The Design and Preparation of Induced Transmission Filters

R. J. Holloway and P. H. Lissberger

Induced transmission (IT) filters are compared with other types of narrow-band filters and their ad-

vantages outlined on the basis of a general expression for spectral transmittance. A method of design

is described which avoids the use of admittance charts and can be applied to optimize the performance

of an IT system at various stages during its preparation. Finally, methods of monitoring are discussed

with emphasis on the theory of a technique particularly useful for IT filters for the ir.

IntroductionInterest in induced transmission (IT) filters first

described by Berning and Turner' has increasedrecently2-6 because it is now clear that they provide ameans of combining some useful advantages of simpleFabry-Perot (F.P.) and all-dielectric types of narrow-band interference filters. Specifically, IT systemscan be designed with the freedom from sidebands usuallyassociated only with F.P. systems as well as with peaktransmittances comparable with, though not usuallyquite as high as those of all-dielectric Fabry-Perotfilters (A.D.F.P.) and double half-wave (D.H.W.)systems. Furthermore, the transitions between pass-band and stop bands are relatively rapid for IT systems;in this respect, they compare favorably with D.H.W.systems. These advantages are examined in the lightof a slightly different theoretical approach from thatdescribed previously.

The method of design to be described here differs intwo main respects from the techniques described pre-viously. First, the main design criterion is to maximizethe transmittance of the filter as a whole, as well as itseffective refractive index,6 and not the potential trans-mittance of the half-filter.' Second, the design param-eters are established on the basis of simple expressionsevaluated in specific cases by the use of a digitalcomputer, i.e., without the use of admittance charts.2-4

This leads to a method of preparation which allows thefilter properties to be optimized at the halfway stage ofits construction.

The problems of the preparation of IT systems areconcerned mainly with the control of the thickness ofthe metallic layer and of the first dielectric layer oneach side of the metallic layer. The difficulty with themetallic layer is that its transmittance on glass is

The authors are with the Department of Pure and AppliedPhysics, University of Salford, Salford 5, England.

Received 13 September 1068.

usually uncomfortably small. For instance at X = 2 ,the transmittance of 500 A of silver on glass is only0.15%. For the two specifically mentioned dielectriclayers, the difficulty is due to the fact that their opticalthicknesses are nonintegral quarter-waves at the designwavelength. A description is therefore given ofmethods of monitoring the thickness of these layerswhich enables filters to be constructed with transmit-tance peaks located with good precision at the desiredwavelength.

General Theory of Interference Filters Appliedto IT Systems

The two effective interface approach of A.D.F.P.and D.H.W. systems discussed by Smith' can begeneralized by minor modifications (Appendix A) toinclude filters with absorbing layers. The resultingexpression for the spectral transmittance, which is thenapplicable to all interference filters, has the form of thewell-known Airy formula, namely,

T = o/ (1 + F sin2 k), (1)

where

To = f'fr,( -R,) (1 - R2)/[1- (R R2) 1/2]2

F = 4(RR 2) 112/ - (RiR2 ) "12]2,

0 = (27r/X)bDdi cos6D + [(p1 + p2)/2)],

R = r 12 and R2 -=I r2 12.

A/1 and q2 are the potential transmittances of systems 1and 2 (Fig. 1) in the directions indicated, andri I exp (ipi) and I r2 I exp(ip2) are complex amplitude

reflectances at the boundaries of the dielectric referencelayer.

In general, the parameters ro, F, and p are functionsof wavelength, their specific spectral variations beingcharacteristic of the type of filter. In the case of aspacerless induced transmission filter, the following twoconditions apply if the dielectric layer adjacent to the

March 1969 / Vol. 8, No. 3 / APPLIED OPTICS 653

I 1 ~I q.R~wrl

I w

.Sy.-

l,-

,U

By . _ _ _ _ _-

Sytem, 2 SyItem 2

' t~~~~~~- ----- ---------- ~~~~~~~~~~~><~~~~

..

IAir k

Fig. 1. General representation of multilayer filter.

