Birefringent Laser Mirrors

William Kahan

Narrow band mirrors composed of a polarizer, a stack of lossless birefringent crystals, and a conventionalmirror are described. Curves showing the reflectance characteristics of the mirrors are included. Designdata are tabulated for narrow band mirrors that have appreciable between-peak reflectances because ofthe Gibbs phenomenon as well as for slightly broader band mirrors with lower between-peak reflectances.Computations are presented in a series of graphs to demonstrate sensitivity of the response to the angularadjustment of the crystal elements. Electric tuning of the birefringent mirror is possible and can beenhanced by a double element design. The choice of electrooptic birefringent materials and the tem-perature sensitivity of the mirror's peak wavelength are discussed along with applications. An appendixprovides computational aids for the synthesis of birefringent filters.

IntroductionA class of birefringent optical filters can be con-

structed by placing N equally thick lossless birefringentcrystal elements between a polarizer and an analyzeras shown in Fig. 1. The anglesl, 2, * * , AN betweenthe slow axes of the birefringent elements and the passaxis of the polarizer, as well as the angle ON+1 betweenthe pass axis of the analyzer and that of the polarizerdetermine the transmission characteristics of the filter.Conversely, given a desired amplitude transmission vsfrequency for white light input, one can determine, bymeans of a synthesis procedure developed by Harriset a.,' the angles i of a filter with transmission char-acteristics that approximate those given. A briefreview of the birefringent filter synthesis proceduresufficient for the purposes of this article follows.

The synthesized filter's transmission is a periodicfunction of frequency with a frequency period P,given by

P,,, = 27r/a, (1)

where a is the difference between the propagation timethrough one of the birefringent crystal elements for alight pulse polarized parallel to the crystal's slow axisand for a light pulse polarized parallel to the crystal'sfast axis, so that

a = nL/c, (2)

where An is the difference between the crystal's slow andfast refractive indices, L is the thickness of the crystal,

and c is the speed of light in a vacuum.wavelength, the period of the filter is

P ) X2/bnL.

In terms of

(3)

The synthesis procedure for a birefringent filterbegins by expanding the desired amplitude transmissionfunction G in terms of the functions: eiM4, m = .1, . . ., N, . . ., where

6=_ aw, (4)viz.:

G(,) = A E Cme-im4.m=0

(5a)

For an N-element filter, G(4) is replaced by thetruncated series ( (I) given by

N

e(,P) = A CmeimO.m=0

(5b)

In Eq. (5), A may be any complex number of the form:

A = Ao exp[i(kPt + 1)], (6)

where Ao, k, and I are arbitrary real numbers. For thetype of birefringent filter discussed in this article, theCi coefficients are restricted to be real numbers, arestriction satisfied if ( is either an odd or an evenfunction of At, i.e., if e can be expanded in a Fourierseries with terms of only sinmi, or terms of only cosmVt.For example, if C is the odd function of ,

M

,(_) = E 'yk sinkP,

The author is with the Electro-Optics Group, Sperry GyroscopeDivision, Sperry Rand Corp., Great Neck, New York 11020.

Received 5 Juily 1968.

where all of the -yk values are real, then2M

e() = exp{i[MV + (3wr/2)]} ECreir,r=O

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 985

(7)

(8)

SLOW AXIS

INPUT-LIGHT FAST AXIS

Fig. 1. A birefringent filter containing seven crystal elements.The optical axes of the crystals are perpendicular to the lightray and the slow axis of the ith crystal element makes an angle0 with the pass axis of the input polarizer. In Ref. 1 the angle

0N±I is denoted by U.

Fig. 2. A N = 7 folded birefringent filter. The last half of thefilter has been replaced by a conventional mirror and the input

is coincident with the output.

withCr =Ym zIr) 0 < r<M- 1,

Cm = 0,

Cr =-r-lIr + < r < 2M.

We can neglect the multiplier of the summation ir(8) and consider the amplitude of the signal timitted by the output analyzer to be

2I

C k) = E3Cre-ir.r=0

(9a)

its mirror image, the result of transforming 0i to -i.

