RICHARD D. MATTUCK

at angle of incidence S°1P= 81°, to determine xoP.[Set rla'= 0 in Eqs. (1) and (2) to obtain conditionfor X 0s and XoP.]

(4) Two additional double layers of BaSt are castsimultaneously over both steps, and procedure (3) isrepeated to find Xbs and X. [Condition for Xs andXbP is same as for Xos and XoP except that d3 in Eq. (1)has now changed.]

(5) The substrate parameters, caP and asS are cal-culated from X0o and Xbe using Eq. (29).

(6) The unknown film is cast simultaneously overboth steps and procedure (3) is repeated to determineXas and aP. [For derivation of conditions for Xas andXap, see reference 1, Eqs. (3), (4), and (11), and Eq.(3) of this paper.]

(7) n2, the refractive index of the unknown and d2,its thickness may be calculated over the range 1.1 <n 2<2.0, d2 <250A, from the following: (a) Eqs. (15)and (16), giving n2 to <2% and d2 to <10%, (b) Eqs.(8)-(12), giving n2 to <1% and d2 to <5%, and(c) Eqs. (23) and (24), with variables as defined in(20)-(22), (10)-(12), giving both n2 and d2 to threesignificant figures.

(8) If the effect of birefringence is desired, smallcorrections may be obtained using (44).

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Preliminary experiments indicate that the lengthiestpart of the procedure is the preparation of the films;the photometric measurements and subsequent calcula-tions may be done in -2 hr.

As pointed out in the sections on sensitivity and onerror, the method should give indices correct to nearlythree significant figures and thicknesses correct towithin -5% for films of thickness 20 A up to 250 A.

(It is to be noted that the substrate material neednot be barium stearate; evaporated films of index1.4-1.6 built up to the equivalent thickness of thetwo steps by using a mask should be quite satisfactory,provided they are of known index, and suitable changesin the angle of incidence and constants used in thecalculations are made.)

The experimental validation of the proposed methodis discussed in a separate paper.

ACKNOWLEDGMENTS

The author wishes to acknowledge the stimulatingadvice and encouragement of Dr. J. B. Bateman, whosuggested the problem, and R. D. Petti who participatedin the discussions and helped with some of thecalculations.

VOLUME 46, NUMBER 8 AUGUST, 1956

Semireflecting Silver Films for Infrared Interferometry*URI OPPENHEIM

The Weiznann Institute of Science, Reltovotli, Israel(Received December 12, 1955)

Several important uses have now been found for the Fabry-Perot interferometer in the near infrared.It has therefore become necessary to make a search for suitable semireflectors in this region. Silver hasbeen found to be the most suitable metallic semireflector for near infrared interferometry. The opticalproperties of silver films have been measured for the 1-4 -micron region. These properties have been in-terpreted in terms of the physical structure of the films.

INTRODUCTION

SEVERAL important uses have now been found forthe Fabry-Perot interferometer in the near infrared

region. The instrument has been applied successfullyto high-resolution absorption spectrometry,'-3 wave-length determinations,4 6 and measurements of refractiveindices.6 9

* A summary of a Ph.D. thesis submitted to the HebrewUniversity of Jerusalem, Israel.

I J. H. Jaffe, Nature 168, 381 (1951).2 Jaffe, Wiggins, and Rank, Nature 175, 908 (1955).3Jaffe, Rank, and Wiggins, J. Opt. Soc. Am. 45, 636 (1955);

Blaise, Chabbal, and Jacqinot, J. phys. radium 15, 749 (1954).Rank, Rix, and Wiggins, J. Opt. Soc. Am. 43, 157 (1953).Rank, Bennett, and Wiggins, J. Opt. Soc. Am. 43, 213 (1953).

6 J. Ramadier-Delbes, J. phys. radium 11, 622 (1950).7 J. H. Jaffe, J. Opt. Soc. Am. 41, 166 (1951).8 J. H. Jaffe and U. Oppenheim, Bull. Research Council Israel

2, 297 (1952).9 Rank, Shull, Bennett, and Wiggins, J. Opt. Soc. Am. 43, 952

(1953).

In view of the fact that the performance of a Fabry-Perot interferometer depends critically upon the natureof the semireflecting films with which its plates arecoated, it has become desirable to make a study of theproperties of semireflectors so that the best choice canbe made for infrared interferometry.