In terms of the optical parameters of system 2 (Fig. 2),the potential transmittance Q is in general given by'

1-1 r 12 -2r r sinpexp(2q) - r 2 exp(-2q) -2r r sin(p + 2p) '

(5)where = k/n, p = Dn, q = Dk, D = 27rd/X, andn + ik is the complex refractive index of the metalliclayer. The maximum value of ,6 is given by

where

L~~~~~~d p L Ir, Ji

d, Ala Altl La e

Fig. 2. Detailed structure of system 2 for an IT filter.

metallic layer on the side of incidence is taken as thereference medium.

(1) Because system 1 (Fig. 1) contains only non-absorbing dielectric layers, the potential transmittancet', is equal to unity.

(2) The potential transmittance i, of the system as awhole is equal to the potential transmittance of thesingle absorbing layer, i.e., , = V12.

Since the potential transmittance of any system isdefined as the ratio of the emergent to the incidentPoynting fluxes, it follows that for the filter as a whole,

, = T/(1 - R), (2)

where R = ro 12 and represents the intensity reflec-tance of the filter on the side of incidence of the radia-tion. Under the above conditions and from Eqs. (1)and (2)

1-R = To'/(1 + Fsin2 ), (3)

where

Tro = (1 -R1) (1 - R2) /[1 - (RR 2) 2 ]2 .

It is interesting to note that the maximum value of thefunction 1 - R is unity corresponding to R = 0 and isobtained when p = nr (n being an integer) andro' = 1, the latter condition corresponding to R = R2.

According to Eq. (2), the maximum transmittanceTmax of the filter as a whole is obtained when theproperties of system 2 (Fig. 2) are chosen in such a wayas to maximize the potential transmittance, i.e., so that? = ,max, and the properties of system 1 so that R = 0.Then,

Tma = max- (4)

cosh2q + r2 cos2p(1 + r2 )

Moreover, it can be readily shown (Appendix B) thatthis value is attained if system 2 (Fig. 2) is designed insuch a way that

r = [qD exp(2q) - 11/E1 -m exp(-2q)]}"12 (8)

and

p = tan-'{ [ AI cos (2p) -1]/[6IIm sin2p] }. (9)

Equations (8) and (9) are very simple expressionswhich, combined with Eqs. (6) and (7), give the ampli-tude reflectance at the boundary of the metallic layerrequired to maximize its potential transmittance interms of the optical constants of the metallic layer andthe wavelength of the incident radiation.

Comparison of IT Systems with Other Types ofFilter

Equation (1), which is the general expression for thespectral variation of the transmittance of an interferencefilter, may be used to outline quantitatively the differ-ences between the common forms of interference filterand thus to emphasize the advantages of IT systems.This is done most conveniently by examining the proper-ties of the individual systems as follows.

F. P.

Here R, R2, V/,, and V/2 do not vary appreciably withwavelength. In that case, both Tr and F are constantsand the filter properties are due basically to the singleresonance function 1/(1 + F sin2 4) whose width de-pends on the value of F and therefore critically on thoseof R and R2 . The peak transmittances are relativelylow since no attempt is made in this type of filter tomaximize the potential transmittances ftx and V/2.These facts are illustrated in Fig. 3(a).

A.D.F.P.

Since there are no absorbing layers in this type ofsystem, 5bj and 1/2 are both equal to unity for all valuesof wavelength [Fig. 3 (b) ]. The other essentialdifference between this type of filter and the simple F.P.type is that R and R begin to vary rapidly at wave-lengths well removed from the design wavelength.This explains the existence of undesirable sidebands

654 APPLIED OPTICS / Vol. 8, No. 3 / March 1969

biur = 0- - (u2 1)1/2, (6)

(7)

I

-- Wavelength(mi-ronE

(a)

g1.a.IHLHLJHHJLHLHglI .1.0 '' ' -'_' ,

0.5. I +F8n.'

0.0 1'5 ""'"''"" 2.0 2.5 30 3

-Wavelength(microne)

(b)

(c)

which constitute the major disadvantage of A.D.F.P.systems.

D.H.W.7

Again 1/, and zP2 are constant and equal to unity.However, RI is made to vary rapidly with wavelengthin the region of the peak transmittance of the filterwhile R2 is relatively constant in the same region. Thisleads to a resonance type of variation in the functionro' [Eq. (3)] as well as in 1/(I + Fsin2') [Eq. (1)].Consequently, the spectral transmittance of D.H.W.systems [Fig. 3 (c) ] can be described by the product oftwo resonance functions and the passbands are cor-respondingly sharper than those of F.P. and A.D.F.P.systems.