Consequently, a filter synthesized from a set of D.coefficients has three other trivial variations with thesame transmission characteristics. More than onepossible set of D, coefficients may satisfy Eq. (12),each set resulting in a different set of Oi but with thesame filter transmission. While some sets of D, giverise to trivial variations of the filter, others may giverise to totally different sets of Oi. In summary, bymeans of the synthesis procedure of Ref. 1, the N + 1numbers C0, C1, , CN are used to determine N + 1angles l 2, O ,N+1 of an N-element birefringentfilter with a transmission that approximates the desiredfunction of frequency.

SymmetryAnalysis of the symmetry relations obeyed by lossless

birefringent filters shows2 that if C = -CN, C1 =-CN-1, etc.; i.e., if C is an odd function of S/, then01 = N, 02 = ON- 1, and N+1 = 90g. If ON+1 were 00,then it would be possible to replace half the straightfilter by a mirror3 as shown in Fig. 2. The folded filterwould employ approximately half as many elements asthe straight filter and would be suitable for use as alaser cavity mirror. But because N+1 = 900, thereflected amplitude of the folded filter is not C(f) asdesired but D(Q), the amplitude of the signal rejectedby the output analyzer of the straight filter. However,one can design a folded filter3 by choosing an oddfunction e( ,) such that the resulting D (v) is theamplitude of the desired filter transmission.

To insure that the last half of the filter can be re-placed by a mirror, we require (t't) to satisfy

(() = -(3(-), (13)

(9b) and hence to be expressible in a Fourier sine series. Ifall the yi values of Eq. (7) are nonzero, then M terms

(9c) in Eq. (8) yield the (2M + 1)Ci coefficients given inEq. Eq. (9), and the synthesized filter, capable of being

rans- folded, has an even number (N = 2M) of birefringent

(10)

If the white light output intensity is Io2, then con-servation of energy gives

D,&) 12 = J2- C(Q) 12, (11)

where D (VI) is the amplitude of the signal absorbed bythe output analyzer. Appendix A of Ref. 1 gives analgorithm for finding D. coefficients of

N

D ()) = Dse-Wk,s=0

(12)

so that the resulting D () satisfies Eq. (11). Once theD, coefficients are known, it becomes a matter of simpletrigonometry to calculate the angles 4y.

The transmission properties of the birefringent filterare not changed by interchanging its entrance and exitfaces, nor are they changed by replacing the filter with

0

0Nz3

.5

7r p fir 'p

Fig. 3. Graphs show reflectance characteristics of NBM'ssynthesized from square wave of Eq. (18).

986 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

l.

11�

Z

I

J

1�

crystal elements. If in addition to Eq. (13), 0((6)satisfies

( (r/2) - 4] = (/2) + 4'], (14)

then evenand

harmonics in the sine series of Eq. (7) vanish

(3( = >3 -Y2,-1 sin (2r - 1) ,r=1

so that2R- 1

C (') = E Cmeim11,m=O

where

causing the reflected intensity between peaks of theNBMI to approach 16% for large N.

Note that because of the factor 2 in the exponent ofEq. (16), the abscissa values of Fig. 3 and of similarfigures in this article are not given by Eq. (4). If anoptical frequency interval Aw is to correspond to aportion of a plot that extends over an abscissa intervalAX, then L, the thickness of the crystal elements, isgiven by L = [2cAX]/[anA)w].

At the expense of increasing the mean square errorbetween the final result and an ideal square wave,' theovershoot can be eliminated by changing the form ofG(4') to make it continuous at ,6 = mor. Two suchchanges of G(4') were tried. In the first, G(,p) wasreplaced by the trapezoid function

o<m<R- 1,

R < m < 2R -1.

(17a)

(17b)

From the summation in Eq. (16), one can see that thestraight birefringent filter synthesized from C(4') hasan odd number of elements (2R - 1); therefore, thefolded filter consists of a polarizer, R - 1 crystals ofthickness L, one crystal of thickness L/2, and a mirror.

Narrow Band MirrorTo synthesize a narrow pass filter that upon folding

results in a narrow band mirror (NBM), the amplitudetransmission function G is chosen to be a square wavewith a period of 27r; i.e.,

G(O = +1,

G( = -1,

o < 4 < r,

or < < 2r.

At points 4' = mor, where m is an integer, G(4') isdiscontinuous and the corresponding Fourier seriesconverges to 0 (Ref. 4). The intensity transmissionof the straight filter C () 12 is then zero at 4' = m-rand approaches unity elsewhere. Therefore, D(4') 12,the reflected intensity of the folded filter, is a narrowpulse of unit height at 4 = mor and approaches 0elsewhere.