Multiple-layer dielectric films are of course suitablefor infrared work, and in fact MgF 2 -ZnS layers havebeen used extensively in the 1.5-,u region.9 However, aserious disadvantage for some application is that phasechanges on reflection are so large as to make theirdetermination extremely difficult." For metals on theother hand these phase changes are very small.6

In this paper we investigate the possibilities of thinmetallic layers. We first predict the optical propertiesof thin metallic films on the basis of simple electro-

'0 D. H. Rank and H. E. Bennett, J. Opt. Soc. Am. 45, 69 (1955).

628 Vol. 46

SEMIREFLECTING SILVER FILMS

magnetic theory. Experimental values are then foundto be very different from these predictions. It is neces-sary to take into account the microstructure of thefilms in order to resolve the discrepancy.

THE FABRY-PEROT INTERFEROMETER

The Fabry-Perot interferometer consists of twooptically flat plates, coated with semireflecting filmsand set parallel to each other. Fringes are formed byinterference of beams which are multiply reflectedbetween the plates. The intensity distribution I(6) ofthe fringes, expressed in terms of the phase differencea between successive beams, is given by

T2 1I(a) =

(T+A)2 4R1 + sin23/2

(1-R)2

QOO5 opIo 0p15

_ t/k

FIG. 1. Dependence of the reflectance R and the efficiency on film thickness. The curves were computed for n = 2, andvarious values of k.

authorsform:

Here R is the reflectance, T the transmittance, and Athe absorbance of the films. The sum R± T+A is equalto unity. The sharpness of the fringes is determinedby the quantity 4R/(1-R) 2 which depends solelyupon the reflectance R. The higher R, the sharper thefringes." The peak intensity of the fringes is given bythe factor'2 T2/(T+A)2. This quantity is referred toas the efficiency and is denoted by r. When there is noabsorption the efficiency has its maximum value ofunity. With increasing absorption the efficiency goesdown rapidly. Two main factors, then, determine thequality of the fringes. The reflectance defines theirsharpness and the efficiency defines their brightness.

In general, for a giyen metal, the reflectance increasessteadily whereas the efficiency decreases with the thick-ness of the layer. A very thin layer has low reflectanceand high efficiency whereas a thick layer has a highreflectance and low efficiency. For the purposes of inter-ferometry a compromise has to be found by choosinglayers of such thicknesses that they will have usefulvalues of R and good or tolerable efficiencies. In practicethis choice depends on the particular application

THEORETICAL

The optical properties of metals can be calculateddirectly from electromagnetic theory. Refined formulasfor the reflectance and transmittance of layers havebeen developed by Hadley and Dennison." Their treat-ment is applicable for all angles of incidence of the lightupon layers and in addition takes into account theeffect of the supporting substrates. For our purposeshowever it is sufficient to take the formulas of Barnesand Czerny,'4 which are applicable only to the caseof light normally incident on unsupported layers. These

"K. W. Meissner, J. Opt. Soc. Am. 31, 405 (1941).12 S. Tolansky, Multiple Beam Iterferometry (Oxford University

Press, New York, 1948), p. 147.1" L. N. Hadley and D. M. Dennison, J. Opt. Soc. Am. 37, 451

(1947).14 R. B. Barnes and M. Czerny, Phys. Rev. 38, 338 (1931).

where

give expressions for R and T in the following

(en -e9)±+4 sin'aR=R

(en-R,,,e- )2 +4R,, Sin2 (a+ *)

(n-1)2+k2

(n+ 1)2+k2

2irnta=-;

(1)

- (2)

2ktank=

n2+k 2 1

27rkt/3=

X

Here, n and k are the refractive index and absorptioncoefficient, respectively, defined so that the complexindex is in the form: n- ik. Combining the expressions(1) and (2) we write an explicit formula for theefficiency:

r= [ (1-R )2 +4R,, sinl 12

(I1- R,,)(e` - e20) +4R,, sin' (a Q)4R,,, sin2a(3)

For given values of n and k, theoretical curves canthus be made showing the dependence of R and ont/X. We have actually computed a large series of suchcurves covering the range of values of the opticalconstants which obtain in the 1 to 4-,v region. A typicalfamily for n = 2 and various values of k is shown inFig. 1. From these data curves have been constructedwhich show explicitly the dependence of the efficiencyT of a layer, upon the optical constants. Figure 2 showsthe dependence of on k for a fixed value of n andvarious values of R. In Fig. 3, is plotted as a functionof n for a constant k.