An alternative approach to the analysis of D.H.W.systems is to consider them as two A.D.F.P. systems inseries. This would lead to the same conclusion, namely,that the transmission band of D.H.W. systems can bedescribed by the product of two resonance functions.

Spacerless IT

In this case 1, = and 412 = A, where ,6 is the poten-tial transmittance of the single absorbing layer. If thelatter is deposited onto a dielectric multilayer systemconsisting of a number of quarter-wave layers combinedwith a single non-quarter-wave phase adjusting layer,and if this dielectric system is designed to maximize thepotential transmittance at the design wavelength, thespectral variation of the potential transmittance isfound to be in the form of a resonance function [Fig3(d)]. Moreover, the spectral variations of thereflectances R and R2 at the boundaries of the referencemedium (Fig. 2) are such (Fig. 4) that the spectralvariation of ro' is also a resonance function.Consequently, the spectral transmittance of the ITsystem as a whole can be represented by the pro-duct of three resonance functions, namely, of At, TO',

and 1/ (1 + F sin 2'). The transmission bands of ITsystems are correspondingly sharp. Furthermore, thefact that the potential transmittance function and ro'remain at low values outside the region of the passbandresults in the suppression of sidebands associated withall-dielectric systems.

- Wavelength(microns)

(d)

Fig. 3. Calculated spectral variations of the functions Ap, To',

and 1/(1 + F sin2 o) for common types of filter. (a) F.P.,NAY = 0.075 + 3.41i, pH = 2.367, lg = 1.5. dAz = 340 A, S =

7.17 H. (b) A.D.F.P., AH = 2.748, ILL = 1.32, pa = 1.5. (c)D.H.W., H = 4.1, L = 2.29, L' = 2.426, pa, = 1.5. (d)Spacerless IT, NA, = 0.67 + 13.9i, PH = AM = 2.29, PL =

1.423, A, = 1.5, dA, = 515 A, M = 0.9074 H.

Wavelength (microns)

Fig. 4. Calculated spectral variations of R, and R2 for a spacer-less IT system. NAY = 0.67 + 13.9i, AH = PM = 2.29, AL =

1.423, A, = 1.5, dA, = 515 A, M = 0.9074 H.

March 1969 / Vol. 8, No. 3 / APPLIED OPTICS 655

Principles of Design

The design of narrow-band spacerless IT systems isbased on the following important criteria.

(1) The filter should have its maximum transmit-tance at the design wavelength.

(2) The effective refractive index0 of the filter shouldbe as large as possible.

(3) The design should be as simple as possible con-sistent with the other design criteria.

If the peak transmittance is specified to suit aparticular requirement, the thickness of the absorbinglayer, usually of silver, is immediately fixed implicitlyby Eq. (6) and the values of I 1 I and p explicitly byEqs. (8) and (9). The designs of dielectric structureon either side of the absorbing layer are then restrictedby criterion 1 to give these particular values of r and p at both boundaries of the absorbing layer.2 Thechoice of the design of the dielectric multilayers isnevertheless extremely wide and must therefore befurther restricted by the remaining design criteria.For instance, to meet criteria 1 and 2, the optical thick-ness of any of the dielectric layers can be chosen from aset of values which differ by integral numbers of designhalf-wavelengths. For filters with narrow passbandsit is necessary to use relatively large optical thicknesses,in particular for layers close to the absorbing laver.Finally, the criterion of simplicity is met by dielectricstructures consisting of layers all with optical thick-nesses equal to integral numbers of design quarter-waves with the exception of the two layers adjacent tothe absorbing layers. The simplest system is onewhich is absolutely symmetrical about the absorbinglayer, but detailed symmetry is not necessary to satisfythe first criterion.

In practice, the design procedure has to take accountof some important practical considerations. Forinstance, it may not be possible with the limited numberof available dielectric materials, to obtain the desiredvalues of r and p in the simple type of dielectricmultilayer structure indicated above. Moreover, re-fractive indices of layers are known to depend on theconditions of deposition and are likely to deviate fromthe assumed values. To minimize the effects of thesedifficulties, it is necessary to obtain expressions forquantities which are easily measurable during theconstruction of the filter in terms of the importantparameters of the system. It is then possible to moni-tor the filter in a way which optimizes its performanceat a number of stages in its construction. The followingcalculations indicate how the transmittance of thesystem at a number of stages in the construction isrelated to the desired values of r and p.