A series of NBM filters with N = 3, 5, 7, 9, 11, and13 were synthesized. Only odd values of N could beused because G(4') of Eq. (18) and the corresponding6((4) also satisfy Eq. (14). The fraction of theincident energy reflected by these folded filters as afunction of 4, was calculated by means of the Jonescalculus5 and the results are plotted in Fig. 3. As Nincreases, the full width at half height decreases and is(N + 1)-' of the peak-to-peak separation. Betweenthe peaks, the transmission does not go to zero becauseof the Gibbs phenomenon' which predicts that if G(4')is discontinuous at 4o, i.e., [G(4'o+) - G(4/o) = J],then in the limit, the infinite Fourier series representa-tion of G(4') approaches a function GI(4') that has a9% overshoot at 4'o (Ref. 6), i.e.,

GF(4'o+) = G(4+o+) + 0.09J,

GF(4o = G(4' 0j - 0.09J.

G(V,) = /Pa,G(4') = +1,

G(4') = (r - l)/la,

G( 4) = -1,

G( 4) = ( - 2r)/a,

O• < • < a, (20a)a4 < P <• r- a, (20b)

Ir-a• < i• < r 4 a, (20c)

iX + a < 4 < 27r - a,

(20d)

27r - a <4' < 2r. (20e)

Although the first derivative of the trapezoid functionis discontinuous, the function itself is continuous, and,as Fig. 4 shows, NBM filters synthesized with a = vr/8show no pronounced Gibbs phenomenon. Doublinga yields mirrors with broader peaks but with lessintensity reflected between peaks as can be seen fromthe curves in Fig. 5.

The second change tried was to make G (4) con-tinuous by connecting portions of a sinusoidal wave to

0

U

0.-U

C

0N=7 4.3

05

.0

N'9 4.4

(I

0 ir p 27r

1.0

0.5

(19a)

(19b)

Consequently, the truncated Fourier series C(A') of thefunction of Eq. (18) overshoots at 4 = mr by 9%,

0

Fig. 4. Graphs show reflectance characteristics of NBM'ssynthesized from trapezoidal wave of Eq. (20) with a = i/

8.

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 987

Cm = 2(R-m)-li

Cm = 2(m-R)+1,

I

For the square wave of Eq. (18),

Ck = (Zk) , O< k <N. (23)

For the trapezoid wave of Eq. (20) with a = or/8,

CA = sin (rZk) /Zk , < k< N. (24)

For the sine plus square wave of Eq. (21) with a = or/8,

Ch = cos(lirZs)/[16Zk - Zs ],

1o0 AN.

0.5

0

1.0

0.5

0 7r ' 2,r

Fig. 5. Graphs show reflectance characteristics of NBM'ssynthesized from trapezoidal wave of Eq. (20) with a = r/4.

01.0

a:Io Q

0

1.0

05

the square wave at points a away from the discon-tinuities, thereby obtaining the function:

G(4) = sin[ (V/2) (r/2) ],

G( = +1, a< A<

0<4< a,

or - a,

(21a)

(21b)0

G(.P) = sin{E(wr - ')/a](r/2)},X- a < < 7r + a, (21c)

< k < N. (25)

AN=5

N.7

N.9

I IA

G(') =-I, 7r + a < < 27r- a,

G(4') = sin { [(4 - 27r) /a](7r/2) },

2r - a < 4' < 27r.

(21d)

(21e)

Both this function and its first derivative are con-tinuous. Figure 6 shows the intensity reflected froma NBM synthesized using G(4) of Eq. (21) witha = r/8. As before, doubling a results in mirrors thathave broader peaks but less intensity reflected betweenthe peaks as can be seen from the curves in Fig. 7.

A comparison of the three sets of NBM's shows thatthe narrowest peaks are obtained when G (') is a squarewave. Although the Gibbs phenomenon produces arelatively large between-peak reflectance, these mirrorsmay still be most useful as laser mirrors when the gainof the laser is smaller than the between-peak reflectance.For lasers with higher gain or for other purposes wherelow between-peak reflectance is desired, the NBM'scorresponding to the G(4) of Eq. (20) or Eq. (21) canbe used, thereby trading off narrow peaks for lowbetween-peak reflectance. Table I lists two differentsets of i angles of the NBM's whose reflectance char-acteristics are shown in Figs. 3, 4, and 6, except forN = 3, where only one set of i was found. Theconstants C used in the synthesis of the NBM's ofTable I are given in the following equations, where

ZkS-UN-2k. (22)

Fig. 6. Graphs show reflectance characteristics-of NBM'ssynthesized from sine + square wave of Eq. (21) with a = r/8.