The following important conclusion is to be drawnfrom Figs. 2 and 3 and similar curves, with regardto the behavior of r as a function of the optical con-stants: For a given R the efficiency goes up with in-

629August 1956

(I - R,, )2 +4R,,,, sin2�/

(eft-R,,,e7#)1+4R,,, sinl(a+q/)

URI OPPENHEIM

10 20 30 40K

FIG. 2. Theoretical efficiency r of a metal film as a function of theabsorption coefficient k, for various values of the reflectance R.

creasing k and goes down with increasing . In otherwords, the more "metallic" the metal, that is, thehigher the absorption coefficient, the better its effi-ciency as a semireflector.

This result may seem somewhat surprising as it isoften assumed that a high absorption coefficient willlead to strong absorption of light falling on a metallayer. It is true, of course, that the amplitude of a wavewhich travels inside a metal decays exponentially withk but for a wave which falls from the outside onto asolid metal the larger part is reflected. This part in-creases with k and only very little remains to beabsorbed. For thin films there is in addition a trans-mitted part which complicates matters, but the abovecalculations show that the interplay of R, T, and A issuch that the efficiency is highest for those metalswhich absorb most strongly.

1.00 20

.75R=0.05

1 ~~~~~~~~~.15

.50

.25

1 2 3

- nFIG. 3. Theoretical efficiency of a metal film as function of the

refractive index n, for various values of the reflectance R.

MEASUREMENT OF OPTICAL PROPERTIES

Having due regard for the rule italicized above, it isa simple matter to select the most efficient metal froma table of optical constants. A search through thetables shows that silver, gold, copper, and platinumare the most favorable for use in the I to 4-hz region.'5 InFig. 4 the efficiencies of films with 60% reflectance areplotted against wavelength for these metals. Gold isthe best and it was therefore provisionally concludedthat gold was the most promising semireflector for thespectral region under consideration.

Experiments with gold however, were disappointing.Observed values for the efficiency were much lowerthan those shown in Fig. 4. This failure to obtain a highefficiency was to be explained by taking into accountthe well-known fact that the optical constants of thinfilms are radically different from those of the bulkmetal. The thinner the films the greater is this de-

1.0

0.8

7T 0.6

0.4

01 2 3

X ( /L)4 5

FIG. 4. Efficiency T of various metals in the 1 to 5-u region, com-puted from their bulk optical constants, for a constant value ofthe reflectance R.

parture. This phenomenon is particularly significantfor infrared semireflectors because the latter havethicknesses of 200 A and below, whereas the corre-sponding thickness for the visible region is in theneighborhood of 700-800 A.

In the light of these experimental facts we add to ourrule which states that the most efficient semireflectorsare those which have the highest k values, a riderstipulating that these k values are to be those of thinlayers and not those of the bulk materials.

Extensive determination of n and k of thin films inthe 1 to 4-, region have been made by Oldham." "7 Ex-amination of these experimental data shows that infact silver is the most promising material for ourpurposes.

15 International Critical Tables (McGraw-Hill Book Company,Inc., New York, 1929), Vol. 5, p. 248.

16 M. S. Oldham, Ph.D. thesis (Iowa, 1949).17 M. S. Oldham, J. Opt. Soc. Am. 41, 673 (1951).

I I I I I

R 0.60

l, I I l I I _ _

630 Vol. 46

SEMIREFLECTING SILVER FILMS

0.0

0.6

20 40 60 80 100 120 140 160 180

_ t (A )

FIG. 5. Measured reflectance R of silver films as a functionof thickness (in angstrom units).

Oldham made measurements of R and T. Values forthe efficiency expressed as T 2 /(1-R) 2 could be calcu-lated from his data, but the accuracy was poor sinceboth T and (1- R) are small quantities for the thicknessrange of interest. For this reason we have measuredagain more carefully the reflectance and transmittanceof silver in the 1 to 4 -p region and computed the effi-ciency therefrom.