It is easy to show that the amplitude reflectancerM exp (ipmf) of the dielectric stack, in terms of the

values of r and p (Fig. 2) required to maximize thepotential transmittance of the absorbing layer, is givenby

m rF 12 + r 12-2 rF r cos(p -) (10)+ rF 12 r 12 - 2 'F || t I COs(p + PF)

and

-4 l r I ( - I rF 1) sinp -|rF (1 - r 11) sinPF|p,11~~~ = tan I ( 7-|F 12) CSp- IrF (1 + r) CSpFJ

(11)where

rF = 'F exp(ipF) = (N - MM)/(N + gm), (12)

N = n + ik,

PM = (27r/X\) 2 1MMdm + Sir,

S = 0 if M = H,and

S = 1 if M = L,

IH and AL are the refractive indices of the two materialsused for the dielectric layer and MMz > AL. If the opticalthickness of the matching layer is MXo/4, then accordingto Eq. (12),

M = (POM/7) -S. (13)

where Por is the value of pf at wavelength X0.Given the value of MH, the value of I rm I at wave-

length Xo obtained from Eq. (10) can be used to deter-mine the number of layers in the quarter-wave system.The necessary formulas are given by Born and Wolf.8Equations (11) and (13) specify the thickness d ofthe non-quarter-wave layer.

Now the transmittance of the dielectric stack alonefor radiation incident in vacuo is given by

T = T(I - rf2) ( 1 rM I o1s)I + rf2 r 12 + 2rf rm cospm

(14)

where rf = (1 - M) /(1 + Mm). For the quarter-wave system alone, the transmittance is simply givenby Eq. (14) when dM = 0. Finally, the transmittanceafter the deposition of the absorbing layer is

TA = 1,VM(1 - I rA [2), (15)where

2 I rF 12 + I rT 12 2 rp I r' cos(p' - PF)

2 =1 + I rF 12 | rT 12 2 rF 1 T' cos(p' + pF)

r' = I e-f q

and

p' = + 2p.

By the use of Eqs. (6)-(15), it is now possible totabulate as functions of absorbing layer thickness thetransmittances at three stages of the filter constructionrequired for the optimization of the filter performancein accordance with the first criterion discussed above.An example is given in Table I for the case of a silverabsorbing layer in the visible region of the spectrum,as well as in the near ir. The application of this tablefor monitoring the thickness of various layers isdescribed in the next section.

656 APPLIED OPTICS / Vol. 8, No. 3 / March 1969

Fig. 5. Schematic diagram of monitoring system.

MonitoringTwo methods of monitoring the thicknesses of the

layers during the preparation of an IT system aredescribed here, one for filters for the visible region ofthe spectrum and one for the ir. In both cases thetransmittance is monitored on the substrate where thefilter is to be constructed. The uncertainties in thethicknesses of the filter layers associated with the useof separate monitor plates for individual or groups oflayers are thus avoided.

In the Visible SpectrumThe quarter-wave layers in system 2 (Fig. 1) are

deposited in the conventional manner, i.e., by theobservation of extrema in the transmittance, the wave-length of the monitoring beam (Fig. 5) being set to thedesign wavelength. Initially, a reading is taken of themonitor signal corresponding to the transmittance ofthe uncoated substrate. When the quarter-wave stackis completed, another such reading is taken and theratio of the two corresponds to the transmittance[T(dM = 0) in Table I] of the stack. Table I is thenused to indicate the thickness of the silver layer andother important parameters corresponding to the filteroptimized to that particular value of T(dM = 0). Inparticular, the table gives the value of the transmittancerequired after the deposition of the next dielectriclayer, i.e., the first non-quarter-wave layer. That layeris then deposited according to the prescribed value ofthe transmittance T. The silver layer is deposited in asimilar way to the prescribed value of TA. Finally, allthe dielectric layers in system 1 (Fig. 1), including thenon-quarter-wave layer adjacent to the silver, aremonitored simply by the observation of the extrema inthe transmittance.5