1.0

0.5

SN3

1.0

N=5

0,5

01.0

N=7

I.0

0

00 - i r , Si 2r

Fig. 7. Graphs show reflectance characteristics of NBM'ssynthesized from sine + square wave of Eq. (21) with a = r/4.

988 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

,0

.3

0.5k

.

05

l.0

N=9

05

0

I" -

l I

I

I.,

.iZ

I

1�

W

I

N-3

A ,

Table I. The Angles 0i of the Birefringent Elements in the NBM's with Reflectance Characteristics as Shown in Figs. 3,4, and 6 are Tabulated with Their Corresponding Values of N

N -pi 42 43 04 45 )6 7

NBM derived from square wave

3 22°30' 67030'5 13°25' 34057' 770111

70025' 54°05' 1°27'7 47°10' -89°301 36049' 12030'

65023' 83°53' 22051/ 34013'

9 29039' 78°40' 75053/ 20036/ 19032/

46°00 -84°52' 60031' 15°40' 40038'11 30°53' 82°27' 87009' 41013' 15002' 45028'

19042' 61049' 87°32' 58°26' 12003' 24043'

13 21°24' 67011' -85°57' 690471 280421 16015' 49020

34°25' 810481 860421 57039' 18018/ 17°55' 67005'

NBM derived from trapezoid wave with a 7r/8

3 20°57' 69003'5 10°19' 33001' 84°52'

14°19' 36°19' -85°30'7 2°49' 14027' 39031' -84051'

15002' -0o07' 75020/ 51°35'

9 -31°22' 76008' 30050' 41037/ -86°52'

-47054' 60008' 30035/ 49049/ -68°54'

11 -59°05 1 56032' 85052' -81°04' 36050' 33054'

-69°11' 56°00 -79049' -89°43' 29°54' 48051'

NBM derived from sine + square wave with ce = 7r/8

3 21°39' 68021'5 11°53' 33051' 81021'

16040/ 36005' -88°35'7 52°06' -81°27' 35°11' 11044'

77°30' 64053' 38030' -22°071

9 31009' 89049/ 78010' 10031' 23003'

43053/ -76°59' 67043' 7034' 38058'

11 8°22' 50°33' -80°40' - 65058' 0011, 23018'

16°23' 74049/ -71°58' 48053' 3°16' 48°23'

Table II. oi of NBM's with Reflection Characteristics as Shown in Figs. 5 and 7

N 0) 42 4)3 4 4) 06

NBM derived from trapezoid wave with a = ir/4

3 15°00' 7500'5 -3°28' 9029/ 61002'7 -29°12' 78015' 41055' 88°53'

-2°14' -3046' 12026' 67°52'

9 73°28' -41°32' 89050' 47014/ 86018'

15°10' 41°48' 5034? -70°48' -57057'

NBM derived from sine + square wave with a =7r/4

3 18053/ 71°07'5 4002/ 26015' 84011'

77041' 45014' -25°02'7 -53034/ 47031' 33053' 88015'

-1°21' 4015' 26°25' 84018'

11 65°05' -39041' -70°48' 48045/ 43023/ -80°55'

7032' 30022/ 51018' -9036' -81°43' -45024'

Table II lists i for NBM's whose reflection char-acteristics are shown in Figs. 5 and 7.

Tolerance of the oi ValuesAn important practical consideration in constructing

an NBM is the error tolerated in the setting of the q4

angles, especially if the crystal elements are to bepermanently wrung together. To study the effects ona NBM if the i are offset by an angle e, reflectancecharacteristics for the N = 7 and N = 11 NBA filtersof Table I were computed with the 4i angles changedfrom their correct values by amounts e = 2, E = 1°,

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 989

0

10 JN 2

0 ir IJI Sir

1.0

0.5

0

(a)

0.5 ez0.5'

0 10 7r 2ir

05

(b)

and e= 2°. In all cases, the angle was added to01, 0), 05, ... and subtracted from 0)2, 4, 06, . .

thereby ensuring that the errors would add to producean error of 2 in the angle between the fast axes ofadjacent crystal elements.