The silver films were deposited on fused silica slidesby evaporation in vacuo. The time of evaporation wasshort: between 2 to 10 seconds, depending on thethickness of the film. By carefully controlling experi-mental conditions the optical properties were foundto be reproducible to within 1%. Thickness was meas-ured interferometrically by Tolansky's method usingfringes of equal chromatic order.2 ,"8 The accuracyobtained was h 10 A.

The reflectance and transmittance were measuredwith the help of a reflectometer which plotted theabsolute values of R and T automatically as a functionof wavelength.'9 The accuracy of these measurementswas

A series of about 20 films covering the thickness rangeup to 200 A was prepared and measured. Values of R

80 100 120 140 ,60 ,80

t(A )

FIG. 6. Measured transmittance T of silver films as a functionof thickness (in angstrom units).

"8 Scott, Mclauchlan, and Sennett, J. Appl. Phys. 21, 843 (1950).19 J. H. Jaffe and U. Oppenheim, J. Sci. Instr. (to be published).

0.20 0.40 0.60 0.80 1.00

-RFIG. 7. Experimental efficiency T of thin silver films as a function

of reflectance R in the 1 to 4-1A region. The curves were computedfrom the experimental values of R and T given in Figs. 5 and 6.

and T expressed as functions of the thickness arepresented in Figs. 5 and 6. From these curves theefficiency was computed and plotted against R forfour different wavelengths (Fig. 7). Figures 5, 6, and 7then, present in convenient form the essential data onthe optical properties of thin silver films in the 1 to 4-itregion, for the purposes of interferometry. Further,since it has been established experimentally that silveris the best metal for this region, it follows that theefficiencies represented in Fig. 7 are the highest that areobtainable with single metallic layers.

These data, however, are substantially different fromthose derived theoretically from the optical constants.For comparison with Fig. 7,. values of T computeddirectly from Eqs. (1) and (2), using the bulk values ofn and k, are plotted in Fig. 8.

-r

0.20 0.40 0.60

-R0.80 1.00

FIG. 8. Theoretical efficiency r of thin silver films computedas a function of refletance R. Optical constants were assumedto be those of the bulk material.

-r

631August 1956

URI OPPENHEIM Vl4

-

+

0

-J

6

0100 200

t (A)

FIG. 9. Electrical resistivity p of thin silver filmsa function of film thickness t.

THE PHYSICAL STRUCTURE OF SILVER FE

The explanation for the difference between t'film constants and those of the bulk material ifound in a study of the physical structure of the

Evaporated thin films are not homogeneous anbut possess a spongy structure. Very thin layers care made up of isolated aggregates of theThicker layers have a connected structure butquite porous. Even layers which are several theof angstroms in thickness do not behave quite Ibulk material. This picture of the structurefilms has been well established from extensiveinvolving electron microscopy,2 02 2 electrontion,23-2 ' resistivity measurements2 -2 8 andother methods. 29

Electron microscope studies were carried outlaboratory with an RCA microscope, Model EThe films were deposited directly on Formcollodion membranes mounted on grids. A consieffort was made to establish that these filmessentially the same in structure as those deposglass or silica. Scott had shown that the structisimilar for silver on Formvar, and for silver membranes of silica.2 ' His experiments left und

20 H. Levinstein, J. Appl. Phys. 20, 306 (1949).21 R. S. Sennett and G. D. Scott, J. Opt. Soc. Am.

(1950).2 2A. Colombani and G. Ranc, Compt. rend. 232, 134423 A. G. Quarell, Proc. Phys. Soc. (London) 49, 279 (1'21 L. II. Germer, Phys. Rev. 56, 58 (1939).25 W. Lotmar, Helv. Phys. Acta 20, 441 (1947).23 W. Reinders and L. Hamburger, Ann. Physik 10, 64927 J. P. Borel, Helv. Phys. Acta 24, 389 (1951).28 C. E. Ells and G. D. Scott, J. Appl. Phys. 23, 31 (1S2

9 J. C. Steinberg, Phys. Rev. 21, 22 (1923).

whether or not the latter was equivalent to a solidsilica or glass substrate. In order to dispose of thispoint, films deposited on glass were examined directly.A grid covered with Formvar was pressed onto a silvermirror and stripped off with Scotch tape. Silver filmsremoved in this way had the same structure as thoseevaporated directly onto Formvar.