A number of important points should be noted here.(1) Since this monitoring scheme relies in two

instances on absolute measurements of transmittance,

Table I. Design Parameters Used for Monitoring

Thickness T (dM = 0) Monitoring

of first non- Transmit- T Transmit- wavelength

Maximum quarter-wave tance of tance after TA Transmit- of first non-

Silver potential layer dielectric first non- tance after quarter-wave

Design thickness trans- (quarter- quarter-wave quarter-wave silver layer

parameters (A) mittance waves) stack layer layer (p)

= 0.5460 u 650 0.774 0.6354 0.464 0.137 0.106

n = 0.075 655 0.770 0.6348 0.458 0.135 0.103

k = 3.41 660 0.765 0.6342 0.453 0.133 0.102

PM 2.35 665 0.760 0.6336 0.447 0.131 0.0999

670 0.756 0.6331 0.442 0.129 0.0978

675 0.752 0.6325 0.437 0.127 0.0958

680 0.747 0.6320 0.431 0.126 0.0939

500 0.695 0.9086 0.230 0.0494 0.0343 1.928

Xo = 2.000 u505 0.688 0.9082 0.227 0.0487 0.0335 1.928

n = 0.67510 0.681 0.9078 0.225 0.0481 0.0328 1.928

k = 13.9515 0.675 0.9074 0.222 0.0475 0.0320 1.927

Ptm = 2.29520 0.668 0.9071 0.219 0.0468 0.0313 1.927

PH = 2.29525 0.661 0.9067 0.217 0.0462 0.0305 1.927

AIL = 1.423530 0.654 0.9064 0.214 0.0456 0.0298 1.927

March 1969 / Vol. 8, No. 3 / APPLIED OPTICS 657

Table II. Measured Properties of Spacerless IT Filters Compared with Design Specificationsa

EffectiveTransmittance Wavelength refractive Half-width

Filter (%) (pU) index ()

1 38.5 2.025 1.82 i 0.01 0.0772 71 2.008 1.88 4 0.05 0.0583 92 1.916 1.78 + 0.02 0.1644 28 1.998 2.0 + 0.2 0.0255 69 1.956 1.84 4 0.03 0.0616 73 2.003 1.9 i 0.3 0.0387 61 1.970 1.80 i 0.03 0.0578 82 1.935 1.79 4t 0.04 0.0699 70 2.008 1.87 i 0.06 0.053

10 75 1.995 1.89 i 0.05 0.04811 70 2.008 1.89 t 0.04 0.04512 73 1.987 1.81 i 0.01 0.052

Theoretical filterd = 515 67.5 2.000 1.814 0.051

a The filter structure isglass HLHLHL M M Ag M LHLHLH I glass,

where H and L denote quarter-wave layers.PHsr = P, = 2.29, AL = 1.423, NA, = 0.67 + 13.9i, A = 1.5, dAz

it is essential that the electronics and other componentsof the monitoring system should be stable and that thedetector should respond linearly to changes of radiantflux.

(2) The intervals of silver thickness in Table I shouldbe chosen so that the corresponding intervals in thetransmittances are smaller than the uncertainties inthe measurement of transmittance.

(3) The precision with which the silver thickness canbe monitored is better than is indicated by the differencebetween T and TA because, as the silver deposition pro-ceeds, the transmittance rises to a maximum before itfalls to TA.

(4) The silver is deposited by the flash evaporationof small silver grains. In this way each grain isevaporated very rapidly, which is necessary if theresulting film is to have the desired optical constants.At the same time, this technique allow the depositionto be terminated with good precision.

In the InfraredThe noticeable feature of IT systems in this region

of the spectrum is that the optical thickness of thenon-quarter-wave layers adjacent to the absorbinglayer is very nearly equal to an integral number ofquarter-waves. In fact, for systems with relativelythick silver layers, it is easy to show that for MM = MHand provided that k >> M and k >> n,

M = (2K + 1) - (1/7r) tan-(2MuA/k), (16)

where K is an integer. This means that it is possibleto obtain a simple analytical expression for the changein wavelength (AX)m of the monitoring beam requiredfor the transmittance during the deposition even of thefirst non-quarter-wave layer to be monitored to anextremum.

= 515 A, M = 0.9074 H.