From the curves in Fig. 8, it is apparent that off-setting 4)i by degree produces no serious defects in theNBM. Larger errors in i affect the narrow peakmirror synthesized from the square wave of Eq. (18)more than they affect the broader peak mirrors syn-thesized from the continuous functions of Eq. (20) andEq. (21). In all three cases, the NBAI is sufficientlyinsensitive to errors in the 4)i values to allow the crystalelements to be positioned by uncomplicated mechanicalmethods without the need for further adjustment.

Electrooptical TuningIf the birefringent elements used in the filter have

sufficiently large electrooptic coefficients, the reflectionpeak of the NBMXI can be tuned over the entire peak-to-peak separation. A change of Ahn in the refractiveindex difference An results in a change of AX, in X,, thewavelength of the reflection peak, where

AX, = X(Aan/3n). (26)

The quantity AIn produced by an electric field appliedto a negative uniaxial crystal is"

Abn = Ano - An = (-no3r13 E3 ) + (n e3r33E3). (27)

In the above equation, the subscripts o and e denoteordinary and extraordinary, E3 is an electric field appliedacross the crystal parallel to its optic axis, and the r 3are the appropriate electrooptic coefficients. Anelectric field E3 applied to the crystal elements of aNBM, therefore, produces the change in the wave-length of the reflection peak

AX, = X, (n03r0 E3/2n), (28)

where ?'c = ?33 - (no/ne) 3r13 is the transverse electro-optic coefficient.

For electrooptic crystals usually employed, AX,/X,would be of the order of 0.1% with E3 = 104 V/cm.To enhance the tunability of a NBM, each birefringentelement of thickness L can be composed of two crystals,one of thickness L = D + L, the other of thicknessL2 = D. The two crystals are then put together sothat the fast axis of one is parallel to the slow axis ofthe other. In the absence of an electric field across thecrystals, the NBM behaves as if composed of bire-fringent elements that are L thick. However, theelectric field induced change in X, is given by

AX, = X, (Aa/a), (29)with

Fig. 8. Graphs show reflectance characteristics of NBM's forcases where the hi angles have been changed from their propervalues by an amount e. (a) NBM's synthesized from squarewave (Fig. 3). (b) NBM's synthesized from trapezoidal wave(ca = 7r/8, Fig. 4). (c) NBM's synthesized from sine + square

wave (a = r/8, Fig. 6).

Aa = Asn(Li + L2)/c; (30)

Eq. (30) is valid if the electric fields in the two crystalshave opposite senses with respect to the crystal axes;i.e., if the electric field is directed so that it increases therefractive indices of one crystal, it is directed so as to

990 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

1.0

0.5

0

1.0

0.5

N,7 e .05

1.0

0.5 | 0.5l

i.0k A I

0..5 LO

0n01.0

ir ql ir

(c)

w

a

r?.1

Table Ill. Electrooptic Materials,a Their Transverse Electro-optic Coefficients r,, and the Wavelength Shift of an NBM for an

Electric Field of 104 V/cm

A~Xp/Ap

r,; (for E = 104(10-'° cm/V) V/cm) Reference

BaTiO, 108b 6 X 10-3 9LiNbO3 20 1 X 10-3 10

LiTaO3 28 37 X 10-3 11KO.6Lio.4 NbO3 68 3 X 10-3 12Ba2NaNb5O, 5 35 2 X 10-3 13

Sro.75Bao 25 Nb2O6 1410 630 X 10-3 14

a An extensive review is given in Ref. 8.b r measured at 0.546 tz; all other r. measured at 0.63 ,.

decrease the refractive indices in the other crystal.It then follows that

AX, = X,,(Asn/Sn)(Li + L2)/(L - L2) ]- (31)

Consequently, the previous AX1,, given in Eqs. (26) and(28) is amplified by the factor (2D + L) IL, whereL is chosen to achieve the desired Px.

