Treatise on Optics (G. Y. Van

Debogert, Schenectady, N.Y., 1852).2. F. J. Duarte, "Prism-Grating System for Laser Wavelength

Measurements," J. Phys. E. 16, 599 (1983).3. M. M. Fejer, G. A. Magel, and R. L. Byer, "High-Speed, High-

Resolution Fiber-Diameter-Measurement System," Appl. Opt.24, in press (1985), to be published.

4. M. M. Fejer, J. L. Nightingale, G. A. Magel, and R. L. Byer, "LaserHeated Miniature Pedestal Growth Apparatus for Single CrystalOptical Fibers," Rev. Sci. Instrum. 55, 1791 (1984).

5. S. A. Myers, "An Improved Line Narrowing Technique for a DyeLaser Excited by a Nitrogen," Opt. Commun. 4, 187 (1971).

6. E. D. Stokes, F. B. Dunning, R. F. Stebbings, G. K. Walters, andR. D. Rundel, "A High Efficiency Dye Laser Tunable from theUV to the IR," Opt. Commun. 5, 267 (1972).

7. D. C. Hanna, P. A. KdrkkAinen, and R. Wyatt, "A Simple BeamExpander for Frequency Narrowing of Dye Lasers," Opt. Quan-tum Electron. 7, 115 (1975).

8. L. G. Nair, "On the Dispersion of a Prism Used as a Beam Ex-pander in a Nitrogen Laser Pumped Dye Laser," Opt. Commun.23, 273 (1977).

9. R. Wyatt, "Comment on On the Dispersion of a Prism Used asa Beam Expander in a Nitrogen-Laser-Pumped Dye Laser," Opt.Commun. 26, 9 (1978).

10. R. Wyatt, "Narrow Linewidth, Short Pulse Operation of a Ni-trogen-Laser-Pumped Dye Laser," Opt. Commun. 26, 429(1978).

11. G. K. Klauminzer, "New High-Performance Short-Cavity DyeLaser Design," IEEE J. Quantum Electron. QE-13,92D (1977);U.S. Patent 4,127,828 (28 Nov. 1978).

12. J. Krasifiski and A. Sieradzau, "A Note on the Dispersion of aPrism Used as a Beam Expander in a Nitrogen Laser PumpedDye Laser," Opt. Commun. 28, 14 (1979).

13. F. J. Duarte and J. A. Piper, "A Double-Prism Beam Expanderfor Pulsed Dye Lasers," Opt. Commun. 35, 100 (1980).

14. A. F. Bernhardt and P. Rasmussen, "Design Criteria and Oper-ating Characteristics of a Single-Mode Pulsed Dye Laser," Appl.Phys. B 26, 141 (1981).

15. B. Racz, Zs. Bor, S. Szatm6ri, and G. Szabo, "Comparative Studyof Beam Expanders Used in Nitrogen Laser Pumped Dye Lasers,"Opt. Commun. 36, 399 (1981).

16. D. A. Greenhalgh and P. H. Sarkies, "Novel Geometry for SimpleAccurate Tuning of Lasers," Appl. Opt. 21, 3234 (1982).

17. F. J. Duarte and J. A. Piper, "Comparison of Prism-Expander andGrazing-Incidence Grating Cavities for Copper Laser PumpedDye Lasers," Appl. Opt. 21, 2782 (1982).

18. F. J. Duarte and J. A. Piper, "Narrow Linewidth, High prf CopperLaser-Pumped Dye-Laser Oscillators," Appl. Opt. 23, 1391(1984).

19. F. J. Duarte and J. A. Piper, "Multi-Pass Dispersion Theory of

Prismatic Pulsed Dye Lasers," Opt. Acta 31, 331 (1984).20. F. J. Duarte and J. A. Piper, "Dispersion Theory of Multiple-

Prism Beam Expanders for Pulsed Dye Lasers," Opt. Commun.43, 303 (1982).

21. H. J. Kong and S. S. Lee, "Dual Wavelength and ContinuouslyVariable Polarization Dye Laser," IEEE J. Quantum Electron.QE-17, 439 ( 9~1).

22. T. Kasuya, T zuki, and K. Shimoda, "A Prism AnamorphicSystem for Gaussian Beam Expander," Appl. Phys. 17, 131(1978).

23. F. J. Duarte, "Variable-Linewidth High-Power TEA CO 2 Laser,"journal to be published.

24. C. S. Zhou, "Design of a Pulsed Single-Mode Dye Laser," Appl.Opt. 23, 2879 (1984).

25. Quantaray, Molectron.26. Lambda Physik.27. J. R. M. Barr, "Achromatic Prism Beam Expanders," Opt.

Commun. 51, 41 (1984).28. We note here that intracavity laser applications, for example,

generally require high temperature stability but could actuallybenefit from some dispersion, which could act to further decreasethe laser linewidth somewhat. Thus a device with ay (N)/9T O at all relevant wavelengths but with large ay (N)/OX, constructednecessarily of more than one material could be quite useful.Whether the appropriate materials exist to fabricate such a deviceis unknown to this author.

29. The arrangement of this example may actually be of practicalinterest. Using a finite number of identical glass (n = 1.5) prisms,each with a magnification of 2, yields a device with magnification

2 N, dispersion equal to 1 /2 N-1 of that of a single prism (perhapssmall enough for many applications), and a reflection loss of only2% per prism.

30. A far from optimal three-prism achromatic expander can beconstructed easily and cheaply from already coated off-the-shelf45-45-90 BK-7 prisms, yielding a magnification of 20.1 and atransmission of 65%. The required incidence angles are 80, 77,and 530, respectively. The use of smaller apex angles will moreclosely approach optimality and hence will yield higher trans-mission. The above nonoptimal case is probably of practicalvalue, however.

31. The question of thermal stability due to PBE dispersion is greatlycomplicated by the issue of the thermal stability of the mechanicalmounts used for the optics in the cavity which can cause as muchas -0.5 cm-'/'C drift in the dye-laser wavelength. 3 2 The use ofthermally stable or compensated construction for such mountsis critical even in multimode devices. Taking such care, we haveobserved mechanical mount-induced thermal drifts of <0.1cm-'/ 0 C near room temperature. Commercial designs, in gen-eral, do even better.

32. F. J. Duarte, "Thermal Effects in Double-Prism Dye-LaserCavities," IEEE J. Quantum Electron. QE-19, 1345 (1983).

0

1138 APPLIED OPTICS / Vol. 24, No. 8 / 15 April 1985