If the first quarter-wave stack is monitored at thedesign wavelength X0 and the wavelength is thenchanged to Xm, the phase change on reflection from thequarter-wave stack, as seen from the medium of thefirst non-quarter-wave layer (AM = H) is9

ZLiAi+ Sir, (17)

where

(AX) m = Xm - X,,

Aj = 1 - (o/Mj) (Aj/ax),

P is the number of quarter-wave layers in the stack andLj can be considered as a weighting factor for the con-tribution of the layer labeled j to the total phase change.Thus, the value of PM [Eq. (12)] at a wavelength X.is given by

PM (Xm) = (27r/Xm) 2AM (Xm) dm

P+2

+ Sir - r[ (AX)m/X\O] L1A1 ,j-2

(18)

where M(m) is the refractive index of the non-quarter-wave layer at wavelength X.. Since M() dM=M (X0/4) by definition,

PM (m) = ir{M + S -E (AX) m/XO] (MA + LjAj)},

(19)where A = 1 - (Xo/tM) (aMM/aX).

For an extremum in the transmittance T EEq. (14)],PM = Sr, where ' is an integer. Therefore, if theextremum is to be used as a means of terminating thedeposition of the first non-quarter-wave layer, the

658 APPLIED OPTICS / Vol. 8, No. 3 / March 1969

10

Fig. 6. Spectral transmittance curve for filter No. 9.

monitoring wavelength must be chosen so thatPM(Xm) = S'ir. Accordingly,

(AX)m _ M - (S' -S) (20)

Xo MAm + XLjAj (

In the absence of dispersion of the layer materials,AM = Aj = 1. Moreover, if the number P of quarter-wave layers is large,0 YL -- L/ (MH - ML). In theseconditions Eq. (20) is simplified to

(AX) m

Xo

M - (S' - S)

M + AL/ (AH - AL)(21)

The results of the application of Eq. (21) to the caseof AM = MH (S = 0) and of a minimum (' = 1) in thetransmittance T are given in Table I (last column),and these are used for the monitoring of the first non-quarter-wave layer. The procedure is similar to thatdescribed above for filters for the visible spectrum; ameasurement is taken of the transmittance of thequarter-wave stack at the design wavelength, and therequired wavelength change (AX). corresponding tothat transmittance is obtained from the table. Withthe new wavelength setting Xm, the transmittance of thenext dielectric layer is monitored to a minimum and thewavelength is then reset to its original value, i.e., thatof the design wavelength Xo. The procedure for theremaining layers is exactly as described previously.

This technique may also be applicable in the visiblespectrum, but the required wavelength changes aremuch larger and much more tedious to calculatebecause the approximations 9 in the derivation of Eq.(21) are no longer applicable when the optical thicknessof the non-quarter-wave layers differs by more thanabout 10% from integral quarter-waves.

Conclusions

The details of the measured performance of twelvespacerless IT filters are given in Table II. These filterswere designed to have a peak transmittance at a wave-length of 2 M and a half-width of about 0.050 A. Theywere prepared with dielectric layers of zinc sulfide(Mu = 2.29) and cryolite (ML = 1.423) and a singlelayer of silver (n = 0.67, k = 13.9). The results con-

firm that filters of this type can be made successfullywith the methods described.

Figure 6 shows the measured spectral transmittancecurve for filter no. 9 and illustrates the basic advantagesof IT systems. These are the relatively steep transi-tions between passband and stop bands and the highdegree of freedom from side bands.

Acknowledgment is made to the Ministry of Tech-nology who supported this work.

Appendix A

The method of superposition, as described byBerning,0 for the calculation of the optical propertiesof two superposed multilayer systems can be veryeasily generalized to include the case where the dielec-tric reference layer (Fig. 1) has a refractive indexdifferent from that of the massive medium on the sideof incidence (o) and from that of the substrate medium(Ms). Thus, the amplitude transmittance and reflec-tance are given by

r = [ro' + r2'(t'tl - ro'r,) / (1 - rlr 2 ') (A-1)

and

t = tt 2 ( 1 - rr 2 ') (A-2)

in terms of the parameters illustrated in Fig. 1. Thesetwo equations correspond, respectively, to Eqs. (130)and (131) of Ref. 10.

Now the intensity transmittance r of the combinationis given by

T = (s/Mo) I t I2. (A-3)

Moreover, since intensity transmittance of system 1must be the same in both directions:

(MD/MO) t 12 = (MO/MD) I tl 12.