Materials

Table III lists a number of birefringent crystals andtheir transverse electrooptic coefficients (see Refs. 9-14).Although the listed r, is based upon measurements at0.63 A, experiments 9 on one of the crystals listed, LiNbO3,show that its r, is not very dispersive in the visible andnear ir. Also, the absence of dispersion in electroopticcoefficients has been found in other materials, 5 and istentatively supported by theory.'6 Therefore, AX,/X,at a wavelength Xv near the visible portion of thespectrum is approximately the product of AX,/XI, at0.63 ,4 and [(n0 '/6n)x/(n 3

/sn)o.63 {J.

Unfortunately, the birefringent crystals with thelarger AX,/X, are not presently produced with thehighest quality. They are usually not lossless, strainfree, and homogeneous. The optical quality of somecrystals deteriorates rapidly when the crystal istraversed by a laser beam. This laser damage canusually be cured and prevented by keeping the crystalat a sufficiently elevated temperature.' 7 An NBMmade of crystals that are damaged by laser beamswould, therefore, have to be kept at a temperaturehigh enough to prevent laser damage. Furthermore, ifthe crystal's nonlinear polarizability and transmissionare such as to generate second harmonics efficiently,the NBM would have to be kept at a temperaturesufficiently far removed from the crystal's indexmatching temperature to prevent loss of laser powerthrough second harmonic generation.

A small temperature change AT produces a change inthe wavelength of a reflectance peak AX, given by

AX, = X_L-(dL/dT) + (sn)-(dn/dT)]AT. (32)

The wavelength change attributed to the linear expan-sion term in Eq. (32) is of the order of 0.1 A/'C-' at awavelength of 1 ,u for a typical crystal linear expansion

of 10-' 0 C-'. No typical value can be assigned to3_ (dsn/dT) (1/3n) because changes drasticallyfrom material to material; e.g., /3 of calcite is 6 X 10-5'C-' (Ref. 19), of LiNbO3 is 5 X 10-4 C-' (Ref. 20),and,3 of LiTaO3 is 8 X 10-3 0 C-' (Ref. 21). Neglectingthe linear expansion of LiTaO3, which is approximatelytwo orders of magnitude smaller than its A, results inAX, = 80 . for a transmission peak at 1 and for atemperature change of 10C. The value of 13 for LiNbO3also dominates over its linear expansion in the directionperpendicular to its optic axis, (dL0 /dT) (1/L) =2 X 10-5 C-' (Ref. 22), and X, shifts only 5 A for atemperature change of 10 C at 1 . Consequently,stringent temperature control of a LiNbO3 NBM is notrequired if the transmission peak is several angstromswide. Changing the temperature of LiNbO3 slightlyaffects its electrooptic response because of concomitantchanges in the crystal length, electrode spacing, bire-fringent and electrooptic coefficients. The net result ofincreasing the temperature of LiNbO3 is to increase itselectrooptic effect by approximately 6 X 10-4 0 C-1

(Ref. 23).Although the NBM's crystal elements may be non-

absorbing at the wavelength of interest, the Fresnelreflection losses at its numerous surfaces must beavoided to enable the NBM to be employed in a lasercavity. However, because of the relatively largetolerances permitted in the Xi angles, the polishedsurfaces of the crystal elements can be permanentlywrung together in their proper positions, therebyeliminating the Fresnel losses of surfaces betweencrystal elements. Also, for use in a laser cavity, a lowloss Brewster angle window can serve as the requiredpolarizer of the NBM.

Applications

The NBM provides a means for frequency modulatingthe output of a laser. For example, the tunable laserNBM can be used to chirp the laser output; i.e., bysweeping the narrow reflectance response of a NBMacross the gain curve of the laser, the output frequencycan be made to change continuously in frequency.Also, the NBM can be used to Q-switch a selectedportion of the laser's gain curve, thereby generating anarrow output from an otherwise broad frequencylaser.

The author wishes to thank I. Itzkan for manyhelpful discussions and comments and for suggestingthe configuration that led to Eq. (31).

Appendix

The choice of Io2 in Eq. (11) is limited primarily inthat it must be at least as large as Cmax, the maximumof C (4') 12, so that energy is conserved. If Io2 is chosenlarger than Cmax, the resulting straight birefringentfilter will not transmit 100% at the 4' corresponding toCmax. However, when synthesizing a NBM, choosingIo2 larger than Cmax still results in a 100% peak reflec-tance but in a nonzero reflection away from the centralpeak, not a serious shortcoming for a laser mirror.