We may summarize the results of our microscopestudies as follows. The film structures were similar tothose found by Scott, who published a series of photo-graphs showing the dependence of structure on thick-ness. The main difference was that of a factor of twoin the thickness scale. That is to say, we found a givenstructure to correspond to a thickness of about one halfthat reported by Scott. We may speak of a criticalthickness t in connection with the dependence ofstructure on thickness. Films thinner than t consist ofunconnected aggregates and those thicker than t arecoherent. We have found t to be about 60 A.

300 Resistivity measurements on the films were alsomade, and they were found to be consistent with themicroscope studies.3 0 The dependence of the resistivity

Ls on the thickness is shown in Fig. 9. It is seen thatbeneath the critical thickness the resistivity is ex-tremely high, indicating the effect of aggregation,

.LMS whereas layers thicker than t, are made up of aggregateshe thin of sufficiently dense packing that there is a continuouss to be electrical path across them. Nevertheless even fairlylayers. thick layers (1000 A) have resistivities which are

d solid, greater than that of the bulk value.if silvermetal. GARNETT'S THEORY

ire still Many workers have described the optical behavioriusands of aggregated films with the help of Garnett's theory 3like the which provides formulas for the effective optical con-of thin stants n' and k' of a film made up of spherical con-studies ducting particles embedded in a dielectric medium.diffrac- The constants n' and k' are related to n and k (i.e.,several bulk constants) through a single parameter q. Thisin this parameter is defined as the volume of metal per unitU II. volume of film and may be described as a packing

var orderable5S were

ited onire wason thinecided,

40, 203

4 (1951).937).

(1931).

952).

1.0

q

1I

0.6

0.2

20 40 60

> (A)

80 100

FIG. 10. The "packing fraction" q of thin silver films as deducedfrom measurements of electron micrographs.

-0 U. Oppenheim and J. H. Jaffe, J. Appl. Phys. 24, 1521 (1953).31 J. C. M. Garnett, Trans. Roy.Soc. (London) 203, 385 (1904).

I I I I I

I I I I

632 Vol. 46

SEMIREFLECTING SILVER FILMS

fraction:

where

3qb(1- qa)2+4q2b2

1.0(4)

0.8(k 2-n2+1) (k 2-u2- 2)+4n2k2

(k 2 - n2 -2) 2+4n2 k2

3nk

(k 2 - n2 -2) 2+4n2k2

In particular the Garnett theory may be used toexplain qualitatively the relationship between thecurves of efficiency versus reflectance R as observedexperimentally (Fig. 7), and as computed from thestraightforward electromagnetic theory (Fig. 8).

The value of n'k' given by Eq. (4) may also be ex-pressed32 approximately in terms of A / T:

Xno An'k=-.-,47rt T

where no is the refractive index of the substrate. Con-sequently the efficiency may be written thus:

T )2 I 2

1+ -n'k'nox

This expression for r depends upon the packing fractionq through Wk'.

We have estimated the packing fractions by a directexamination of the electron micrographs, measuringgraphically the area covered with silver. Figure 10shows the dependence of q on the thickness. Values ofq from Fig. 10 were substituted into the expression (4).Comparison between theory and experiment at 2 is

32 H. Wolter, Z. Physik 105, 269 (1937).

-r 0.6

0.4

0.2

0.20 0.40 0.60 0.80 1.00

R

FIG. 11. Efficiencies of thin silver films: Curve A is the experi-mental efficiency (see Fig. 7). Curve B is the theoretical efficiencyfor a continuous layer. (See Fig. 8.) Curve C shows the theoreticalefficiency taking into account the aggregated structure of thefilms (Garnett theory).

made in Fig. 11. Curve A is experimental, taken fromFig. 7. Curve B shows the dependence of r on reflec-tance R computed without taking into account thefilm structure. That is to say, Eqs. (1) and (2) wereused. The values of n and k were taken as 0.7 and 13,respectively. Curve C is computed using the same bulkvalues but including the Garnett correction. It is seenthat there is a qualitative agreement between thecurves A and C. Similar results were obtained fordata referring to other wavelengths. The Garnetttheory therefore does indeed explain in some measurethe main features of the r versus R curves.

ACKNOWLEDGMENT

I should like to thank Dr. Joseph H. Jaffe who sug-gested the problem and under whose direction thework was carried out.

633August 1956