Therefore, from Eqs. (A-2), (A-3), and (A-4)

1r =(o/MD) I t' I2(S/MD) I t2

I1 -rlr 2 ' 12

(A-4)

(A-5)

Now, (o/MD) I t 1 represents the intensity transmit-tance for system 1 from the medium of refractive indexMD to air (o). Thus,

(Mo/MD) I t 12 = 4p1 (1 - I r 12).

Similarly,

(S/MD) t2 12 = &2 ( 1 - I r2 12) .

From Eqs. (A-5)-(A-7), it then follows that

r = ro/(1 + Fsin2 p),

(A-6)

(A-7)

(A-8)

where o, F, and p are as defined in Eq. (1).

Appendix BIt is easily shown' from Eq. (5) that the values of

I r I and p that give rise to a particular value of thepotential transmittance ,6 lie on a circle in the xy plane,

March 1969 / Vol. 8, No. 3 / APPLIED OPTICS 659

o12 1:)

_ Wav-1-lgth(--cron),.

where x = r I cosp and y = r I sinp. In that case,the center of the circle has coordinates:

sc= rPisin2p/[l - ,exp(-2q)] (B-1)

and

Yc = r[f cos (2p) - ]/[1 - t/'exp (-2q)].

(B-2)

When the potential transmittance assumes its maximumvalue 41M, the circle degenerates to a point with the co-ordinates of the center of the circle corresponding toAt M. It then follows from Eqs. (B-1) and (B-2) thatthe unique values of r and p are those given by Eqs.(8) and (9).

References1. P. H. Berning and A. F. Turner, J. Opt. Soc. Amer. 47, 230

(1957).2. V. R. Costich, Ph.D. Thesis, University of Rochester, 1965.3. P. W. Baumeister, V. R. Costich, and S. C. Pieper, Appl.

Opt. 4, 911 (1965).4. R. L. Maier, Thin Solid Films 1, 31 (1967).5. D. J. Hemingway, M.Sc. Thesis, University of Salford, 1967.6. D. J. Hemingway and P. H. Lissberger, Appl. Opt. 3, 471

(1967).7. S. D. Smith, J. Opt. Soc. Amer. 48, 43 (1958).8. M. Born and E. Wolf, Principles of Optics (Pergamon Press,

Inc., New York, 1965), Chap. 1, Eqs. (96) and (99).9. P. H. Lissberger, J. Opt. Soc. Amer. 49, 121 (1959).

10. P. H. Berning, in Physics of Thin Films-Advances in Re-search and Development, Georg Hass, Ed. (Academic PressInc., London, 1963), Vol. 1.

UCLA Residence Halls Coordinator, Rieber Hall, 310De Neve Circle, Los Angeles, California 90024. Telephone: (213) 825-5305or 825-5306. If requested, a list of off campus housing is mailed (see enroll-ment form).

ENROLLMENTEnrollment may be made by individuals or companies. Any number of personsfrom a single company may enroll as long as there are vacancies. Reservationsmay be requested for industry and government employees requiring time toobtain authorization. Companies may enroll for a given number of individuals,supplying names latei. To insure enrollment, individual names must be receivedby the University before June 3, 1969. For additional applications, useseparate sheet giving information requested on enrollment form. Indicate ifparking permit is desired

WHEREBoelter Hall, Roomversity of California,

4442 / Uni-Los Angeles

FOREngineers and electronics engineersworking in aeronautical industries,space sciences, or motion picturesElectronics engineers are becomingmore and more involved in electro-optics in which lenses play an im-portant part. A knowledge of theaberrations and. other properties oflenses is often of great value tothem even if they never have oc-casion to actually design a lens.Knowing lens-design procedureshelps them judge an existing lensand decide whether it could beusefully improved for their par-ticular applications.

PURPOSETo explain the nature of the princi-pal aberrations of lenses . . . discuss

d~ ~~: - - - -ZI 11 methods used to design lenses fori ~~~~~~~~~~~specific purposes, and the limita-

IIIIL 111 1 1111)11112S1111141 N tions within which they can be

I AN R IHYSICAL SIN EENNslINY, IF WALIFO lmade to operateENGINEERING AND PHYSI~~AL SCIENCES EXTENSION / UNIVERSITY EXTENSION, UCLA

660 APPLIED OPTICS / Vol. 8, No. 3 / March 1969