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 991

For the NBM's discussed in this paper, o2 was setequal to C.a., which in turn was computer determinedusing the Newton-Raphson method. 9 In the synthesisof other filters, I62 was increased if the solution of Eq.(A-8) of Ref. 1 yielded an odd-multiple real root y with-2 < y < +2, a condition that prevents the use of thealgorithm to find a set of Di values that satisfies Eq.(11) and Eq. (12).

The Bi and Ai of Eqs. (A-5) and (A-6) in Ref. 1 canbe related to each other through

NBk = E Vk,,A 0, c = 0,1, *.*,N, (A-1)

q=O

where the coefficients Vk, are zero except

Vq~q = 1, q = O 1, ... , N (A-2a)

Vow = [(-l)q + 1](-l)q/2, q = 2,3, * * N.

VAq = Vk-1,q1 - Vk,q-.2

k=1,2,--,N;q=k+1,kc+2,.-.,N. (A-2c)

References1. S. E. Harris, E. 0. Ammann, and I. C. Chang, J. Opt. Soc.

Amer. 54, 1267 (1964).2. E. 0. Ammann, J. Opt. Soc. Amer. 56, 943 (1966).3. E. 0. Ammann, J. Opt. Soc. Amer. 56, 952 (1966).4. H. S. Carslaw, Introduction to the Theory of Fourier Series and

Integrals (Dover Publications, Inc., New York, 1930), p. 230.5. W. A. Shurcliff, Polarized Light (Harvard niversity Press,

Cambridge, 1962).6. Reference 4, Chap. 9.

7. R. Courant and D. Hilbert, Methods of Mathematical Physics(Interscience Publishers, Inc., New York, 1953), p. 52.

S. cf. I. P. Kaminow and E. H. Turner, Appl. Opt. 5, 1612(1966).

9. P. H. Smakula and P. C. Claspy, Trans. Metal. Soc. AIME239, 421 (1967).

10. Reference 8, p. 1626.11. J. F. Ward and P. A. Franken, Phys. Rev. 133, AiS3 (1964).12. A. Ashkin et al., Appl. Phys. Lett. 9, 72 (1966).13. R. C. Miller, G. D. Boyd, and A. Savage, Appl. Phys. Lett.

Book 6, 77 (1965).14. The value of /3 for calcite was obtained from data of W. L.

Wolfe, S. S. Ballard, and K. A. McCarthy, in A merican In-stitute of Physics Handbook, D. E. Gray, Ed. (McGraw-HillBook Company, Inc., New York, 1963), p. 6-18.

15. The value of for LiNbO3 was obtained from the data ofG. D. Boyd, W. L. Bond, and H. L. Carter, J. Appl. Phys.38, 1941 (1967).

16. The value of /3 for LiTaO3 was obtained from data of R. C.Miller and A. Savage, Appl. Phys. Lett. 9, 169 (1966).

17. S. C. Abrahams, H. J. Levinstein, and J. M. Reddy, J. Phys.Chem. Solids 27, 1019 (1966).

18. J. D. Zook, D. Chen, and G. N. Otto, Appl. Phys. Lett. 11,159 (1967).

19. J. B. Scarborough, Numerical Mathematical Analysis (JohnsHopkins Press, Baltimore, 1962).

20. A. R. Johnston, Appl. Phys. Lett. 7, 195 (1965).21. P. V. Lenzo, E. H. Turner, E. G. Spencer, and A. A. Ballman,

Appl. Phys. Lett. 8, 81 (1966).22. L. G. Van Uitert, S. Singh, H. S. Levinstein, J. E. Geusic,

and W. A. Bonner, Appl. Phys. Lett. 11, 161 (1967).23. J. E. Geusic, H. J. Levinstein, J. J. Rubin, S. Singh, and

L. G. Van Uitert, Appl. Phys. Lett. 11, 269 (1967).24. P. V. Lenzo, E. G. Spencer, and A. A. Ballman, Appl. Phys.

Lett. 11, 23 (1967).

Information Office USSR Embassy

Nikolai Basov (left) and Alexander Prokhorov of the P. N.Lebedev Physics Institute of the USSR Academy of Sciences, who

shared with Charles H. Townes the Nobel Prize for the laser.

992 APPLIED OPITICS /-Vol. 8, No. 5 / May 1969